' TL YY"\ Y IT* VY"* T At A T% 1 ^**^ 1 lS,jSir*' r t^lr N DU STRJ AL AR1THMET1 Southern Branch of the fniversity of California Los Angeles L 1 QA 103 R69 This book is DUE on the last date stamped below MY 1 4 19201 Z Form L-9-15m-8,'26 INDUSTRIAL ARITHMETIC ROR AY AN ELEMENTARY TEXT FOR BOYS IN INDUSTRIAL, TECHNICAL, VOCATIONAL AND TRADES SCHOOLS, BOTH DAY AND EVENING BY NELSON L. RORAY DEPARTMENT OF MATHEMATICS, WM. L. DICKINSON HIGH SCHOOL, JERSEY CITY, N. J. WITH 86 ILLUSTRATIONS PHILADELPHIA P. BLAKISTON'S SON & CO. 1012 WALNUT STREET COPYRIGHT, 1916, BY P. BLAKISTON'S SON & Co. XHK MAPLK FKKSS YUMK PA PREFACE The following pages presuppose a knowledge of the ordi- nary course of Grammar School Arithmetic. They are intended: First. To review and give drill in the mathematical tools needed by boys in the shops during the first year of the Industrial High School. Second. To give some of the problems the boys must handle in the school shops and may have to handle in practical life. Many of the problems have been taken from the shops of the Wm. L. Dickinson High School. Third. To introduce the idea of general positive number, its use in formulae and in simple equations, thereby, incidentally, giving some preparation for the course in algebra. No formal approach to algebra, however, is intended. Fourth. To give the boy who leaves school through neces- sity during the first year of his Industrial High School course some of the practical applications of the most used geometrical formulae. The manuscript of this book has been used with about thirty different classes under several teachers in the Industrial Department of the Dickinson High School during the past five years. In order that the problems will be expressed in the language and data of the shops and also that they will satisfy the actual mathematical needs of the first year shop work, consultations were held from time to time with the shop teachers. VI PREFACE The drawings were made under the supervision of Mr. Stewart by the pupils of the Industrial Department, most of them by Mr. Rossback of the Junior Class. The author especially acknowledges his indebtedness to his colleagues Messrs. Burghardt, Steele, Stewart, Loomis and Wagner for valuable suggestions and assistance and also to Mr. Mathewson for his encouragement and help in the preparation of this book. THE AUTHOR. DICKINSON HIGH SCHOOL JERSEY CITY, N. J. CONTENTS PAGE. LESSONS I-IV. Reviews 1-7 LESSON V. General Number 8 LESSON VI. Formulae n LESSON VII. Angles and Polygons 13 LESSON VIII. Measurement, Woodworking. .' 18 LESSON IX. Measurement Drawing 20 LESSON X. Screw Threads 23 LESSON XI. Machine Shop Measurements 25 LESSON XII. Decimal Equivalents ' . 27 LESSON XIII. Review 29 LESSON XIV. Area of Rectangle 31 LESSON XV. Review 33 LESSON XVI. Square Root < 35 LESSON XVII. Review 37 LESSON XVIII. Fractional Review 38 LESSON XIX. Area of Parallelograms 39 LESSON XX. Area of Triangles 40 LESSONS XXI-XXII. Reviews 42-44 LESSON XXIII. Circles Area and Circumference 46 LESSON XXIV. Speed 48 LESSON XXV. Speeds of Pulleys and Gears 50 LESSON XXVI. Cutting Speed and Feed . . . 54 LESSON XXVII. Area of Cylinder 56 LESSON XXVIII. Review 57 LESSON XXIX. Volume of Prism 59 LESSON XXX. Review 62 LESSON XXXI. Review 63 LESSON XXXII. Review of Percentage 65 LESSON XXXIII. Board Measure 66 LESSON XXXIV. Review 67 LESSON XXXV. Woodworking Problems 68 LESSON XXXVI. Volume of Cylinder 71 LESSON XXXVII. Review 72 vii Vlll CONTENTS PAGE. LESSON XXXVIII. Review 73 LESSON XXXIX. Forge Shop Problems 74 LESSON XL. Review 76 LESSON XLI. Simple Equations 78 LESSON XLII. Theorem of Pythagoras 79 LESSON XLIII. Review 81 LESSON XLIV. Factoring _ 83 LESSON XLV. ^/ab = \/ a . Vb 84 LESSON XLVI. Review 85 LESSON XLVII. Triangles 86 LESSON XL VIII. 30 Right Triangle 88 LESSON XLIX. Review 89 LESSON L. 60 Right Triangle 91 LESSON LI. Review 92 LESSON LII. Equilateral Triangle, Altitude and Area . . 94 LESSON LIU. Regular Hexagon 95 LESSON LIV. Screw Threads 97 LESSON LV. Review .....: 99 LESSON LVI. Thread Cutting 102 LESSONS LVII, LVIII. Tapers and Taper Turning .... 110-113 LESSON LIX. Review 114 LESSON LX. Ratio 116 LESSON LXI. Sectors and Segments ... 119 LESSONS LXII, LXIII, LXIV. Reviews 120-124 LESSON LXV. Area of Pyramids and Cones 125 LESSON LXVI. Volume of Pyramids and Cones 127 LESSONS LXVII-LXXII. Reviews 129-139 LESSON LXXIII, LXXIV. Alloys 140-144 LESSON LXXV-LXXVIII. Print Shop 145-153 LESSON LXXIX. Fractional Review 154 INDUSTRIAL ARITHMETIC LESSON I The expression 5 2 means 5X5; the expression 3 4 means 3X3X3X3. In the expression 5 2 , 5 is called the base and 2 the exponent. The exponent shows how many times the base is to be taken as a factor, that is multiplied by itself. 1. Name the base and the exponent of each of the following expressions, also state the meaning of each: 3 2 ; 4 3 ; 5 4 : 6 5 ; io 2 ; io 3 ; io 4 ; 8 4 ; 7 5 . 2. Find the value of each of the above expressions. 3. Expand each of the following: io 2 ; io 3 ; io 4 ; io 5 ; io 6 . 4. Note the number of zeros in the expansion of each expres- sion in problem 3. How many zeros are in the expansion of any power of io? 5. State the power of io of each of the following: 1000; 100; 1,000,000; 100,000,000; 10,000; io; 100,000. 6. Multiply each of the following by io; by 100; by 1000: 24; 24.3; 2.43; 3.1415. 7. Compare the position of the decimal point in the product with its position in the multiplicand in each example of problem 6. What effect upon the position of the decimal point of a number has multiplying it by io? By 100? By io 4 ? By io 8 ? State the principle for multiplying quickly any number by io or any power of io? 8. Write down the value of each of the following: .0323 X io 3 ; 3.141592 X io 7 ; .000323 X io 2 ; 32.30 X io 4 ; 3I4I-592XI0 8 ; 2 INDUSTRIAL ARITHMETIC 323,000 X io 5 ; 1.732 X io 6 ; .01414 X io 2 ; .507984 X io 3 ; 52 X io 8 ; .00005 X io 6 ; 8976 X io 5 ; .001 X io 4 ; 5.00004 X io 6 ; 8.976 X io 4 . 9. State a quick, easy way for multiplying a number by 20; by 300; by 4000; by 50,000. 10. State the result of multiplying each of the following numbers by 20; by 400; by 5000; by 6000: 123; 3.24; .0122; 83.016; 12,140. 11. Since 25 = 100/4 then 12 X 25 = 12 X 100/4 = 1200/4 = 300. That is to multiply a number by 25 we move the decimal point two places to the right and divide the result by 4. 12. Since 33% = 100/3, what is the principle for multiplying a number by 33^? 13. State the principle for multiplying a number by 12^; by 16%; by 50; by 66%; by n^. 14. Use the above principles and multiply each of the fol- lowing numbers by 25; by 33^; by 16%; by 12^; by n^; by 50; and by 66%: 17.28; 1.448; 981; 1892; 1658; 3.1416; 2.0524; 78.6; 4.32; 1214.0678; 3.572. 16. At 12^^. per ft. what will 1448 ft. of concrete cost? At i6%?<. per ft.? At n}^. per ft.? 16. How many feet will a line 12 in. long represent if the scale is i in. = 12^ ft.? If i in. = 33)^ ft.? If i in. = 25 ^.? 17. If a cubic foot of iron weighs 480 Ib. how many pounds will a plate weigh that contains 16% cu. ft.? 25 cu. ft.? 12^ cu. ft? 66% cu. ft.? 75 cu. ft? n>^ cu. ft.? LESSON II REVIEW Divide 175.6 by 10; by 100; by 1000. Compare the position of the decimal point of the quotient with that of the dividend. State a short easy method for dividing a number by 10 or any power of 10. 1. State the quotient: 56.34 -T- 10; 76.54 -r- 1000; 587 -r- 100; .057 -f- 100; 563.4 -j- 100; 893.2 -7- 10; 68.3 -r 1000; .003 -r- 10; 5.634 -f- 1000; 8932 -T- 10,000; .78 -T- 10,000; .056 -f- 1000; .789 4- 10; .0683 -f- 100; .06 -T- 100; .56 -r- 10. 2. State a short easy method for dividing a number by 20; by 300; by 4000; by 3000; by 80. 3. State the quotient: 248 -H 20; 67.5 -T- 500; 7593 -f- 5000; 89 -T- 400; 248 -T- 40; 7-86 -7- 200; 759.3 -4- 5000; 8.9 -T- 40; 246 -T- 600; 3675 -7- 2000; 7.593 -T- 500O; .89 -5- 4000; 1728 -7- 1200; 8958 -7- 5000; 7593 * 5; 7 6 * J 900- 4. At 90 j. per 1000 cu. ft. what will 2486 cu. ft. of gas cost? 5. How many tons in 5240 lb.? In 137,801 lb.? In 756 lb.? 6. At $25 per 1000 what will 12,280 bd. ft. of lumber cost? 7. At $12^ per 1000 find the cost of 16,300 bricks. 8. At $5 per ton, how much must be paid for 12,340 lb. of coal? 9. A gallon of paint will cover 200 sq. ft. of surface two coats. State a rule for finding the number of gallons of paint required to give any surface two coats. 3 INDUSTRIAL ARITHMETIC 10. Find the number of gallons of paint required for two coats for 2578 sq. ft.; for 3780 sq. ft.; for 160 sq. ft. 11. A quart of grass seed is sufficient for 300 sq. ft. of lawn. How many quarts will be required to seed 2970 sq. ft.? LESSON III REVIEW 100 Since 2< = then 4 100 1200 X 4 1 200 -T- 2<; = 1 200 -s - = - - = 48. 4 100 That is, to divide a number by 2$, move the decimal two places to the left and multiply result by 4. 1. Divide each of the following by 25: 1660; 72.86; 56; 56.4; 1728; 3564; 3.1416. 2. Study the above and find a short, easy way for dividing a number by 33^; 50; 16%; 11%; 14^; 66%. 3. State the quotient: 1800 -T- 33^; 12.34 -4- 12}^; 35.81 -f- 50; 1234 -i- 50; 27.16 *- 33^5 35-7 * "M; 17-60 -5- 14^; 75-*6 -s- 25; 87 -i- 33^; 3.46 -T- 16%; 18.32 ^- n^; 1765 -T- 33^; 128 4- 33^; 2468 * 16%; 175-3 -s- 66%; 18.32 -i- 14^. 4. In problem 3 change each division sign to a multiplica- tion sign and state product. 6. It is required to cut a i6-ft. bar of steel into pieces 12^6 m - long; how many pieces can be cut from the bar, allowing Ke-in. waste for each piece in cutting? 6. At i6%j. per Ib. how many pounds of copper wire can be bought for $18? For $2400? For $300? 7. How many taper shanks 24% in. long can be cut from a round piece 12 ft. long, if ^ in. to each piece is wasted in cutting? 8. A man bought a building lot at $16% per ft. What did he pay for a 68-ft. lot? 9. Find a short easy way for multiplying a number by 99, by 101. Hint. 99 = 100 i. s + + = ? + - H 54 = ? -H- 2. - 2 29% + H - 15 + H - uK = ? 3. Multiply 15% by 12; 18% by 8; 17% by 8; 28^ by 8. 4- % x 7 72 /> /2 /8 A x25 ^ x64 ' X % X % = ? 5^ X 3 X 2 = ? X 2 H X%= ? 2% X 5 X 8 = ? 6. Divide !% 4 by 2; 3^ by 3 7 ^; by 25; by 16%; 11. Change to decimal fractions: *-/n *-/A *-/a */* n a /o n * "ri? ^ 72 > 74 > 78 > 716) 73 2 > 764- 12. Change to common fractions in their lowest terms: .15; .015625; .0625; .125; .03125; .235. LESSON V GENERAL NUMBER In our previous work all the numbers we have used have had particular values and were represented by a definite symbol. For example the symbol 5 stands for a group of five units. We shall now use other symbols to represent numbers which may have any values whatever, or numbers whose values are, as yet, unknown, e.g., we speak of a rod a ft. in length meaning any number of feet, of x marbles meaning any number of marbles, etc. 1. If John has $6 and Henry has $5, together they will have $6 + $5. 2. If John has $a and Henry has $b, together they will have $a + $b. 3. In which of the above problems are numbers used that are represented by particular values? That are represented by any value whatever? 4. If in problem 2, a stands for 10 and b stands for 5, how many dollars have they together? If a = 7 and b = 8? If a = 12 and b = 10? If a = 13 and b = 17? 5. Read each of the following and tell what operation is in- cluded: (a) 7 + 5; 8X3; 9-*- 35 8-7J 9 + 55 5 + 35 9-6; 7 X 55 16 -^ 4; 7 - 5; 8 -^ 7; 8 + 10. o (b) a + b; a b; a X b; a * b; t ; 3a; 3b; 4c; 2a + 3c; b 4x X 3 b J 3 X n; 5 -=- k; |; 7ab; gef; (x - i) -4- p; a X c + d; n 5; 6 + 8b; 3a sb + cd. 8 GENERAL NUMBER 9 Express with the proper symbol of operation the solution of each of the following: 6. Tom has 10 marbles and Frank has 7 marbles, how many marbles have the boys together? If Tom has a marbles and Frank has r marbles? 7. What is the perimeter of a square if a side is 6 ft.? S ft.? Tft? Xft.? Bft.? 10 ft.? 8. By Jiow much does n exceed 8? 12 exceed 9? 7 exceed 5? 10 exceed k? k exceed 3? e exceed f? r exceed t? y exceed 2p? 9. If the age of a boy is now 16 years, how old was he 7 years ago? 8 years ago? What operation is used to solve this problem? How old was he a years ago? d years ago? q years ago? 10. At $3 each how many hats can be bought for $18? For $24? For $36? For $a? For $f? For $h? 11. At $a each how many books can be bought for $15? For $13? For $b? For $v? For $x? For $i? 12. At $2 each what will 6 hammers cost? 8 hammers? a hammers? c hammers? k hammers? 13. How many feet longer is a i2-ft. stick than a lo-ft. stick? A i5-ft. stick than a i2-ft. stick? An a-ft. stick than a b-ft. stick? Make a word problem for each of the following: 14. 7 + 5; a .+ b; a - b; $a X b; $a *- $G; $a 4- $y; 3 a ; S b - 15. What is the sum of 3 ft. and 5 ft.? Of 3a and $a? Of 7a and 6a? Of 3m and 5m? Of 3a and 5a? 16. What is the difference between 8 books and 5 books? Between 8x and 3X? Between 10! and 3!? Between 8s and 53? Between 7r and $r? 17. sa + 6a - 2a + 4a = ( )a; izy - 8y + icy - $y = ?; 3 b + 7 b - 5 b + 8b = ( )b; 3 k + 4 k - 5 k - 2k = ?; 4* + 5X 8x + iox = ?; 8s + 75 IDS 55 = ?; 8^4! + 2^1 10 INDUSTRIAL ARITHMETIC 2 ^1 = ? 3 m + iom - 8m - 2m = 18. Find the value of each of the following expressions if a = i, b = 2, c = 3: a + b; c d; c + b a; (20 + b) -J- 2; (8a 2) -f- b; c + b + a; aXbXc; b X c a; 2c ^ b -f- 5; 40 + 5 - 2 be be b 19. The expressions ab, a-b, a X b all mean that a is to be multiplied by b. The expression a(b + c -f- d) means that b, c and d are to be added and the sum multiplied by a. When 2 or more num- bers have no sign of operation between them like abed, it is to be understood that the numbers are to be multiplied. LESSON VI FORMULAE The statement of a rule by means of general numbers and other mathematical symbols is called a formula. For exam- ple, the rule for finding the area of a circle is, multiply the square of the radius by 3.1416. This rule stated by means of a formula, if S = area of circle, T = 3.1416 and r radius of the circle, is S = Trr 2 . o p 1. If r and -T be any two fractions, state by means of a b d formula the rule for the multiplication of two fractions; for the division. 2. State as a formula the following: The area of a rectangle is equal to its base multiplied by its altitude. Let b = base and h = altitude. 3. If b = the area of the base, h = the altitude and v = the volume of a cone, from v = ^bh, state in words the rule for finding the volume of a cone. 4. The formula for finding volume (v) of a sphere whose radius is r is v = ^jTrr 3 . Find volume of a sphere whose radius is 5, 6, 8, 10. 6. L = length of open belt in feet (approximate). D = distance between centers of pulleys in feet. R and r = radii of two pulleys in feet. r) + 2d. The distance between the centers of two pulleys is 20 ft. and the radii of the pulleys 18 in. and n in. Find the approximate length of the_open belt required for the pulleys. 12 INDUSTRIAL ARITHMETIC 6. The approximate distance a body will fall from rest in any number of seconds is given by D = i6t 2 . Find the distance a body will fall from rest in 4 sec.; 10 sec.; 15 sec.; i min. 7. C = Circumference of a circle and r = its radius. C = 27rr. What is the velocity of a point on the rim of a wheel, radius 3 ft. making 12^ revolutions per second. LESSON VII ANGLES AND POLYGONS 1. The figure BAG is called the angle BAG. The point A is called the vertex of the angle and the straights AB and AC are called the sides of the angle. In reading an angle the vertex is always read between the other two letters, as angle BAG; writ- FIG. i. FIG. 2. ten ZBAC. An angle is often named from its vertex letter only, as the above is called the angle A; written Z A. 2. Read the angles of Fig. 2. These angles may be called Zi, Z2, ^3, etc. Right Angle FIG. 3. FIG. 4. 3. The angles a and b are called vertical angles. Name another pair of vertical angles in the above figure. Name pairs of vertical angles in the figure for example 2. 4. The angles a and d are called adjacent angles. Name other pairs of adjacent angles in the above figures. 13 14 INDUSTRIAL ARITHMETIC 6. Two angles are equal if their sides can be made to coincide. 6. If two straights so intersect that any pair of adjacent angles formed are equal, the straights are said to be perpen- dicular to each other and the angles formed are right angles. If the ZABE = ZDBA (Fig. 4), then the straight AC is perpendicular to the straight DE and the angles formed are right angles. Name the right angles in Fig. 4. 7. Right angles and perpendiculars are often constructed by means of the T square or a right triangle. 8. An angle is often measured by the number of degrees it contains. An angle of one degree (i) is one of the 90 equal angles into which the right angle can be divided. How many degrees in one right angle? Two? Three? Four? % right angle? >? i}? %? ^? A protractor is used to measure an angle in degrees. 9. If the sides of the angle lie in the same straight the angle is called a straight angle, e.g., ZDBE is a straight angle (Fig. 4). Name other straight angles. How many degrees in a straight angle? How many right angles? Obtuse B FIG. 5- FlG - 6 - 10. An angle less than a right angle is called an acute angle. 11. An angle greater than a right angle but less than a straight angle is called an obtuse angle. 12. Straights in the same plane that have no common point are called parallel lines. ANGLES AND POLYGONS 15 Parallel lines are constructed by means of the parallel ruler or the T square. 13. A curved line is a line no part of which is straight. Parallels FIG. 7. FIG. 8. 14. A line not straight but no part of which is curved is called a broken line (Fig. 8). B ED DC Polygon Parallelogram FIG. 9. FIG. io. 16. If the end points of a broken line coincide the figure formed is called a polygon. AB, BC, CD, etc., are the sides of the polygon. The angles FAB, ABC, etc., are the angles of the polygon. The broken line is called the perimeter of the polygon. Exercise. If each side of the above polygon is 7 in. what is the length of its perimeter? If 5 in.? If io in.? If 3 ft.? If 5 ft. 4 in.? 16. A polygon of three sides is a triangle. A polygon of four sides is a quadrilateral. 17. If the opposite sides of a quadrilateral are parallel the figure is called a parallelogram. Exercise. How many sides has a parallelogram? Why? Can the sides of a parallelogram be curved? Why? 18. Facts relating to a parallelogram: Draw a parallelo- i6 INDUSTRIAL ARITHMETIC gram and letter it A, B, C, D. Use the compasses and com- pare AB with CD. Are they equal? Compare AD with BC. Are they equal? Draw another and make the same comparison. Compare the opposite angles in each parallelogram. The above exercises illustrate the following principles: 1. The opposite sides of a parallelogram are equal. 2. The opposite angles of a parallelogram are equal. Exercise. If in the parallelogram ABCD, AB = 10 in., CD = ? Why? AD = 3 in., BC = ? Why? CD = 15 ft., AB = ? Why? BC = 3 ft. 4 in., AD = ? Why? B = 78, D = ? C = 102, A = ? THE RECTANGLE A B C D Rectangle FIG. ii. 19. A parallelogram having one of its angles a right angle is called a rectangle. Exercise. Are the opposite sides of a rectangle equal? Why? Are the opposite angles of a rectangle equal? Why? A FIG. 12. At least how many of the angles of a rectangle are right angles? Find by means of a right angle whether the other angles of a rectangle are right angles. ANGLES AND POLYGONS 17 20. Any one of the sides of a parallelogram is called its base; e.g., AB, or BC or AD or CD is a base of the parallelo- gram. Name the bases of the rectangle in example 19. 21. The perpendicular from one side of a parallelogram to the opposite side is the altitude of the parallelogram, e.g., AF in the figure for example 20. Name the altitude of the rectangle in example 19 if DC is the base; if AD is the base. THE SQUARE FIG. 13. 22. A rectangle with a pair of intersecting sides equal is a square. Exercise. Is a square a parallelogram? Why? How many degrees in at least one angle of a square? Why? In each angle of a square? Why? Are the sides of a square equal? Why? Name the bases and altitudes of the above square. If one base of a square is 2 in., what is the length of its altitude? Draw a square one of whose sides is 12 in., letting i in. = 6 in. Draw a rectangle with base 20 ft. and altitude 4 ft. (scale i" = 4' o".) Draw a i-in. square, or i sq. in.; a i-ft. square or i sq. ft. I LESSON VIII MEASUREMENTS Measuring a line consists in finding the number of standard units of length it contains. There are many standard units of length in common use, such as the inch, foot, yard, etc. The number of standard units of length a line contains is the length of the line in terms of that unit; e.g., a line is 5 ft. long if it contains the foot five times. In the wood working shops the ruler used for measuring lengths is divided into inches and each inch divided into 16 equal parts, the wood worker thus measures to ^Q in. In the machine shops one of the instruments used enables the machin- ist to measure to ;Kooo m - This instrument is called a microm- eter and is used for measuring fractional parts of an inch. This instrument records its measurements in the decimal scale instead of in terms of common fractions. The ruler of the wood worker reads % in. whereas the micrometer of the machinist would read .125 in. for the same length. The ordi- nary micrometer registers accurately to .001 in. The experi- enced machinist can measure very closely with it to .0001 in. WOOD WORKING MEASUREMENTS 1. Measure as accurately as possible the top of your desk. 2. Find the number of feet in the length of the schoolroom ; the number of yards. 3. Measure the length and width of your schoolroom. How many feet of baseboard are required for the room, making deductions for all openings in the baseboard? 18 MEASUREMENTS 4. Find the cost of the chalk trays of the blackboards in your schoolroom at 12%^. per ft. 6. Determine the number of feet of moulding in the panels of the doors of the schoolroom and its cost at $}^>i. per ft. 6. How many strips of floor moulding 16 ft. long will be necessary for the schoolroom? 7. How many feet of picture moulding will be required for the room if the moulding is at the top of the walls? If 3 ft. from the top of the walls? -5'- Floor Plan -20' FIG. 14. 8. The above is a floor plan. (a) Find cost of the floor moulding for it at 8^. per ft. (b) Find cost of the baseboard at 20^. per ft. 9. What is the perimeter of a rectangular room M ft. long and R ft. wide? S ft. long and 12 ft. wide? K ft. wide and 2K ft. long? LESSON IX DRAWING MEASUREMENTS The inch of the scale used for measuring in the drawing room is divided into 32 equal parts. To what fraction of an inch can the draughtsman measure? In the drawing room working plans for the machinist, the carpenter, the pattern maker, etc., are made. These drawings are seldom made full size but are drawn to a scale, that is, each inch of the drawing represents i ft. or 2 ft., etc., of the actual size of the object drawn. When making drawings it frequently happens that a line must be drawn whose length contains ^2 i n - or even 3^4 in. In ordinary drawings %4 in. is neglected, no at- tempt being made to draw any line shorter than ^2 m - Scale K" = 4' o" FIG. 15. Notice that the plan has marked on it the actual lengths of the lines represented, but the lengths of the drawing are made in accordance with the scale selected. Exercises 1. If the scale is i" = 3' o", how long a line must be drawn to represent each of the following: DRAWING MEASUREMENTS 21 g: in.; 3^ in.; 5.5 in.; 6.25 in.; 9 ft. o in.; 12 ft. o in.; 15 ft. o in.; 6 ft. o in.; 18 ft. o in.; 100 ft. o in.; 8 ft. 6 in.; 7 ft. 5 in.; 15 ft. 8 in.; 18 ft. ^2 i n -J 25 ft. K2 in.; 9 in.; 15 in.; 2 in.? 2. If the scale is i" = o'. 8", determine the length of each of the following for the drawing: i in.; % in.; % in.; 2^ 4.125 in. 3. In the above figure what should be the length of each line? 4. Fill in the following, the scale being i" = 3". Length of line in Actual length represented drawing by drawing 6 in. ? 9 in. ? < 2 in. ? in. ? in. ? ? 16.5 in. 5. Calculate the length of each line on the drawing for the miter box if the scale is half size; if one-quarter size. (Fig. 16.) FIG. 1 6. 6. What is the length of each line in the drawing for the mortar box, if the scale is W = o'. i"? (Fig. 17.) 22 INDUSTRIAL ARITHMETIC FIG. 17. 7. The scale for the saw horse is 3" = i' o". Calculate the length of each line for the jlra wing. (Fig. 18.) FIG. 1 8. 8. The side of a filing cabinet is to be 14 in. long 10% in. high, the stile and rail each i ^ in. wide. Calculate the dimen- sions for a drawing* with scale %" = i' o"; }/' = i' o". LESSON X SCREW THREADS 1. If a cylinder is spirally grooved or has a thread wound spirally around it, a screw is formed. The nature of a screw depends upon the diameter of its body, the number of threads per inch, and the shape of the thread. 2. Upon the number of threads per inch depend the pitch of the screw and the lead of the screw. The pitch of a screw is the distance between the top of one thread and the top of the next. For example, if a screw has 8 threads per in., the pitch is evidently ^ in. 3. The lead of a screw is the distance it advances in one revolution. In a single-thread screw with 10 threads per in. the lead is J^o in. Exercises (The following problems refer to screws of single thread.) 1. A screw has 12 threads per in. What is its pitch? Its lead? 2. Find the pitch and lead for the following number of threads per inch : 3; 4; 5; 6; 8; n; 20; 13; 40 3. A screw that has 40 threads per inch will advance how far in one complete revolution of the screw? 4. Find the number of threads per inch for the following leads: K in.; % in.; % in.; ^ in.; % in.; % in.; ^ in. 23 24 INDUSTRIAL ARITHMETIC 5. What is the pitch of a screw that has a threads per in.? Its lead? 6. Does the lead of a single-thread screw equal its pitch? Why? 7. If the lead of a screw is }/ in., what distance will it advance in ^ revolution? Y revolution? }/ 5 revolution? % revolution? 8. A screw has 40 threads per in.; what distance will it advance in ^5 revolution? % 5 ? K? ^5? 2 ^ 5 ? !J 5 ? 2? 10? 15? 20? 40? 9. Express as a decimal, correct to three places, the lead of each of the following number of threads per inch: 40; 20; 10; 8; 12; 7 10. Express each of the results in problem 8 as a decimal correct to three places. LESSON XI MACHINE SHOP MEASUREMENTS The measuring instruments of the machine shop are a 6-in. scale graduated on one edge to ^4 in. and the micrometer. The 6-in. scale is used for rough measurements and the micrometer for more accurate measurements. The readings of the mi- crometer, as already stated, are in the decimal scale. The micrometer is essentially a screw with 40 threads per in. What is its lead? How far in the decimal scale will it advance for ^5 of a revolution? How then is .001 in. measured with the micrometer? .002 in.? .006 in.? .oi5in.? .026 in.? .Thimble FIG. 19. A micrometer screw. The names of the different parts of a micrometer are given in the figure. The spindle is made to move backward and forward within the barrel by turning the thimble, to which the spindle is fastened. The concealed end of the spindle is a screw con- taining 40 threads to the inch. On the edge of the thimble are 25 divisions equally distant. By turning one division of the thimble, the spindle is made to move through ^5 of %Q in- or 25 26 INDUSTRIAL ARITHMETIC .001 in. Each revolution of the thimble is indicated on the barrel by means of a small mark and every fourth revolution of the thimble by a longer mark. The barrel is thus divided for the space of i in. Exercises 1. How many divisions of the thimble must be turned in order to have the micrometer measure each of the following: .003 in.; .005 in.; .010 in.; .025 in.; .018 in.; .075 in.; .125 in.? 2. What part of an inch is the distance between any pair of consecutive small divisions of the barrel? Between any pair of consecutive large divisions of the barrel? 3. What part of an inch is the distance between 6 consecu- tive divisions of the barrel? 8? 10? 15? 2O?4O?4i? 4. How many revolutions of the thimble for each of the following: .025 in.; .050 in.; .075 in.; .150 in.; .200 in.; .3 in.; .5 in.? 5. What decimal part of an inch do the following barrel readings represent: 5 large divisions; 3 large divisions; 6 large divisions; 10 large divisions; 4 large divisions; 4 large divisions and 2 small divisions; 3 large divisions and 3 small divisions; 2 small divisions? 6. Express as the decimal part of an inch the following readings: 5 large and 3 small of the barrel and 12 of the thim- ble; 3 large and 2 small barrel, and 16 thimble; i large of the barrel and 8 thimble; 6 large and 3 small barrel, and 14.5 thimble; 9.5 thimble. LESSON XII DECIMAL EQUIVALENTS As we have already learned the machine shop's fractional measurements are in multiples of ^4 in.; also, if these meas- urements are to be accurate the micrometer is used. Hence, in order to measure ^4 in. its equivalent as a decimal must be known. Why? Show that ^54 = .015625. (Learn this result.) In setting the micrometer for ordinary work three decimal places only are considered, that is J^4 = - OI 5- If it is required to set the micrometer for 3^2 or %4 we consider 3^54 = .0156, because to get ^54 we multiply ^4 by 2, %2 %4 = -031; that is ^34 = .0156 whenever it is to be multiplied by any number. By means of the following it is possible to state quickly the decimal equivalent of any number of 64ths. = 25 = -125 = .015625 Express as a decimal 1 J^4- Solution. 1% 4 = i% 4 + ^4 = K + ^4 = -25 + .015 = -265. Express as a decimal 6 %4. Solution. 6% 4 = 64^ 4 _ ^ 4 = x _ . 4. Find the number of square units in each of the circles in problem 3. 6. Find the number of square units in the ring between two concentric circles of radius 10 and 8 respectively. 6. A circular walk 4 ft. wide around a flower bed 20 ft. in diameter. Find the number of square feet in the walk. 7. Inscribed within a square is a circle whose diameter is 10 ft. Find the number of square feet between the circle and the sides of the square. 8. If the inner side of a circular running track is % mile and the track is 20 ft. wide what is the length of the outer side of the track? 9. If the diameter of a circle is increased by i ft., how much is the circumference increased? 10. The diameter of the earth is about 8000 miles at the equator. Suppose an iron band lying everywhere upon the equator were stretched so that it would be eyerywhere % ft. from the surface of the earth, how many feet would its length be increased? 11. If the radius of a circle were multiplied by 2, by 3, by 4, by 5, by 6, by 7, by 8, by 9, by 10, by what number would its circumference be multiplied? Its area? 12. What is the number of square feet in the surface of the track in problem 9? LESSON XXIV SPEED Speed is rate of change of position. Speed is measured by a number of units of length per unit of time. For example, a body moving through 1 20 ft. each minute has a speed of 1 20 ft. per min. or 2 ft. per sec., or 7200 ft. per hr. Exercises 1. A point is moving at a speed of i mile per min. What is its speed per hour? Per second? 2. What is the speed per minute of a train running 60 miles per hr.? 58 miles per hr.? 63 miles per hr.? 3. If the circumference of a wheel is 31.4 ft., what is the speed of a point on its rim, if the wheel is making 100 .R.P.M. (revolutions per minute) ? 4. What is the speed of a point on the rim of a cast-iron flywheel i ft. o in. in diameter, when the wheel is making 1680 R.P.M.? 5. A lathe spindle is running 1500 R.P.M. What is the speed of a point on the surface of a 6-in. cylinder placed in the chuck? 6. Ordinarily the maximum safe rim speed of cast-iron fly- wheel with solid rim is about 85 ft. per sec. Determine if any of the following are exceeding the safe speed: i ft. diam. making 2000 R.P.M.; 15 ft. diam. making 90 R.P.M.; 6 ft. diam. making 280 R.P.M.; 7^ ft. diam. making 220 R.P.M.; 9 ft. diam. making 185 R.P.M.; 8^ ft. diam. making 200 R.P.M. 48 SPEED 49 7. Speeds for grindstones: Machinist's tools, 800-1000 ft. per min. Carpenter's tools, 500-600 ft. per min. The maximum safe speed for a grindstone is ordinarily about 3400 ft. per min. For grinding machinist's tools how many R.P.M. should a grindstone 3 ft. in diameter make? For carpenter's tools? What is the maximum safe number R.P.M. ? 8. If S = surface speed of a point on a, the circumference of a revolving wheel or cylinder, D = the diameter of the wheel or cylinder and R.P.M. = number of revolutions per minute of the wheel or cylinder, express the formula for S in terms of D and R.P.M. 9. Use the formula derived in problem 8 and find S when D = 10 and R.P.M. = 20; D = 5; R.P.M. = 100; D = 8; R.P.M. = 1200; D = 20; R.P.M. = 950. 10. If S = 500, D = 5; find R.P.M.; if S = 1000, R.P.M. = 500, find D; if S = 2500, R.P.M. = 1500, find C; if S = 3000, R.P.M. = 1000, find C; if S = 4000, C = 30, find R.P.M.; if S = 5000, C = 40, find R.P.M. LESSON XXV SPEEDS OF PULLEYS AND GEARS If two pulleys are connected by a belt and one of the pulleys set in motion, the belt will cause the other pulley to move also, that is, motion is transmitted from one pulley to the other by means of the belt. The pulley that transmits motion to the belt is called the driving pulley. The pulley that is set in motion, or to which motion is transmitted, by the belt is called the driven pulley. FIG. 32. Motion is also transmitted by means of gears. In this case the motion is transmitted by direct contact, the teeth of the driving gear being made to mesh with the teeth of the driven. Should it, however, happen that the two gears are too far apart to mesh, an intermediate gear is used to transmit motion from the driving gear to the driven gear. Three or more gears meshing together form a train of gears, one or more of which is always an intermediate. Gear B is the intermediate. Speed of pulleys or gears means the number of revolutions per minute (R.P.M.) the pulleys or the gears are making. So SPEEDS OF PULLEYS AND GEARS 51 Exercises 1. If the driving gear has 24 teeth and the driven gear 8, when the driving gear has made one revolution, how many will the driven gear have made? 2. Driving gear 36 teeth, driven gear 9 teeth; driving gear 40 teeth, driven gear 5 teeth, driving gear 18 teeth, driven gear 7 teeth; driving gear 21 teeth, driven gear 6 teeth. How many revolutions will the driven gear in each of the above make for one revolution of the driving gear? 3. If the driving gear contains 32 teeth and is making 40 R.P.M. what is the speed of the driven gear with 9 teeth? 4. The front sprocket of a bicycle contains 24 teeth and* the rear sprocket 8 teeth, how many revolutions will the pedals make in going i mile? The wheels of the bicycle being 28 in. in diameter? 5. The driving gear has 30 teeth and the driven gear 10 teeth. If they are connected with an intermediate of 40 teeth, what number of revolutions does the driven gear make for each of the driving gear? Explain why this is true. What effect upon the speed of the driven gear has the intermediate? 6. The diameter of the driving pulley is 1 2 in. and its speed 300 R.P.M. What is the speed of the driven pulley whose diameter is 4 in.; 3 in.; 5 in.? 7. The diameter of the driving pulley is 10 in. and its speed 900 R.P.M. Required the speed of the driven pulley of diameter 4 in. 8. On the driving shaft is a 24-in. pulley making 300 R.P.M. What is the speed of the driven shaft which has a lo-in. pulley belted to the driver? 9. Find the surface speed of each pulley in problem 8. 10. If the surface speed of 6-in. pulley is 3000 ft. per min. ; how many R.P.M. is it making? 11. The maximum safe surface speed of a grindstone is 52 INDUSTRIAL ARITHMETIC 2400 ft. per min. Find the maximum safe number R.P.M. a 6-ft. stone may make. 12. If the surface speed of an emery wheel making 600 R.P.M. is 4000 ft., what is its diameter? 13. A driven pulley 5 in. in diameter has a speed of 2500 R.P.M. If the speed of the driving pulley is 500 R.P.M., what is its diameter? 14. The speed of the driving pulley is N = R.P.M. and its diameter D. Find the formula for the speed S of the driven pulley whose diameter is d. 16. Use formula of problem 14 and find S if N = 120, D = 12, and d = 6; if N = 1200, D = 20, d = 5, find S; if S = 500, d = 10, D = 25, find N; if S = 2500, d = 8, D = 24, find N; if N = 3000, D = 18, d = 5, find S; if N= 1800, D = 40, d = 8, find S; if S = 900, D = 20, d = 2, find N. 16. If D = number of teeth in driving gear, d = in driven gear N = R.P.M. of D and n = R.P.M. of d, show that D _ n d ~N 17; Use formula of problem 16 to find D when d = 10, n = 30, N = 60; to find N if D = 40, d = 24, n = 12; to find d if D = 80, N = 4, n = 8; to find D if N = 12, n = 6 and d = 20; to find N if D = 80, d = 28 and n = 2. 18. FIG. 33. In the above train, E and F are keyed to the same shaft. D has 60 teeth, E 40, F 30 and G 45. Find the number of revolutions of G for each revolution of D. SPEEDS OF PULLEYS AND GEARS 53 Suppose an intermediate of 60 teeth is placed between F and G, then what is the result? 19. Let D = 100, E = 90, F = 70 andG = 55. If R.P.M. of D = 40 find R.P.M. of G. 20. FIG. 34. E and F are keyed to the same shaft, likewise C and D. A = 40, B = 20, C = 35, D = 50, E = 45, F = 55 and G = 60. For each revolution of A, find number of revolutions of G. 21. Let G = 80, F = 70, E = 60, C = 50, D = 55,6 = 20 and A = 40. When G has made one revolution, how many has A made? LESSON XXVI CUTTING SPEED AND FEED When turning pr cutting a cylindrical piece in a lathe the number of lineal feet of the surface of the piece cut by the tool in i min. is called cutting speed of the tool. For example, if the piece is making 20 R.P.M. and its circumference is 4 in., the cutting speed is evidently 4 in. X 20 = 80 in. = 6% ft. per min. The distance the tool advances along the work in each revolu- tion is called feed. It is expressed in ordinary work as a fractional part of an inch. For example, if the tool advances ^fg m - along the work in one revolution the feed is H$ in. The width of the chip is equal to the feed. Exercises 1. Is the surface speed per minute of a revolving cylinder the same as the cutting speed of the tool? Why? 2. A piece of round stock 2 in. in diameter is making 40 R.P.M. What is the cutting speed? 3. If the feed is ]/{Q in. and the work has a speed of 42 R.P.M. how long will it require to turn a piece 3 in. long? 4. A piece of work revolves 50 times while the tool advances i% 6 ". Find the feed. 6. If the feed is ^2 m - an d the cutting speed 20 ft. per min., find the time required for the tool to travel i in. along a piece 4 in. in diameter. 6. When cutting soft steel the speed may be 100 ft. per min. if the depth of the cut is ^ in. and the feed ^2 m - Find tne 54 CUTTING SPEED AND FEED 55 R.P.M. a cylinder of soft steel 3 in. in diameter is making from the above data. 7. If the feed per revolution of a drill is .0075 in., find the time required to drill a hole .25 in. in diameter through a rectangular piece of steel iY in. thick, if the drill is making 735 R.P.M. 8. How many strokes of the shaper will be required to rough cut a rectangular piece 2% in. wide, if the feed is ^2 in.? 9. A round piece 36 in. long and 2 in. in diameter is to be turned to a diameter of i% in. How long will be required to do the work if the feed of the rough cut is 34o in. and the finish cut j^o in. and the work has a speed of 120 R.P.M. 10. If F = feed, N = R.P.M. and D = distance the tool moves find F in terms of D and N. 11. Use the formula of problem 10. If F = Ho in., R.P.M. = 50, find D; if D = 2, F = M 2 in., find R.P.M.; if R.P.M. = 40, F = % in., findD; if D = i, R.P.M. = 32, find F;"if R.P.M. = 32, D = 2, findF; if D = 3, R.P.M. = 32, find F. LESSON XXVII AREA OF SURFACE OF CYLINDER In geometry it is proved that the area of the curved surface of a right cylinder is the circumference of its base multiplied by its altitude. The complete area includes the area of the curved surface and the area of the two ends. Exercises 1. Find the number of square units in the curved surface of each of the following cylinders: Circumference of base 28 in., altitude 10 in.; circumference of base 38 ft., altitude 12 ft.; cir- cumference of base 42.5 ft., alti- tude 22 ft.; circumference of base 15 ft. 8 in., altitude 9 ft. 2. Find the number of square in the curved surface of each of the following cylinders: Altitude 18 ft., radius of base 10 ft.; altitude 10 ft., radius of base 8 ft.; altitude 8 ft., radius of base 12 ft.; altitude 15 ft., radius of base 9 ft. 6 in. 3. Find the number of square units in the complete surface of each of the cylinders in problem 2. 4. Find the cost of painting a cylindrical column 20 ft. high and 3 ft. in diameter at 5f. per sq. ft. 5. How many square feet of tin will be required to line the bottom and side of a cylindrical tank 10 ft. high and 8 ft. in diameter? What will be the cost at ioj. per sq. ft.? 6. Express the formula for area of the curved surface of a cylinder; the complete area of a cylinder; if H = its alti- tude and R = radius of its base. 56 FIG. 35. Figure of a cylinder showing its surface changed to un j S a rectangle. LESSON XXVIII REVIEW 1. Find the side of a square that has the same area as the complete area of a cylinder 2 ft. in diameter and 4 ft. long. TT = 3.1416. 2. A shaft to which are attached two pulleys is making 300 R.P.M. Compare the surface speeds of the two pulleys if their diameters are respectfully 18 in. and 10 in. 3. The outside diameter of a pipe is 2 in. and the inside diameter i% in. Find the difference between the outside and inside areas of a piece 10 ft. long. 4. A grinding wheel 6 in. in diameter has a surface speed of 500 ft. per min. The wheel is attached to a shaft which has a 3-in. pulley, which is belted to lo-in. pulley on another shaft on which is also a 4-in. pulley; this pulley is belted to a i2-in. pulley on another shaft on which is another i2-in. pulley which is belted to a 5-in. pulley attached to a shaft run directly by the motor. Find R.P.M. of motor. 5. Three pipes respectively 3 in., 4 in. and 6 in. in diameter are discharging into a header. What is the diameter of the header, if the rate of flow in all four pipes is the same and all the pipes are full? 6. How many R.P.M. must a cylinder 10 in. in diameter make in order that the tool may advance 2 in. in 3 min., if the feed is %4 in.? 7. If the driving gear has 75 teeth and a speed of a 40 R.P.M. what is the speed of the driven gear with 25 teeth? 8. How will you set the micrometer for .149 in.? .403 in.? .738 in.? 57 58 INDUSTRIAL ARITHMETIC 9. How many revolutions must a thread whose pitch is make to advance .2 in.? % in.? 10. If B = 36, b = 49 and H = 27, find V. V = ? (B + b + VBb). o 11. If R = 12, r = 9 and H = 15, find V. V = (R 2 + r 2 + Rr. LESSON XXIX VOLUME OF A PRISM 1. A prism is a solid whose bases are parallel and whose faces are parallelograms. 2. A prism takes its name from the shape of its base, e.g., if the base is a triangle, it is called a triangular prism; if a quadrilateral, a quadrangular prism, etc. 3. The perpendicular between the bases is called the altitude of the prism. 4. The intersections of the faces are called the edges of the prism. 5. If the edges are perpendicular to the bases the prism is a right prism. We will consider only right prisms. The edge of a right prism is equal to the altitude of the prism. 6. A prism whose bases and faces are squares is called a cube. 7. Measuring a solid consists in finding the number of standard cubes a solid contains. A standard cube is a cube each of whose edges is a standard unit of length; e.g., a cubic inch, cubic foot, etc., is a cube each of whose edges is an inch, a foot, etc. 8. The number of cubes a solid contains is called its volume in terms of the cube; e.g., if a solid contains 15 cu. ft., its volume is 1 5 in terms of a cubic foot. 59 FIG. 36. Right Prism. 6o INDUSTRIAL ARITHMETIC 9. Find the volume of a prism 6 in. long, 3 in. wide and 3 in. high. How many cubes in a section along the base? How many in one layer? How many in three layers? / / xz AS \ A- A /T / \ - ,/r ?r / 1 / 1 ) y __-J_ / / / FIG. 37. 10. Find the number of cubes in each of the following prisms : 8 in. X 3 in. X 4 in.; 15 in. X 12 in. X 6 in.; 9 ft. X 9 ft. X 9 ft.; 7 ft. X 7 ft. X 10 ft; 18 ft. X 20 ft. X 8 ft; 17 ft. X 6 ft X 4 ft; 10 ft. X 8 ft X 12 ft; 9 ft X 8 ft X 6 ft; 25 ft. X 15 ft. X 20 ft; 10 ft. 6 in. X 8 ft 6 in. X 7 ft 3 in.; 15 ft. 8 in. X 12 ft. 9 in. X 10 ft 6 in. 11. If each edge of a cube is 5 in. how many cubic inches in it? If 6 in.? If 10 in.? If 12 in.? If 15 in.? If a in.? If b in.? If c in.? If n in.? 12. How many cubic inches in a cubic foot? How many cubic feet in a cubic yard? Fill in the following table and learn it. cu. in. = i cu. ft. cu. ft. = i cu. yd. 13. Change to cubic inches, 18 cu. ft.; 25 cu. ft.; 33^ cu. ft.; 12^ cu. ft; 100 cu. ft. 14. Change to cubic yards, 5760 cu. ft; 9000 cu. ft.; 1765 cu. ft. VOLUME OF A PRISM 6 1 16. Change to cubic feet, 15,625 cu. in.; 172,800 cu. in.; 2456 cu. in. 16. The dimensions of a rectangular prism are a, b and c, derive the formula for its volume. 17. If a = 4, b = 5, c = 3, V = ? a = 10, b = 9, c = 8, V = ? a = 8, b = 6, c = 12, V = ? 18. If V = 144, a = 4, b = 9, c=? V = 250, a = 6, b = 10, c = ? V = 400, b = 5, c = 12, a = ? V = 920, a = 9, c = 10, b = ? LESSON XXX REVIEW 1. Find the number of cubic yards of earth taken from a cellar 27 ft. long, 25 ft. wide and 5 ft. deep. 2. A ditch i mile long, 10 ft. wide and 6 ft. deep had how many cubic yards of earth removed from it? 3. How many cubic feet of masonry in a wall 40 ft. long, 15 ft. high and 4 ft. 6 in. wide? 4. A wall 2 ft. thick and 8 ft. high is built around a lot 50 ft. X 150 ft. Find the number of cubic yards of masonry in it if there is an opening 10 ft. wide in the wall. (Two solutions.) 5. If your schoolroom is 1 2 ft. high, how many cubic feet of air are in the room? 6. A piece of lumber 16 ft. and 2 in. X 8 in. contains how many cubic feet of wood? 7. What will be the cost of a house 36 ft. X 25 ft. and 20 ft. high at 20fi. per cu. ft.? 8. A tank 10 ft. square and 12 ft. deep will hold how many gallons of water? i cu. ft. contains 7^ gal. 9. How many cubic feet of wood in a pile 8 ft. long, 4 ft. wide and 4 ft. high? 10. Find the number of cubic feet in a box 5 ft. 6 in. long, 3 ft. 8 in. wide and 2 ft. 9 in. high. 11. Use the formula for the volume of a prism and solve for a in terms of the other letters, likewise for b, for c. 12. State in words the results you have obtained in problem n. 62 LESSON XXXI REVIEW 1. A piece of steel % in. square and 7.25 in. long is taken to make a lathe tool. Find its weight if i cu. in. of steel weighs .28 Ib. 2. If the piece in problem i weighs 4 Ib., what was its length? 3. The base of a rectangular prism is 8 ft. X 5 ft. What is its height, if it has the same volume as a 6-in. cube? 4. A cube of steel 6-in edge is hammered when hot into a rectangular prism whose base is 5.6 in. X 3.4 in. Find the length of the prism. 5. A rectangular piece of steel 8 in. X 5 in. X 3.5 in. has each dimension reduced % in. How much is its weight reduced? 6. A liquid gallon contains 231 cu. in. A rectangular measure 4 in. square must be how deep to hold i qt. ? i gal. ? 7. If the I section of problem 6, lesson 22, is 8.25 ft. long, what is its volume? 8. A rectangular tank 15 ft. long and 12 ft. wide contains 500 gal. of water. How deep is the water in feet? 9. If the weight of the water in the tank of problem 8 weighs i ton, how deep is the water? i cu. ft. weighs 62.5 Ib. 10. Two rectangular pieces of metal 5 ft. X 5 ft. X 8 ft. and 6.5 ft. X 3.4 ft. X 10 ft. are melted and cast into a single rectangular piece 8 ft. square. How long is the new piece? 11. A bar of iron 2 in. square and i in. long is drawn out until it is 1^3 in. X i in. What is its length? 12. If a piece of stock 2^ in. square and 8 in. long is forged into a piece 2 in. square, how long is the new piece? 63 64 INDUSTRIAL ARITHMETIC 13. The sides and bottom of an open steel tank 4 ft. square and 6 ft. high outside dimensions are 2 in. thick, find the weight of the tank. 14. A piece of steel 14.5 in. X 16 in. and 2 in. thick has a rectangular hole 8 in. X 4.25 in. cut in it. What is the weight of the piece of steel? 15. If the scale is i" = 3' 4" how long must the drawing be for each of the following lengths: 12 ft. o in; 15 ft. 7 in; 16 ft. 9 in.; 7 ft. 10 in.; 23 ft. 8 in.? LESSON XXXII REVIEW OF PERCENTAGE The expression 6% means .06 or *Koo- The expression 28.3% means .283 or 283 /Looo- 25% = 2 ^00 = M; 33^% = 33 Vloo = 1 A\ 20% = ?/ioo = ?5 50% - ? - ? Exercises 1. Express as a decimal fraction without their denomi- natots each of the following: 12^%; 32%; 35-3%; 96-4%; 76.5%; 87.7%. 2. 25% of 144 = ? 12%% of 840 = ? 20% of 255 = ? 33^% of 175 = ? 75% of 164 = ? 66%% of 930 = ? 3. 28% of what number is 28? 4- 73% f what number is 146? 6. 86.2% of what number is 2586? 6. 28 is what per cent, of 560? 35 is what per cent, of 70? 7. 144 is what per cent, of 770? 29 is what per cent, of 125? 8. How many board feet of flooring must be purchased for a floor 25 ft. X 1 8 ft. if an allowance of 20% is added for waste? 9. In finishing a piece of steel its weight was reduced 1.5%. What was the weight of the finished piece if the rough piece weighed 78.31 lb.? 10. How many pounds of lead are there in 168 lb. of soft solder, if 33^% the solder is lead? 11. A certain grade of steel contains 3.4% nickel. How many pounds of nickel in i ton of this grade of steel? 12. The foot shrink rule used by the pattern maker is 12^ in. in length. The carpenter's foot rule is what per cent, of the length of the pattern maker's foot rule? 13. About .37% of soft steel is silicon. How much silicon is there in 1500 lb. of soft steel? 5 65 LESSON XXXIII BOARD MEASURE ft 1. A board foot is a piece of lumber having an area of i sq. ft. on its flat surface and thickness of i in. or less. A board foot i in. thick contains how many cubic inches? 2. In estimating the number of board feet in a piece of wood it is customary to estimate thickness less than i in. as an entire inch; e.g., a board 12 ft. long, 8 in. wide and % in. thick is estimated the same as if it were 1 2 ft. long, 8 in. wide and i in. thick. 3. Find the number of board feet in a piece of lumber i^ in. thick, 8 in. wide and 16 ft. long. 4. Find the number of board feet in a piece of lumber 6 in. thick, 8 in. wide and 16 ft. long. 5. Find the number of board feet of lumber in each of the following: i in. X 4 in. X 12 ft.; i^ in. X 8 in. X 16 ft.; 5 in. X 4 in. X 18 ft; 2 in. X 6 in. X 16 ft.; % in. X 6 in. X 20 ft.; 2 in. X Qin. X i6ft.;7in. X i8in. X 20 ft.; 7 in. X 8 in. X 12 ft.; i^ in. X 6 in. X 12 ft.; % in. X 8 in. X 16 ft.; i% in. X 10 in. X 18 ft.; 4 in. X 6 in. X 14 ft. 6. How many board feet of flooring i in. thick are required for a floor 20 ft. 6 in. X 15 ft. 9 in. if 25% is allowed for waste? 7. How many board feet in 10 pieces of lumber 2 in. X 8 in. and 1 6 ft. long. 66 LESSON XXXIV REVIEW 1. The outside dimensions of a box are 4 ft. X 3 ft. X 2 ft. If the boards are i in. thick, how many board feet are in the box and its lid? 2. The inside dimensions of a coal bin are 10 ft. X 6 ft. X 5 ft. If the sides are 2 in. thick, how many board feet of lumber are in its sides, if one of the sides is cellar wall? 3. The floor of a veranda is laid of boards i^ in. thick. If the floor is 8 ft. X 23 ft. 6 in., how many board feet of flooring does it require, allowing 10% for waste. Find the cost at $60 per M. 4. How many board feet of lumber will be required to floor a platform 16 ft. 4 in. long X n ft. 6 in. wide, the lumber to be i in. thick, if no allowance is made for waste? If 10% is allowed for waste? 5. Find the area (complete) of a cylinder 10 ft. high and 10 ft. in diameter. 6. How many square feet of tin will be required to make a pipe 4 ft. long and 7 in. in diameter if i in. is allowed for the seam? Find the cost at lojif. per sq. ft. TT = 3%. 7. If a wheel 2 in. in diameter is making 2500 R.P.M., through how many feet will a point on its rim pass in i min.? In i hr.? 8. A chimney 30 ft. high is 18 in. square and has a flue 12 in. square. How many cubic feet in the masonry of the chimney? 9. If the feed is % in. and the work has a speed of 40 R.P.M. how long will it require to turn a piece 4^4 in. long? 10. If the feed of the shaper is ^2 m -> now many strokes must it make to rough cut a piece 3^6 m wide? 67 LESSON XXXV WOODWORKING PROBLEMS Rough undressed lumber for shopwork is sawed into the following standard thickness: i in.; \}^ in.; i^ in.; 2 in.; 2}^ in.; 5 in.; 3^ in.; and 4 in. In dressing lumber about % in. of its thickness is removed, that is a i-in. board when dressed will be % in. thick. If a dressed board i% in. in thickness is wanted a 2-in. thickness must be dressed down Y in. In ordering lumber from the mill the number of pieces wanted, the thickness, the width and length of each piece must be given. A piece of lumber 2 in. X 2^ in. X 40 in. means the thick- ness is 2 in., the width 2}^ in. and the length 40 in. A mill bill is an order on the mill for lumber wanted. It must contain the number of pieces, kind of wood and dimen- sions of each piece. The dimensions are usually given for rough lumber. In estimating the amount of lumber required for a given job, add about 25% of the actual number of board feet in the finished article. This allowance is for waste. Waste in handling lumber results from any or all of the following causes : 1. Saw cuts about % in. 2. Checks or cracks in lumber especially at ends. 3. Knots. 4. Small pieces resulting from sawing. 68 WOODWORKING PROBLEMS 6 9 5. General defects in lumber. 6. Dressing of lumber. Exercises 1. Find the total number of board feet and the cost of the following mill bill for a type stand. Allow 25% for waste. No. of pieces Wood Dimensions Bd. ft. Price Cost 6 legs Chestnut 2%" X zH" X 43" it. 2 rails Chestnut W x S H" x 72" it- 2 rails Chestnut iK" X 2%" X 72" li. 3 rails Chestnut i^" X 3 M" X 22" it 3 rails Chestnut I^" X 2%" X 22" it. i top Chestnut i" X 2%" X 75" it- Total Allowance Grand total. . 2. Find totals for the following used in making step ladders. No. of pieces Wood Dimensions Bd. ft. Price Cost 2 Chestnut H" x s 5 A" x 90" it. 2 Chestnut H" X 2>" X 75" It- 3 Chestnut K" x 2ji" x 36" It- 6 Chestnut H" X 4 H" X 20" it- i Chestnut %" X 9 M" X 22^" it- 3. Find totals for the following used in making case racks for print shop. The waste in this problem was due to an error in cutting and the pieces had to be replaced. 7 o INDUSTRIAL ARITHMETIC No. of pieces Wood Dimensions Bd. ft. Price Cost 24 Chestnut aH" X aH" X 43" 7i. 12 Chestnut iK" X 3 M" X 72" 7i. 12 Chestnut iM" X 2%" X 72" 7i. 18 Chestnut iK" X 3 M" X 22" 7i. 18 Chestnut iK" X 2^" X 22" 7*. 6 Chestnut W X 2%" X 75" 7t- waste 4 Chestnut 2%" X 2%" X 4l" 7t- i Chestnut iM" X 2^" X 72" 7t> i Chestnut iM" X 3 M" X 72" 7*. LESSON XXXVI VOLUME OF A CYLINDER 1. In geometry it is proved that the volume of a cylinder is equal to the area of its base multiplied by its altitude. 2. Find the volume of each of the following cylinders: Altitude 1 5 ft., radius of base 7 ft. ; altitude 10 ft. radius of base 8 in.; altitude 18 ft., radius of base 3 ft.; altitude 18^ ft., radius of base i ft.; altitude 25 ft., diameter of base zojft.; altitude 16 ft. 8 in., diameter of base 3 ft. 3. Find the number of gallons a cylindrical tank 10 ft. high and 8 ft. in diameter will hold, i cu. ft. = 7^ gal. 4. At 25 i. per cu. ft. what will be the cost of a stone column 15 ft. high and 2 ft. in diameter? 6. A cylindrical pipe 25 ft. long and 6 in. in diameter, inside dimensions, has water running through it at the rate of 25 ft. per min. How many gallons will pass through it in i min.? In i hr.? 6. A cylinder 4 ft. in diameter and 10 ft. high contains how many times as much volume as a cylinder 2 ft. in diameter and 10 ft. high? 7. A prism 4 in. square and 2 ft. long must have how many cubic inches cut from it to give a cylinder 4 in. in diameter? 8. Find the volume of a cylinder the radius of whose base is R and whose altitude is H. 9. Use the formula of problem 8 and solve for H in terms of V, TT and R; for R in terms of V, ir and H. 10. If V = 100 TT, R = 5, find H. If V = 250 TT, H = 10, find R. If V = 400, R = 6, find H. If JV = 500, H = 10, find R. 71 LESSON XXXVII REVIEW 1. How deep must a cylindrical cistern 6 ft. in diameter be to hold 500 gal.? 1200 gal.? 2. A dealer was using a cylindrical measure 6 in. in diameter and 7% in. deep. He called it a gallon. Was it? TT = 3.1416. 3. A cylinder whose altitude is h, radius of base 2r, contains how many times as much volume as a cylinder altitude h and radius of base r? 4. A water main is 3 ft. internal diameter and i in. thick. Find the weight of 8 ft. i cu. in. of pipe weighs .26 Ib. 6. A cylindrical tank 8 ft. internal diameter is full of water; if the surface of the water is lowered 3 ft., how many gallons were drawn from the tank? 6. A closed cylindrical can 8 in. deep and 8 in. in diameter is placed inside of a cubical box 12 in. each edge, inside diameter. How much water can be put into the box? T = 3^7. 7. Which contains the more volume, a cylinder 6 ft. deep and 4 ft. in diameter or a cylinder 4 ft. deep and 6 ft. in diame- ter? The greater area? 8. How much air passes into a room through an i8-in. pipe each minute if air is flowing through the pipe 250 ft. per min.? 9. A piece of tin 24 in. long and 10 in. wide is rolled into a can; how many gallons will the can hold if ^ in. is allowed for the seams? (Two solutions.) The top and bottom of can do not come out of the piece 24 in. X 10 in. 10. How many loads of earth must be removed to dig a cess- pool 10 ft. deep and 10 ft. in diameter? i load equals i cu. yd. 11. A rotary air-fan delivers 2000 cu. ft. of air per min. through a pipe 2 in. in diameter. Find the rate of discharge in feet per minute. 72 LESSON XXXVIII REVIEW 1. The following was received for use in the machine shops of this High School. Find its cost at 17 f. per Ib. i cu. ft. of steel weighs 490 Ib. IT = JESSOP'S TOOL STEEL (ROUND) Size Total length Weight Cost . "t^ii I?S 5 ' p l%" 7'5" ? ? iM" 20' 3 " ? ? iHs" i6' S " ? ? i" 9'3%" ? ? X" 7'7" ? ? Total. 2. Find the weight of a piece of round steel i ft. long and 2 in. in diameter. Find its cost at 17^. per Ib. 3. Estimate the cost at 2%. per Ib. of the following bill of machine steel received at this High School: i% in. in diameter and 36 ft. long; i% in. in diameter and 37 ft. long; \Y in. and 18^ ft.; i^ in. and 37^ ft.; i in. and 8^ ft.; % and 26^ ft. 4. A piece of cold rolled soft steel 16 ft. long and 2 in. in diameter is worth how much at $ff. per Ib.? 73 LESSON XXXIX FORGE SHOP PROBLEMS 1. A shovel for the foundry was made from the following stock: % in. X 2 in. X 6^ in. for the blade and J^6 in - round X 21 in. for the handle. Find the weight of the shovel, i cu. ft. of iron weighs 480 Ib. 2. Find the cost of 50 shovels of problem i at 3%i. P er Ib. 3. A piece of stock i in. X x in. X 15 in. was used to make one pair of tongs. Find the weight of 35 pairs of the tongs. < 7 Stoch y'z x l"x 6" FIG. 38. 4. A three-way piece was made from stock % in. X i in. X 6 in. Find the cost of the piece at 3%^. per Ib. 5. A clothes hook was made from a piece of stock % in. in diameter and 6 in. long. Find the weight of 100 of these hooks. 6. Find the weight of the material used to make a three-way piece if the stock is % in. in diameter and 4^ in. long. 74 FORGE SHOP PROBLEMS 75 7. What is the weight of 100 hooks for foundry shovels if the stock is 10 in. X ^ in. in diameter? 8. In Fig. 38 compute the amount of stock in the shouldered piece after 3 in. are cut from each end. 1 i : ^ t J O'l 1 t t( N LJQ i ^ r '/ s_ t r " l/" ^ f I/. . rf ... - 9 ^C . s JT r* r FIG. 39. 9. In Fig. 39 if 3^2 in. are cut from each end of piece, find weight of shouldered piece. LESSON XL REVIEW 1. Show that \/3 = 1.732, also \/2 = 1.414. Memorize these numbers. ? i2\/2 = ? 25A/3 = ? 75\/3 = ? l6\/2 = ? + 5^ + 7*6 X ^ 3M + 6>i + 4 = 4. Find the volume of the cube the area of one of whose faces is 64 sq. in. 5. How many board feet in the cube in problem 4? 6. If a window frame cost $1.65 what will be the cost of the window frames in a house with 15 window openings? 7. How many board feet of lumber i in. thick will be used in laying a floor 28 ft. X 24 ft. allowing 25% for waste? 8. A man paid $35 for repainting a house, which was ]/^Q of the amount paid for the property. Find the cost. If he sold the property for $1500 did he gain or lose and how much? 9. A cylindrical bucket is 18 in. in diameter and 2 ft. high. How many gallons will it hold? 10. The gable of a house is 12 ft. high and 24 ft. wide. How many shingles will be required for the two gables? Allow 750 shingles for each 100 sq. ft. 11. Vi 57 6 32 = ? A/57 -69 = ? A/875. 14 = ? 12. An auger hole i in. in diameter is bored through a piece of wood i ft. long and 6 in. square. Find the volume remain- ing in the wood if the hole is lengthwise and perpendicular to the base? 76 REVIEW 77 13. Find the number of inches of 6-in. round stock required to make a forging 2)^ in. in diameter and 55 in. long. 6 -86- FIG. 40. 14. How long a piece of 6-in. round stock will be required to forge the above? LESSON XLI SIMPLE EQUATIONS Find the value of the letter in each of the following: 1. 3x = 6, 4x = 16, 5x = 25, 6x = 36. 2. yx = 14, sx = 30, 77 = 14, 3 a = 15. 3. lob = 30, i2c = 36, 1 5k = 45, i6p = 48. 4 - 5P = 36, 7 r = 15, 8m = 30, sL = 36. 6. x 2 = 36, y 2 = 25, m 2 = 49, n 2 = 100. 6. 2x 2 = 100, 2x 2 = 50, 3n 2 = 75, 4s 2 = 400. 7. 6x 2 = 216, 5L 2 = 80, 4p 2 = 64, iok 2 = 6250. 8. x 2 = 15, 2 y 2 = 30, 3 d 2 = 48, 4 m 2 = 72. 9. a + 5 = 10, m + 10 = 25, y + 10 = 24. 10. r + 6 = 36, d + 7 = 49, y + 8 = 50. 11. lod + 4 = 44, 553 + 6 = 31, 76 + 10 = 31. 12. 8k + 3 = 27, lorn + 5 = 45, isp + 10 = 55. 13. 6x + 5 = 30, 7y + 4 = 28, lor + 6 = 40. 14. x 2 + 9 = 25, x 2 + 25 = 169, y 2 + 36 = 100. 15. d 2 + 36 = 100, y 2 + 36 = 100, g 2 + 100 = 674. 16. 25 + a 2 = 169, 144 + d 2 = 169, p 2 + 64 = 81. 17. If three times a number is increased by 5, the sum is 14. What is the number? 18. Seven times a number and 12 more is 26. What is the number? 19. What number increased by five times itself will give 30? 20. If John's money were multiplied by 5 and $15 added to the product the result would be $100. How much money has he? 78 LESSON XLII THEOREM OF PYTHAGORAS It is proved in geometry that " The square of the hypothenuse of a right triangle is equal to the sum of the squares of its legs." The legs of the right triangle are usually represented by "a" and "b" and the hypothenuse by "c." Stated as a formula the above theorem is a 2 + b 2 = c 2 . NOTE. It is also proved in geometry that if the square of one side of a triangle is equal to the sum of the squares of its other two sides, the triangle is a right triangle, the right angle being opposite the greatest side. Exercises Find the missing parts of the following right triangles: 79 80 INDUSTRIAL ARITHMETIC 1. a = 3, b = 4, c = ? 6. a = 9, b = ?, c = 15. 2. a = 5, b = 12, c = ? 6. a = 10, b = ?, c = 26. 3. a = ?, b = 24, c = 25. 7. a = ?, b = 15, c = 17. 4. a = 18, b = 80, c = ? 8. a = ?, b = 40, c = 41. 9. What is the hypothenuse of a right angle whose legs are 510 ft. and 680 ft. respectively? 10. A house 25 ft. wide has a gable 10 ft. high. Find the length of a rafter. 11. Find the length of the diagonal of a rectangle 25 ft. X IS ft. 12. Rectangular frames are often braced by a piece from one corner to the opposite lower corner of the frame. Find the length of a brace for such a frame 8 ft. X 6 ft. 13. A telegraph pole 30 ft. high is supported by a guy wire fastened 10 ft. from its top and anchored 25 ft. from its base. What is the length of the guy wire? LESSON XLIII REVIEW 1. Find the distance across corners of each of the following square head screws: K in.; % in.; % in.; % in.; { 6 in. 2. Find the distance across corners of each of the following square bolt heads: \Y in.; 2 in.; 4^ in.; 2% in.; 3% in. 3. A piece of round stock 2 in. in diameter is to be milled square on one end. What is the side of the largest square that can be cut from the piece? 4. Thirty feet from the mast head of a derrick a boom is fas- tened to the mast. The boom is 25 ft. long. One foot from the mast head is a block containing two pulleys and ^ ft. from the end of the boom is a block with a single pulley. How long a rope must be used to allow the boom to stand at right angles with the mast and also to have 10 ft. of rope below point of attachment of boom to mast? 5. The width of a rough, U. S. standard nut or bolt head in terms of the diameter (D) of the bolt is W = i^D + ^ in. Find the width of a square nut for a bolt whose diameter is i % in., also the distance across corners. 6. If the base of a rectangle is three times its altitude and its perimeter is 40, find its base, its altitude, its diagonal and its area. 7. The area of a rectangle is 288 sq. ft. Find its base if its altitude is twice its base. 8. Solve for the letters: 25 + 3 X = 125, 2Y + 10 + 5Y = 79, 2X 2 + 125 = 375. 36 + X 2 = 136, 5 Y + 6Y - 10 = 84, 40 + 3A 2 = 77. 6 81 82 INDUSTRIAL ARITHMETIC 9. Find the length of the straight line from the home plate to second base of the baseball diamond. 10. Two persons M and P start from vertex A to go to the opposite vertex C of the square ABCD. M goes by way of B along the perimeter of the square and P on the diagonal AC. If they both reach C at the same time what is the ratio of their speeds, the side of the square being 100? LESSON XLIV FACTORING 1. The factors of a number are the numbers that multiplied together will produce the number; e.g., the factors of 8 are 2, 2 and 2. 2. What are the factors of 16? Of 49? Of 18? Of 24? Of 100? Of 27? 3. Numbers that are produced by squaring a number are called perfect squares; e.g., 25 the square of 5 is a perfect square. 4. Name all the perfect squares between i and 144. 6. Factor each of the following so that one of the factors is a perfect square: 75; So; 125; 150; 18; 27; 45; 63; 8; 12; 20; 24; 98; 147; 128; 192; 32; 48; 80; 96; 288; 432. 6. Find all the factors of each of the following: 150; 275; 95; 100; 240; 3675; 5625; 2468; 357; 7564; 231; 440; mi. 7. The product of three numbers is 720 and two of the num- bers are 8 and 9; find the other number. 8. What three numbers multiplied together will give 144? 1728? LESSON XLV SQUARE ROOT 1. Does the \/4 . 9 = A/4-A/9? Does the \/25 . 4 = A/25A/4? Does the A/25 J 6 = A/25A/i6? Does the A/36 . 25 = \/36\/25? 2. These problems illustrate the following principle: The square root of the product of two numbers is equal to the product of their square roots. That is \/ab = V'a.v'b. 3. Find the square root of 27. A/27 = A/9 3 = A/9 A/3 = 3 A/3 = 3 X 1.732 = 5.196 4. Find the square root of a 2 b. \/a 2 b = va*vE = a A/b. 6. Find the square root of each of the following: 18; 8; 12; 75; 50; 32; 48; 72; 108; 98; 147. 6. Simplify: Vc 2 xy; Vb 2 d 2 k; VpM; A/a 2 b 2 c; Vm 2 n 2 . 7. Simplify: \/2o; A/ioo; \/20 8. 5 V2 + 3 V2 - 2 A/2 = ? 8Vs + 6Vs ~ 3 A/5 = ? 7\/3 + 8 V3 + 6 V3 = 9\/6 + 3A/6 i2\/6 = _ = ?_A/8o - A/75 + A/27 + \/48 = ? A/24 + A/ISO + A/2i6 = ? \/20o + \/8oo + \/3200 = ? A/432 + A/363 A/3oo = ? A/3 + A/8 + V27 = ? Vs + V20 + Vi2 = ? 84 LESSON XLVI REVIEW 1. Factor so that one of the factors shall be a perfect square : 500; 800; 288; 1440; 112; 250; 360; 490. 2. Solve for the value of the letter in each of the following: Sx + 6x = 22; 8x + 10 = 30; yx - 5x = 24; ;x + 15 = 50; 3x + 5x + 10 = 14; 4x + 7 X - 5 X = 2 55 IOX + 8x. + 12 = 38; 6x - sx + 7x = 36. 3. Simplify: \/49; A/3a 2 ; \/2a 2 ; V$b 2 ; 4. If the hypothenuse of a right triangle is 2 a and one of the legs is a, what is the other side? 5. If each leg of a right triangle is m, what is its hypothenuse? 6. Each side of a square is t; find its diagonal. 7. If the diagonal of a square is i5\/2, what is its side? 8. If the area of a square is 144, find the diagonal. 9. What is the area of a circle whose radius is 1 2 ? 10. The area of a circle is 2571-, what is its radius? Solution. n-R 2 = 2571-; R 2 = 25; R = 5. 11. Find the radius of each of the following circles whose areas are: 367r; 4971-; 6471-; IOOTT; 62571-; 14471-. 12. Find the diagonal of a square, whose area is 100; 400; 900; 625; 169; 576. 13. What is the area of a square whose diagonal is 8\/2; ; 3\/2; i4\/2; i6\/2; 7\/2j io\/2; a\/2; b\/2 ? LESSON XLVII TRIANGLES 1. A triangle with two sides equal is an isosceles triangle. The angles opposite the equal sides are called base angles of the isosceles triangle. A triangle with all its sides equal is an equilateral triangle. 2. Construct from cardboard an isosceles triangle, tear off one of the base angles. Apply to the other base angle. Are they equal? What statement can you make about the angles of an equi- lateral triangle? 3. If one base angle of an isosceles triangle contains 18, how many degrees in the other base angle? If 23? If 45? If 60 ? If 48? 4. Construct from cardboard a triangle with no two sides equal. Cut off two of the angles and place them as in the following figure: FIG. 42. What kind of angle is now formed at B ? How many degrees in it? What then is the sum of the angles of a triangle? The sum of the angles of a triangle is 86 TRIANGLES 87 5. Find the third angle of each of the following triangles: A = 28, B = 48, C = ?, A = 30, B =60, C = ?, A = 45, B = ?, C = 90, A = 40, B = 60, C = ?, A = 60, B = 60, C = ?, A = 100, B = 40, C = ? 6. In a right triangle what is the sum of the two acute angles? 7. If one acute angle of a right triangle is 30, what is the other acute angle? If 20? If 48? If 45? 8. The vertex angle of an isosceles triangle is 100. How many degrees in each base angle? 9. If a base angle of an isosceles triangle is 80, how many degrees in the vertex angle? 10. In a triangle Z A = 2 Z B and Z C = 3 Z B. How many degrees in each angle of the triangle? HINT. Let x = number of degrees in ZB. 11. In a certain triangle ZA = aZB and ZC = 2ZA. Find the number of degrees in each angle of the triangle. 12. Find the number of degrees in the sum of the angles of a quadrilateral. 13. Construct a triangle with a pair of angles equal. Test to find out if the sides opposite those angles are equal. 14. Construct an equilateral triangle. Draw its altitudes. Test to find out if the altitude of an equilateral triangle bisects the base to which it is drawn. LESSON XLVIII THE 30 RIGHT TRIANGLE B FIG. 43. Theorem. In a 30 right triangle the leg opposite the angle of 30 is one-half the hypoihenuse. Given the right triangle ABC with C the right angle, Z A = 30, BC the side opposite 30 and AB the hypothenuse. To Prove. BC = 3^AB. Proof. In the ZC take ZDCA = 30. Then ABAC is isosceles (?) .'. AD = DC. Also each angle of ADCB is 60 (?) .'. BD = DC. /.AD = BD, or BD = 3/AB. But BC = BD (?) .'. BC = KAB Q. E. D. Exercises 1. If in the above figure AB = 100, 120, 200, 300, 400, 18, 6a, yb, 4(0 + d), 8(e + f + g), what is BC? 2. If in a 30 right triangle the side opposite 30 is 15}^, what is the length of the hypothenuse? 3. If a kite string 200 ft. long makes an angle of 30 with the ground, about how high is the kite? 88 LESSON XLIX REVIEW 1. If the hypothenuse of a right triangle with one of its acute angles 30 is 100, what is the length of the side oppo- site the angle of 60 ? 2. A string 200 ft. long attached to the top of a pole reaches the ground at a point P. The angle made by the string and a line joining P with the foot of the pole is 30. How high is the pole? How far is P from the foot of the pole? Also D FIG. 44. 3. In this figure AB = 400 ft. Find BC, CD and DA. find number of degrees in /DBA and ZBDA. 4. The diagonal of a rectangle is 30 ft. and makes an angle of 30 with the base. What is the altitude of the rectangle? What is the base? Its area? 5. A rope is stretched from an upper corner of a room 15 ft. square and 10 ft. high to the opposite lower corner. What is the length of the rope? 6. One rectangle is 40 ft. X 20 ft. and another rectangle is 80 ft. X 10 ft. Find the side of the square that has an area equivalent to the sum of -the rectangles. 7. A circular cistern 8 ft. in diameter and 12 ft. deep is full of 89 90 INDUSTRIAL ARITHMETIC water. If a pipe drains from it 10% of its contents in i hr. and another pipe conducts water to it equal to 8% of its contents in i hr., how many gallons of water will be in the cistern at the end of i hr.? 8. Your Street and Water Board placed 157 cylin- drical street markers at the intersections of many principal streets. They are 9 ft. high and 4 in. in diameter. Find the cost of the paint to give them two coats, allowing i gal. to paint 250 sq. ft. one coat, the paint costing $2.50 per gal. Theorem. In a 60 right triangle the side opposite 60 is one-half the hypothenuse multiplied by \/3. Use Fig. 45 and show that x = a \/3' FIG. 45- 6 FIG. 46. Exercises 1. If a = 10, c = ?, b = ? If b = 8\/3, c = ?, a = ? If c = 20, a = ?, b = ? If a = 15, b = ?, c = ? If b = i8\/3, c = ?, a = ? If c = 40, a = ?, b = ? (Fig. 46.) 2. The diagonal of a certain rectangle makes an angle of 60 with the base. If the base is 2o\/3, what is the diagonal and also the area? 3. A ladder makes an angle of 60 with the ground and the foot of the ladder is i$\/3 ft. from the building against which it is leaning. How long is the ladder and how high does it reach on the wall? 4. If the width of a house makes an angle of 60 with the rafters and the rafters are 18 ft. long, how high is the gable and how wide is the house? 91 LESSON LI REVIEW 1. Fig. 47 represents a lathe center with its dimensions. Find H and the diameter of the small end if the body tapers .6 in. per i ft. ^*ts ^ < r" f-U FIG. 47. . o ^ / /a FIG. 49. FIG. 50. 2. In Fig. 48 AB = AE = CF = FG = i in. If GE = 1.3 in., find the length of the figure, also its area. 3. Find length of AB also area of Fig. 49. 4. When the rocking lever of Fig. 50 is turned 30 from the horizontal line how much farther has P' fallen than P has risen? 92 REVIEW 93 D G A FIG. 51. 5. The diagram shows The Pratt truss used in bridge con- struction DE and BA are called struts. EB topchord. If the height of the truss is 12 ft.; find length of the struts, topchord and length of the bridge. 6. An inclined plane 20 ft. long makes an angle of 30 with the horizontal line. What is the height of the inclined plane? 7. If in the mortar box of Lesson 9, the inside length of the bottom is 6 ft. and the inside height of the sides is 14 in., find the inside top length of the box. LESSON LH ALTITUDE AND AREA EQUILATERAL TRIANGLE 1. Find the altitude of the equilateral triangle whose side is 8, 6, 10, 12, 14, 16, 18, 20. 2. The side of an equilateral triangle is 17. What is its altitude? 3. If the side of an equilateral triangle is a, show that its altitude is \/^. Learn this formula. 4. Use the result obtained in problem 3 and give the altitude if the side is 3, 4, 5, 6, 7, .87, 13.34, 8, 89.34, 25, 13. 6. Find the area of each of the triangles in problem i and 2. 6. Show that the area of the equilateral triangle whose side a 2 ,- is a, is V 3- Learn this formula. 7. Use the result obtained in problem 6 and give the area of each of the following equilateral triangles: 5; 7; 8; n; 13; 15; 2.5; 3.5; 4.68; 12; 16; 17; 22.2; 38; 40; 50. 8. The base of a prism is an equilateral triangle whose side is 5.4. Find the volume of the prism if its altitude is 6. 9. Find the volume of a prism whose altitude is h and whose base is an equilateral triangle side a. 94 LESSON LIII THE REGULAR HEXAGON If six equilateral triangles be ar- ranged as shown in Fig. 52 a regular hexagon is formed. The point (o) that is the common vertex of the equilateral triangles is the center of the hexagon. OB or OC, etc., is a radius of the hexagon. Z A, Z B, etc., are vertex angles of the hexagon. Exercises FIG. 52. 1. How many degrees in each vertex angle of a regular hexagon. 2. Are the sides of a regular hexagon equal? Why? 3. Use problems i and 2 to form a definition of a regular hexagon? 4. What is the perimeter of a regular hexagon whose radius is 8, 9,. 10, a, b, x? 5. How many degrees in Z BOE? Is BOE a straight line? Why? In Z IOH if OH is perpendicular to CB and OI is perpendicular to EF, is IOH a straight line? Why? In screws that have hexagonal heads and also in hexagonal nuts the length of IOH is called the distance across the flats, the length of BOE the distance across the corners. 6. Find the distance across the flats of each of the following standard hexagonal nuts having given distance across the corners: 95 INDUSTRIAL ARITHMETIC n. n.; in.; n.; n. n. NOTE. Use \/3 = *? an d express results as multiples of /- Q 3^4 in., for example, the answer for ^ in. is - which is consid- ered J'le in. 7. Find the distance across the corners of each of the follow- ing standard hexagon nuts having given distance across the flats: in.; in.; in.; 2^ in.; n. 8. The distance across the flats for both a square and a hex- agonal bolthead is 2 in.; find the distance across the corners for each and express the results correct to three decimal places. 9. Find the area of a regular hexagon whose side is 8, 10, 12, 13-5, S%'> a, b, f, m. 10. A regular hexagon whose radius is 10 is inscribed within a circle; find area of part of the circle between the perimeter of the hexagon and the circumference. 11. Find the volume of a prism whose base is a regular hexagon, each side 20. The altitude of the prism is 8.5. 12. A piece of steel 2 in. in diameter and 10 in. long is milled into the largest possible hexagonal piece. Find its weight both before and after being milled. 13. The distance across the flats of the sleeve is 3% in. and is 2% times the diameter of the hole. The hole is one-half the diameter of the cylinder and one-fourth the total length of the sleeve. The hexagonal part is one-third the total length. Find all dimensions and its weight when made of brass, i cu. ft. brass weighs 524.1 Ib. FIG. 53. LESSON LIV SCREW THREADS 1. The depth of a thread is the perpendicular distance from the bottom of the groove to the straight line joining the tops of the thread. Twice this distance is the double depth. 2. The root diameter of the screw is the outside diameter of the screw minus its double depth. 3. In practice there are threads of several shapes. We shall consider but two kinds, the Sharp V thread whose angle is 60 and the U. S. standard form of thread. ! P J Vvl U.S. Standard Thread FIG. 54- 60 V Thread FIG. 55. The U. S. standard has the same angle as the 60 thread, but has its top and its bottom flat. The width of this flat is one- eighth of the pitch of the thread. The depth of this thread is three-fourths of the depth of the 60 V thread of the same pitch. r EXERCISES (These problems refer to Screws of Single Thread) 1. A screw has 15 threads per in. What is its pitch? Its lead. 7 97 98 INDUSTRIAL ARITHMETIC 2. Find the pitch and lead for the following number of threads per inch: 13; 14; 25; 36; 8; 17; 18; 20. 3. Determine the depth of each of the following 60 V threads whose pitches are H2 in-, He in-, Ho in., % in., Y in., K in., K in. 4. Show that the depth of the 60 V thread is equal to its pitch multiplied by 3^ ^3 or about .866. 6. Find the root diamters of each of the screws in problem 3 if the outside diameter of each is 2 in. 6. Determine the depth of thread for the pitches given in problem 3 for the U. S. standard-shaped threads. 7. How many revolutions will a 60 V thread make in advancing 2 in., if its pitch is ^{Q in.? 8. If the flat of a U. S. standard-shaped thread is % in., what is its depth? 9. If the root diameter of a 60 V thread with 10 threads per in. is i Y in., what is its outside diameter? 10. Find the outside diameter of the U. S. standard with 8 threads per in. and a root diameter of i% in. LESSON LV REVIEW 1. How many revolutions will a 60 V thread make in ad- vancing \Y in. if its pitch is 3^o in-? 2. What is the depth of the thread in problem i? 3. If the flat of a U. S. standard-shaped thread is ^fg m -> what is its depth? 4. The root diameter of a 60 V thread with 12 threads to the inch is % in. What is its outside diameter? 5. The outside diameter of a U. S. standard is i in. and its pitch 34 in. If the screw is 2 in. long what is the length of the thread? 6. If the outside diameter of a sharp V thread is ^ in. and its pitch 3"{o m - what is the length of the thread of a screw 3 in. Long? 7. If the feed is ^2 m - an d the cutting speed 20 ft. per min., find the required time for the tool to advance 2 in. along a piece 3 in. in diameter. 8. When cutting hard steel the speed may be 33 ft. per min. if the depth of the cut is 34 m - an d the feed 3^6 m - Find the number of R.P.M. a cylinder of hard steel 2 in. in diameter is making for the above data. 9. If the scale is % in. = 5 ft. o in. what is the length on a working drawing for each of the following: 2 ft. 6 in.; 4 ft.; 10 ft.; 12 ft. 8 in.; 25 ft.; 48 ft.? If the scale is 34 in. = 3 ft. o in.? i in. = 5 ft. 6 in.? % in. = 4 ft. 6 in.? 10. A carpenter has a mitre box 5 in. wide outside dimen- sions. Explain how he will find the points on the box through which he must saw to get a 45 angle; a 60 angle. 99 100 11. INDUSTRIAL ARITHMETIC FIG. 56. The above figure is a side view of an oblique thrust. R, P" and P are in a straight line. Find area of shaded portion, length of SP and S'P'. 12. Find the nearest number of 64ths to which each of the following fractions is equivalent: 15 ii 6 9 21 125 ii 19 13 17 ' 13 ' 7 ' ii ' 23' 127' 15 ' 21 ' 14 13. State the limit of the error for each of your answers in problem 12. 14. If R = 5 and r = 3, find value of V in V = -v(R*-r*). O 15. If TI = 6, r 2 = 3, and h = 2.4 find value of V in 16. Use the following formula 86,670 SCREW THREAD to find the value of T in the table 101 r D 5 P r D 5 P 7 5 100 18 5 1 80 7 5 200 10 5 1 20 8 5 1 60 16 5 140 12 5 2220 9 5 200 17. How many degrees in each angle of the triangle ABC if ^A = 3/5 and ZC = 2 18. Simply: \/io8 \27; \5o \2oo; A/3o - 19. If the feed is + i2; A/32 + X/48- in. and the work has a speed of 30 R.P.M.. how long will be required to cut a piece i foot long? 20. Floor moulding is required for a floor 36' X 24'. If the strips can be purchased 10', 12' or 16' long, which length would you purchase assuming no waste and the price the same per foot? Why? 21. What is the side of an equilateral triangle whose area is 25\/3 ? 2o\/3 ? ioo\/3? 22. Solve for the letter in each of the following: 5-y 2 + io=no; 25 + a 2 = 75; 15 + 62 = 115. 5/> 2 + 7 = 232; 36 + k 2 = 436; 29 + d 2 = 173. 3* 2 + 6 = 426; 10 + e 2 = 210; 17 + g* = 417. LESSON LVI THREAD CUTTING Mathematically, thread cutting is merely a problem of gears. The cutting tool is made to advance along the piece being cut by means of a lead screw. The piece is made, to revolve as the tool advances. If the number of R.P.M. of the piece is the same as the R.P.M. of the lead screw, the tool will evidently cut on the piece the same number of threads per inch as the number on the lead screw; e.g., if the lead screw has 4 threads per in. the tool will cut 4 threads per in. on the piece. If the R.P.M. of the piece is twice the R.P.M. of the lead screw, then 8 threads per in. will be cut. The diagram (Fig. 57) of the head stock of a simple geared lathe will help to make clear how the number of R.P.M. of the work and of the lead screw is accomplished. I, I' and I" are intermediates, hence have no effect on R.P.M. I and I' determine the direction of rotation of the lead screw. C and S' are both keyed to the stud shaft. S' and S in the problems that follow are equal. C and L are the gears that by changing determine the R.P.M. of the lead screw. Standard "change gears "are made in series that have a com- mon difference or either 4 teeth or 5 teeth. The smallest gear of each series contains 20 teeth, and the largest 1 20 teeth and 100 teeth respectively. 20, 24, 28, 32, 36,40, 44, 48, 52, 56, 60, . . 120, . . A series. 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, . . 100, . . B series. Lathes are also supplied with other gears as 46, 66, 69. 102 THREAD CUTTING 103 Idler I Fixed Gear S' on Stud S Gear on Spindle Carries Work I.' Idler 6 Change Gear on Stud I 3 * Intermediate FIG. 57- 104 INDUSTRIAL ARITHMETIC Exercises 1. In the above diagram S is made to revolve; explain how the motion is communicated to L. 2. S = S'. If the number of teeth of C equals twice the number of teeth of L which is revolving the faster and how many times as fast? If L has 6 threads per in., how many would be cut on the work? 3. If C = L and S = ^S', compare number of R.P.M. of C with that of L. 4. If C = YJL and S = ^S' compare R.P.M. of C with that ofL. 5. If C = L, S = S' and I" = 3 L, compare R.P.M. of C with that of L. 6. If the lead screw has 8 threads per in. how many threads per inch are being cut on a piece making twice as many R.P.M. ? 3 times as many? 2% times as many? How many times as many to cut 20 threads per in.? 30 threads per in.? 40 threads per in.? 12 threads per in.? 7. The lead screw has 4 threads per in., and L has 80 teeth. How many teeth must C have in order that 8 threads per in. may be cut? 16 threads per in.? 8. What gears on C and L from the A series may you use to cut 12 threads per in. with a lead screw of 6 threads per in.? 5 threads per in.? 8 threads per in.? 9. The lead screw of a lathe has 5 threads per in. What gears on C and L may you use to cut 10 threads per in.? 12 threads per in.? 15 threads per in.? 3 threads per in.? 18 threads per in.? 10. With a lead screw of 5 threads per in., what is the largest number of threads per inch that can be cut in a simple-geared lathe withS = S'? 11. The standard pipe thread for i^-in. pipe is n^ threads per in. What gear will you use to cut this thread with a lead THREAD CUTTING 105 screw of 6 threads per in.? 5 threads per in.? 4 threads per in.? 12. Let NT = No. of teeth of L, N R = R.P.M. of L, N' T = No. of teeth of S, N' R = R.P.M. of S. Then show that NT /N'R\ Threads per inch to be cut N' T VN R / T Threads of L 13. Use the formula of problem 12 to find number of threads that will be cut per inch when N'T = 30, N T = 20 andL = 6; N' T = 40, N T = 45, L = 4; N T = 65, N' T = 55, L = 8; N T = 120, N' T = 48; L = 5; NT = ioo, N' T = 85, L = 8; L = 6, NT = 60, N' T = 30; N' T = 85, N T = 80, L = 8; L = 6, NT = 5 2 > N 'T = 40- Very often the number of teeth of S and of S' are not equal, the number of S being less than the number of S'. The effect of this is to make S revolve more rapidly than S' and therefore than L provided C and L are equal. Suppose S = 20 and S' = 40; then S will make two revolutions while S' and L are making one. Hence, if L has 6 threads per in., 12 threads will be cut on the work. That is, we get the same result with S = 20, S' = 40 and L = 6, as if we had S = S' and L = 12. This illustrates the following: When S and S' are not equal, we assume C and L to be equal, find the number of threads the given lead screw will cut per inch, use this number as the lead screw and proceed as in the previous work. This number is called the lead number or the calculating lead screw. Example. If S = 30; S' = 40, and L = 6, what gears may be used to cut 12 threads per in.? Solution. Since S/S' = %, S will make i^ revolutions, while L is making one, that is, 6 X 1^3 or 8 threads per in. will be cut. Hence the lead number is 8. Also x % = 4 %2J therefore gears of 48 and 32 must be used, 32 on C and 48 on L. Stated as a formula the above is Threads per inch to be cut _ NT of L Lead number NT of C io6 INDUSTRIAL ARITHMETIC Many lathes are made with the number of teeth of S and of S', 30 and 40 respectively. Exercises 1. If S = 30, S' = 40, find the lead number when L = 4, L = 6, L = 8, L = 5. 2. If S = 30, S' = 40 and L = 6, what gears may be used to cut 10 threads per in.? 15 threads per in.? 16 threads per in.? 20 threads per in.? 4 threads per in.? 40 threads per in.? 3. If S = 30, S' = 40 and L = 6, what gears may be used to cut a standard pipe thread for %-in. pipe? 4. With S = 30, S' = 40, L = 6 and the A series, what is the largest number of threads per inch that can be cut in a simple- geared lathe? The least number? FIG. 58. In the above train of gears B and C are keyed to the same shaft, also D and E. Such a train is said to be compounded. If the train is set in motion by A or A drives C, then A is called the driving gear of the train and F the driven gear. Also A, B, D are called the drivers and C, E, F the driven. Law for Compound Gears. N A = R.P.M. of A and N F = R.P.M. of F A, B, C, etc. = the number of teeth of each gear. THREAD CUTTING IOJ Then NA _ C.E.F NP ~ A.B.D' Expressed in words this formula is: The R.P.M. of the driving gear divided by the R.P.M. of the driven gear of a train is equal to the product of the number of teeth of the driven gears divided by the product of the number of teeth of the driving gears. Exercises 1. If N A = 10, A = 20, B = 18, C = 30, D = 22, E = 26 and F = 40. Kind N F . 2. Solve problem i without the use of the formula. 3. Find NA when A = 10, C = 20, B = 15, E = 30, D = 18, F = 36 and N P = 20. 4. Find N F when N A = 8, A = 12, C = 24, B = 18, E = 36, D = 20 and F = 40. Compound-geared Lathe. A lathe is said to be compound geared when there are two changes of speed between S', the fixed gear of the studshaft, and L, 'the gear of the lead screw. Fig. 59 shows one way of compounding the gears of a lathe. C' is another gear keyed to the same shaft with I". The change gears are L, C', I" and C. I" and C' are together known as the compound and one gear is usually twice the other, as 60 and 30. Compound gearing is used when in a simple-geared lathe a gear with a larger number of teeth than is usually made would be needed or a gear too large for the center distance between the shafts. Law for Compound-geared Lathe. Threads to be cut per inch _ I."L _ I" L Lead number = C'.C = C r . X C The following will illustrate how to apply this formula. io8 INDUSTRIAL ARITHMETIC FIG. 59. Find the gears necessary to cut n3^ threads per in. with a lead screw of 6 threads per in. on a lathe with S = 30 and S' = 40. Lead number = 8; then-^ = o X % = X 6 %o- That is 7,7 = and ~ = \ -, the required C 30 L 64 gears. Since I" and C' are usually 60 and 30, we should try to make COMPOUND-GEARED LATHE IOQ the fraction equal to the product of two fractions, one of which is either ^ or %. Thread Cutting 1. With S = S'; find the compound gearing necessary to cut 32 threads per in., with a lead screw of /^-in. pitch. 2. With S = 30 and S' = 40, find the compound gearing that may be used to cut 22 threads per in. with a lead screw of pitch % in. 3. With S = 30, S' = 40 and L = ^-in. pitch, what simple gearing will cut 23 threads per in.? What compound gearing? 4. Make L = 8 threads per in. in problem 3 and then solve it. 6. Make L = 6 threads per in. in problem 3 and cut 27 threads per in. 6. With S = S' andL =3^-in. pitch, find gears for cutting 24 threads per in. by compounding. 7. Make S = 30 and S' = 40 in problem 6 and solve it. LESSON LVII TAPER A piece of turned work with uniformly increasing diameter is called a taper. The difference of the diameters of the two ends of the taper is the taper of the piece. It is usually given as a certain number of inches per foot or per inch. For example, if a taper is i ft. long with diameters of 2 in. and i in. respectively, the taper is 1 in. per ft. or ^{2 m - P er m - Exercises 1. If the larger diameter of a taper 18 in. long is 3 in. and the taper % in. per in., what is the smaller diameter? 2. The diameters of a taper are 2 in. and i in. If the taper is ^ 6 m - P er i n -> find the length of the taper. 3. What is the difference in the diameters of a taper 10 in. long whose taper is % in. per ft.? 4. What is the taper per foot of a taper, 3^ in. with di- ameters % in. and J^o in.? 6. The taper per foot of a Jarno taper is .6 in. Find the length of a Jarno taper whose diameters are % in. and %Q m - ', 2 in. and 1.6 in.; i in. and .8 in. 6. The length of a Jarno taper is 10 in. and its larger diame- ter 2.5 in.; find its smaller diameter. If the length is g% in. and the smaller diameter 1.9 in., what is the larger diameter? 7. The taper per foot of a Brown and Sharp taper is % in. Find the length if the two diameters are 2.25 in. and 2.58 in. TAPER III If the length is 7% in. and the larger diameter 2.052 in., find the smaller diameter. 8. The diameters of a taper are 1.045 m - an d 1-348 in. and the taper per foot .516 in.; find the length of the taper. 9. What is the taper per foot of a piece 9^6 m - long with diameters of i% in. and 2^ in.? 10. If the length of a taper is L ft., and the diameters of its ends a in. and b in., what is its taper (T) per foot? 11. Use the formula to find T when L = 2, A = i, B = 2; when L = 6 in., A = i, and B = 144; when L = 10 in., A = 2, B = 3.375; when T = .623, A = 1.02 and B = 1.28. Find L. LESSON LVIII TAPER TURNING Tapers are turned in a lathe either by means of the taper attachment or by offsetting the tailstock. When the taper at- tachment is used the taper in inches per foot is determined and the taper attachment set to that number. If the tailstock is offset we must know the taper per foot and the length of the piece expressed in feet. Then if S = offset in inches, T = taper in inches per foot and L = length of piece in feet, we have as the formula for the offset, or NOTE. The proof of this formula depends upon principles of geometry. Exercises 1. A cylinder i ft. long is to be tapered Y in. per ft. How much must the tailstock be offset? 2. Determine the offset for each of the following: Taper % in. per ft., piece 8 in. long; taper 3^ in. per ft., piece 18 in. long; taper .6 in. per ft., piece 10 in. long; taper .602 in. per ft., piece 14 in. long; taper .625 in. per ft., piece 5 in. long; taper .592 in. per ft., piece 25 in. long. 3. The tailstock of a lathe is offset 2 in. Find the taper if the piece is 8 in. long; 10 in. long; 2 ft. long. 4. If the offset is % in. and the taper % in. per ft., what is the length of the piece? TAPER TURNING 5. How much must the tailstock be offset to turn the piece shown in the figure? ^t Sf 7 3 , 7 " 4" > , TI" II FIG. 60. 6. A lathe center 5 mi long is to be tapered .6 in. per it. Find the offset. 7. A taper pin 4^ in. long has the large end .49 in. in diame- ter and the small end .398 in. in diameter. How much was the tailstock offset to turn the piece? FIG. 61. Determine each offset for turning the piece shown. 9. Seller's taper is % in. per ft. Determine the length of a piece being turned to this taper if the set over is % in. 10. The Jarno taper No. 18 is 9 in. long and has diameters of 1.8 in. and 2% in. What must be the offset to turn this taper? 11. How far must the tailstock be set over to taper a piece 15 in. long with American taper? (A. T. = %e m - P er ft.) LESSON LIX REVIEW 1. Find the altitude of the equilateral triangle whose side is 3^ in., % in., % in., correct to three decimal places. 2. A boiler tube is to be made 3 in. in inside diameter, ^f e in- thick and 10 ft. long. How long a piece of brass 3 in. square will be required to make the tube? 3. Find the length of the line ABCD as in the figure. A B^H *$ i tr in' 10 FIG. 62. 4. Find the altitude of each of the equilateral triangles of Fig. 63. 5. In turning a locomotive wheel 78 in. in diameter, what is the proper number of revolutions per minute, in order that FIG. 63. the cutting speed may be 10 ft. per mm.? 6. A piece of brass 4 in. in diameter is making 80 R.P.M- What is the speed of a point on its surface? 7. If a piece tapers .0026 in. per in., what is its taper per foot? 8. The standard pipe thread taper is % in. per ft. How much must the tailstock be offset to turn this taper on a piece 2 ft. long? 9. A round shaft is 3'^ in. in diameter. Find the length of the greatest square end that can be made on the shaft. 114 REVIEW 115 10. If S = 30 and S' = 40 and L = 3^-in. pitch, find the simple gearing you may use to cut 5 threads per in. ; also the compound gearing. 11. If the scale is i" = o' 8" what should be the length of each line for a scale drawing of problem 3. 12. What is the micrometer reading for each of the following: .777 in.? .326 in.? .565 in.? .480 in.? .444 in.? 345 in.? 13. What is the nearest number of 64ths for each number of problem 12? 14. The diameter of a drill is 2 in. Its speed is 92 R.P.M. and its feed per revolution is .015 in. How many cubic inches are being removed per minute? 15. A drill 4 in. in diameter making 46 R.P.M. removes 10.8 cu. in. per min. Find its feed per revolution. LESSON LX RATIO The quotient obtained by dividing a by b is called the ratio of a to b. The ratio of a to b is written as a:b, or a/b or a-f-b. Exercises 1. Find the value of each of the following ratios: 10:5; 16:20; 32:8; 100-^25; 17-5-19; 25-5-5; !% 4 ; 34 i ? ; 172 ^44; a 2 /a; b 4 /b 2 ; c 3 /c 2 ; d/d 3 . 2. Find the ratio of the areas of two rectangles of altitudes 5 and 10 and bases 8 and 16 respectively. 3. What is the ratio of the areas of two triangles of altitudes 10 and 1 8 and bases 22 and 30 respectively? 4. The radii of two circles are id ft. and 15 ft. What is the ratio of their areas? Of their circumferences? 6. Two gears of 60 and 45 teeth respectively. What is the ratio of their speeds? If they have a and b teeth? 6. Find the ratio of the speeds of two pulleys connected by a belt if their diameters are 12 in. and 8 in.; 15 in. and 7 in.; a and b. 7. What is the ratio of the areas of two circles whose radii are p and q? 8. What is the ratio of the surface speeds of the pulleys in problem 6? 9. The ratio of the speeds of the driving gear to the driven gear is 3:2. If the driving gear containd 48 teeth, how many teeth has the driven gear? 116 RATIO 117 10. The efficiency of a machine is the ratio of the units of work given out by the machine and utilized to the total number of units of work put into it. What is the efficiency of a machine that gives out 37 ft.-lb. from 50 ft.-lb. put into it? 20 ft.-lb. from 45 ft.-lb.? 38 ft.-lb. from 48 ft.-lb.? FIG. 64. 11. The ratio a/ioo is called the grade of the slope. The loo ft. is horizontal distance and a the vertical. If a road bed rises 10 ft. each 100 ft. measured horizontally, what is its grade? What is a 7% grade? 10% grade? 12. A vertical rise of 20 ft. in 1000 ft. is what grade? 13. Water consists of 2 parts hydrogen and i part oxygen. What is the ratio of oxygen to hydrogen? Of hydrogen to oxygen? Of hydrogen to water? Oxygen to water? 14. Name several gears that will have a speed ratio of 5:2. 15. The volumes of two sphere's have the same ratio as the cubes of their radii. Find the ratio of the volumes of two spheres whose radii are 2 and 3; 4 and 5; and 7 and 9; 10 and 12. 16. Find the ratio of the areas of two squares whose sides are m and n respectively. 17. What is the ratio of the lead to the pitch of a single- thread screw? 18. If the scale is % in. = 2 ft. o in., what is the ratio of a scale drawing 2 in. long to the length of the line it represents? 19. A speed ratio of 7:3 is required for two gears. If the driven gear has 56 teeth how many must the driving gear have? 20. If pigiron contains 93% pure iron, 3% carbon and 2% sulphur, find the ratios of the different elements given. Il8 INDUSTRIAL ARITHMETIC 21. The heating surface of a certain boiler is 1800 sq. ft. The grate measures 9 ft. X 8 ft. Find the ratio of grate surface to heating surface. 22. If the ratio of two numbers is 9 : 7 and one of the numbers is 1423, what is the other number? Two solutions. LESSON LXI SECTORS AND SEGMENTS The part of a circle between two radii and an arc is called a sector of the circle, as AOB. The part of a circle between an arc and a chord is called a segment of the circle, the shaded part of Fig. 65. The angle at the center of the circle formed by the two radii is called the angle _ of the sector. A "^**LI | , L , -J->^ O Exercises 1. What part of the whole circle is a sec- tor whose angle is 6o? 120? 30? 90? 180? 45? 15? 38? 2. Find the area of a sector of 90, if the radius of the circle is 10, 8, 12, 5, 25, 15, 35. 3. If the radius of a circle is 10, find the area of a sector of 60, 90, 45, 30, 180, 36, 18. 4. If the area of a sector AOB is 28 and the triangle AOB is 20, what is the area of the segment? 6. Find the area of the segment if the area of sector AOB is 98.3 and triangle AOB 76.84. 6. State how we can find area of a segment of a circle. 7. Find the area of each of the segments in the following, having given that the angle AOB is 60: Radius of circle 8; 9; 10; 12; 15; 22; 30; 42. 119 LESSON LXII REVIEW 1. The radius of each circle is 5. between the circles of Fig. 66. Find the area included FIG. 66. FIG. 67. 2. The diameter of a circle is 10. The circumference is divided into six equal parts and lobes formed as in figure. Find the area of each lobe of Fig. 67. 3. The diameter AE of Fig. 68 is divided into four equal parts, and semicircles drawn as indicated. Find the area of each figure. Let AE equal 8. FIG. 68. FIG. 69. 4. Find the area of each lobe of Fig. 69 if the side of the square is 16. LESSON LXIII REVIEW 1. One cubic inch of steel weighs .29 Ib. An I-beam has a cross-section as shown in Fig. 70 and a length of 12 in. Find its weight. -72- FIG. 70. 2. In forging a bolt 24 in. in total length i^ in. in diameter with a head ^ in. thick and i^ in. square, the stock is cut from a bar of iron, i^ in. square in cross-section. How long a piece will it take? Allow i in. in length for waste. 3. A circular disk ^ in. thick and 4 in. in diameter is to be made from the same rod. Find the length required. 4. The external diameter of a hollow cast-iron shaft is 18 in. and its internal diameter is 10 in. Calculate its weight if the length is 20 ft. and cast iron weighs .26 Ib. per cu. in. 6. Find the length of steel wire in a coil, if its diameter is .025 in., and its weight 50 Ib. 6. The larger diameter of a piece of steel is % in., and the smaller 3 ^4 in. Find the taper per foot if the piece is i% ft. long. 7. The larger diameter of a piece of steel is 3^ in. If it is i% ft. long and the taper is % in. per ft., what is the smaller diameter? Find the offset for turning this taper; also the taper of problem 6. 121 122 INDUSTRIAL ARITHMETIC 8. Find the cost of 25 pieces of 2 in. X 8 in. each 16 ft. long at $30 per M. 9. It is required to build a bin that will hold 50 bu. It can be built in a space 7 ft. long and 3 ft. 6 in. wide. How high must it be if a bushel contains \Y cu. ft.? 10. A certain lumber company has a piece of timber 3 ft. square and 80 ft. long. How many board feet in the piece? LESSON LXIV REVIEW 1. Find the area between the tangents to the circle and the arc. The tangents make an angle of 60, the radii are per- pendicular to the tangents and OF bisects angle APB. OA = 12 in. 2. Find the area included between the four circles, radius is 5. ABCD is a square. Each FIG. 72. FIG. 73- 3. The diameter of the circle is 16. Semicircles are drawn as indicated in Fig. 73. Find area of shaded portion. 4. The center of one circle lies on the circumference of the 123 I2 4 INDUSTRIAL ARITHMETIC other. If the radius of each circle is 12, find the area common to both circles. Fig. 74. FIG. 75. 5. Find the area of the two crescents formed as given in Fig. 75. ACB is a right triangle. LESSON LXV AREA OF THE SURFACE OF A PYRAMID AND OF A CONE The line from the vertex of a pyramid perpendicular to its base is called the altitude of the pyramid. The foot of this altitude in such pyramids as we shall study is the middle of the base. Name the altitude of the pyramid (Fig. 76). FIG. 77. The altitude of any one of the triangles that form the faces of the pyramid is called the slant height of the pyramid. What is the slant height of the above pyramid? What is the altitude of the cone? The slant height? The area of the curved surface of a cone is one-half the prod- uct of its slant height by the circumference of its base? How do you find the area of the surface of a pyramid? Find the area of the lateral surface of each of the following pyramids: 1. Base a square 20 in. each side, slant height 18 in. 2. Base a square 18 in. each side, slant height 20 in. "5 126 INDUSTRIAL ARITHMETIC 3. Base a square 4 ft. 5 in. each side, slant height 3 ft. 8 in. 4. Base an equilateral triangle each side a, slant height b. 5. Find the complete area of each of the above pyramids. 6. If each side of the base of a square pyramid is 8 in. and its altitude 6 in., what is its slant height? Its area? 7. Find the area of the curved surface of each of the follow- ing cones: Radius of base 12 in., slant height 16 in.; radiusof base 10 in., slant height 20 in.; radius of base 3 ft., slant height 10 ft.; radius of base 2 ft., slant height 4 ft.; radius of base 5 ft. 4 in., slant height 12 ft. 6 in.; radius of base 4 ft. 5 in., slant height 8 ft. 9 in. 8. The altitude of a cone is 8 in. and the radius of its base 6 in. What is its slant height? Its area? 9. The slant height of a cone is i ft. 8 in., and the radius of its base 6 in. What is the area of its surface? Its complete area? Its altitude? 10. Find the complete area of a cone whose slant height is 24 in., the radius of its base being 8 in. 11. The radius of the base of a cone is 5 in., and its slant height makes an angle of 60 with the radius. Find the com- plete area of the cone. LESSON LXVI VOLUME OF A PYRAMID AND OF A CONE A pyramid is one-third of a prism with the same base and altitude as the prism. A cone is one-third of a cylinder with the same base and alti- tude as the cylinder. How then will you find the volume of a pyramid? Of a cone? 1. Find the volume of each of the following pyramids: Base a square 10 ft. on each side, altitude 15 ft. ; base a square 123^5 ft. each side, altitude 24 ft.; base a square 9 ft. 8 in. each side, altitude 10 ft.; base a square 7 ft. 3 in. each side, altitude 7 ft., base an equilateral triangle each side 10 in. and altitude 20 in. 2. What is the volume of a pyramid whose altitude is 18 ft. 6 in. and whose base is a right triangle, hypothenuse 10 ft. and one acute angle 30? 3. What is the weight of a solid steel pyramid 18 in. long and the base a square 10 in. on each side? 4. Find the volume of each of the following cones: Radius of base 8 in., altitude 12 in.; radius of base 12 in., altitude 15 in.; radius of base 10 in., altitude 3 ft.; radius of base 3 ft. 3 in., altitude i ft. 6 in.; radius of base 3 ft., altitude 3 ft.; radius of base 4 in., altitude 4 ft. 2 in.; radius of base r, altitude h. 6. Find the volume in cubic inches of the following round piece : 127 128 INDUSTRIAL ARITHMETIC FIG. 78. 6. A cylindrical piece of steel i ft. 4 in. long and 4 in. in diameter has a conical hole 4 in. long and 3 in. in diameter bored from one end of it. What is the weight of the piece? 7. Find the volume of a cone 10 in. in diameter if it tapers ^ in. per in. 8. Find the volume of a cone i ft. 6 \n. long if it tapers i in. per in. LESSON LXVII REVIEW 1. A pile of coal of conical shape 10 ft. high lies at an angle of 30 with the horizontal. How many tons in it if i cu. ft. weighs 38 lb.? 2. Find the weight of a conical casting of iron 8 in. in diame- ter and slant height 14 in. 3. The rain which falls on a house 22 ft. X 36 ft. is con- ducted to a cylindrical cistern 8 ft. in diameter. How great a fall of rain would it take to fill the cistern to a depth of 7^ ft. ? 4. How many gallons of water will a 6-in. pipe deliver per hour if the flow is 3 ft. per sec.? 5. A band saw runs on pulleys 48 in. in diameter at a rate of 1 80 R.P.M. If the pulleys are decreased 18 in. in diameter, how many R.P.M. will they have to make to keep the band saw travelling at the original speed? . 6. A shaft has upon it two pulleys, each 8 in. in diameter. The speed of the shaft is 400 R.P.M. What must be the size of the pulleys of two machines if, when belted to these shaft pulleys, one of them has a speed of 300 R.P.M. and the other 900? 7. A %-in. drill, cutting cast iron, may cut at the rate of 40 ft. per min. How many R.P.M. may it make? 8. Find the cost at 40$. per lb. for sheet copper to line bottom and sides of a cubical vessel 7 ft. each edge, if the sheet copper weighs 12 oz. per sq. ft. 9. If the feed is % Q in. and the work has a speed of 164 ft. per min., how long will it take to cut a piece 2 in. in diameter and i ft. long? 9 129 , 130 INDUSTRIAL ARITHMETIC 10. If S = S' and L = %-in. pitch, find the simple gearing that may be used to cut 15^ threads per in. 11. If the threads of problem 10 is a 60 V thread, find its depth. 12. The ratio of the areas of two circles is i : 4 and the radius of the smaller circle is 6. What is the radius of the larger circle? 13. Studding for partitions is 2 in. X 4 in. and 16 ft. long. It is set 1 6 in. between centers. How many pieces must be bought for a partition 8 ft. high and 12 ft. long? What will it cost at $30 per M ? 14. A right cone of altitude 10 ft. has a slant height of 18 ft. Find its complete area and also its volume. . LESSON LXVIII REVIEW 1. A main line shaft runs 176 R.P.M., a pulley on this shaft is 36 in. in diameter and is belted to a pulley on the counter- shaft 12 in. in diameter. Another pulley on this same counter- shaft is 16 in. in diameter and is belted to a pulley 4 in. in diameter on a grinder. What is the speed of the counter- shaft? Of grinder spindle? If the grinding wheel is 10 in. in diameter, what is its surface speed? 2. The diameter of a driving pulley is 9 in. and its speed is 1000 R.P.M. What is speed of driven pulley whose diameter is 4 in. ? If this speed is too fast, what should be the diameter of the diven pulley to have a speed 250 R.P.M. less than the 4-in. pulley? If the speed of the 4-in. pulley is too slow by 250 R.P.M., what size driving pulley should be used instead of the 9-in. pulley? If we keep both pulleys (9 in. and 4 in.) and make our speed changes by changing speed of 9-in. pulley, what would be the speed of 9-in. pulley to give 1125 R.P.M. of 4-in. pulley? 3. How many gallons of water in a railway track tank 1200 ft. long, 19 in. wide and 7 in. deep, if the water is 2 in. below the top of the tank? 4. The diameters of the steps of a step cone pulley are 8 in., 5^ in. and 4 in. respectively. Find the ratio of their surface speeds when the shaft to which the pulley is attached is making 900 R.P.M. 5. A pump has a water cylinder of 6 in. and a stroke of 16 in. How many gallons of water are pumped in i hr. if -the pump makes 60 strokes per min.? 132 INDUSTRIAL ARITHMETIC 6. A smokestack 90 ft. high is to be held in place by five guy wires attached 30 ft. from the top of the stack. The wires are anchored 55 ft. from the bottom of the stack on a level with its bottom. Find the number of feet in the guy wires allowing 35 ft. for fastening. 7. One formula for making concrete is i part cement, 2 parts sand and 4 parts crushed stone. How many cubic feet of each will be required to make a concrete wall 100 ft. long, 2 ft., thick and 4 ft. high. 8. What is the micrometer reading for a piece of iron whose diameter is .422? About how many 64ths of an inch? 9. Each edge of a pyramid of four faces is 8 in. Find its complete area. 10. Find the value of the letter in each of the following: 3X + 8x - 6x + 4X = 25; >^x + >x + %x = 10; yx + 2X + 3x - 8x = 15; MX + >t + Kox = 30; 2X + 3X + 8x 2X = 14; Y& + %x + x =28. LESSON LXIX REVIEW 1. A steel plate 5 ft. long, 3 ft. 6 in. wide and i in. thick, has a hole 10 in. in diameter cut through it. Find weight of plate, allowing .29 Ib. per cu. in. 2. A tubular boiler has 124 tubes each 3% in. in diameter and 1 8 ft. long. What is the total tube surface? 3. A room is heated by steam pipes. There are 240 ft. of 2-in. pipes and 52 ft. of 5-in. pipes and 2 ft. of 4^-in. feed pipe. What is the total heating surface of the room? 4. A hollow steel shaft 10 ft. long is 18 in. in external diame- ter and 8 in. in internal diameter. Find weight of the shaft. 5. If the depth of a sharp V thread angle 60 is % in., what is its pitch? 6. Three circles each of g-in. radius are tangent to each other. Find area between the three circles. 7. Find the volume generated by an equilateral triangle whose side is 8 if it revolves about its altitude as an axis. Find area also. 8. If the back stay of a suspension bridge is 125 ft. long and is anchored 1 20 ft. from the base of the pier, what is the height of the pier? 9. The distance across the flats of a 2-in. Whitworth hexagon nut is 3^2 m - Find the distance across the corners. 10. A pipe has a sectional area of 125 sq. in. at i part and 80 sq. in. at another. If 6000 cu. ft. of water flow past each section per hour, find the velocity of the water in feet per second, at each point. 11. A triangular piece of steel is % in. thick and two of its sides are 28 -in. and 32 in. If these sides form an angle of 30, what is the weight of the piece? 12. Solve each of the following: 6a 2 + 10 = 160; 2b 2 + 3b 2 = 500; 8b 2 100 = 700; ion 2 10 = 90; 8k 2 15 = 225; 6x 2 + 4 =40. 133 LESSON LXX REVIEW 1. Find the weight of 100 steel planer bolts with heads i in. square and % in. thick, body 5 in. long and diameter % in. 2. FIG. 79. Calculate the amount of stock in the angle weld (Fig. 79). 3. FIG. 80. Calculate the length of stock J^ in. X i in. required for the forging shown in the diagram. REVIEW 135 4. FIG. 81. Find the weight of the forging and the length of stock 2 in. X 4 in. required to make it. i cu. in. weighs .2779 Ib. 5. .._ T } xl" J < n" * ' 'i 1 ' 7 ' * 2 ' "* 4 * "* 70 * FIG. 82. Find the weight of the forged crankshaft as per diagram. 6. FIG. 83. Calculate the amount of stock in the eye bend. 136 7. INDUSTRIAL ARITHMETIC -*%- FIG. 84. The link is forged from % 6 -in. round stock. Find the weight of a chain of 50 steel links. LESSON LXXI REVIEW 1. An engine pumping water from a cylindrical tank 10 ft. 6 in. in diameter lowered the surface of the water 2 ft. 4 in. What was the weight of the water pumped out? 2. What is the weight of a piece of steel shafting 12 ft. long and 3 in. in diameter? 3. A brass plate 2 in. thick, in the shape of a semicircle with a radius of 14.5 in., has four holes through it, each % m - i n diameter. What is its weight? i cu. in. of brass weighs .3031 Ib. 4. A rectangular box is 8 ft. 8 in. long, 5 ft. 2 in. wide and 4 ft. 3 in. high. Find the cost at i2). per Ib. of lining the sides and bottom with lead weighing 7 Ib. per sq. ft. 5. Two pulleys each 18 in. in diameter are connected with a belt. How long is the belt if the distance between their centers is 10 ft.? 6. The water cylinder of a pump is 6^4 m - m diameter and the length of the stroke is 15^ in. Find the time such a pump would require to empty a rectangular cistern 30 ft. 6 in. long, 13 ft. 9 in. wide and 12 ft. 4 in. deep, if the pump is making 90 strokes per min.? 7. The angle of elevation of the top of a flag staff is 30 at a point P. If the distance from P to the foot of the staff is 60 ft., how high is the staff? 8. When a house is heated with a hot air furnace the cross- sectional area of the cold air box should equal three-fourths the total cross-sectional area of the hot air pipes supplying the rooms? What should be the area of the cold air box for eight hot air pipes, two being 10 in. in diameter and the others each 8 in. in diameter? 137 LESSON LXXII REVIEW 1. A poker handle is made from ^-in. stock. The ring is i in. in diameter inside measurement. How much stock must be allowed for the making of ring? The length is calculated from center of stock. 2. The wheels of a band saw are 36 in. in diameter. What is the speed in feet per minute if the wheels are making 500 R.P.M.; if 3 ooR.P.M.? 3. If the small diameter of a taper shank is % in. and the taper is .6 in. per ft., what would be the large diameter of the shank if it were 4 in. long? 4. If a forge uses 30 Ib. of coal per day and there are 23 forges in the shop, what is the cost of fuel at $4.50 per ton to run the shop 200 school days? 6. How many feet per minute does a point on the surface of a 2^-in. cylinder travel, if the cylinder is making 1200 R.P.M.? 600? 3000? Find the cost of the following order for lumber used in mak- ing imposing tables. No. of pieces Wood Dimensions Bd. ft. Price Cost 16 Chestnut 3^"X 3^"X 3 7" it- 8 Chestnut %"XioM"X 3 4" / it- 8 Chestnut %"XioM"X22" it- 8 Chestnut %"X 3 ^" X34" it- 8 Chestnut %"X 3 M" X22" 7*. 4 Chestnut %"X2 4 " X36" It. 8 Chestnut %"X6M" X2 4 " it- 8 Poplar %"X6M" Xa 4 " 6i. 4 Poplar %"X6W' X2 4 " 6i. 4 Poplar M"X2 4 " X2 4 " 64. 138 REVIEW 139 Find the total number of board feet and the cost of the fol- lowing mill bill for doors for a case in the stock room. No. of pieces Wood Dimensions Bd. ft. Price Cost H Chestnut i"X2^"X66^" it- 8 Chestnut i"X2K"Xso" it. 8 Chestnut i"X2^"X37" it- 22 Chestnut i"X2K"Xi4K" it- IS Chestnut i"Xs" Xi4K" It- Find the total number of board feet in the following mill bill for indian clubs: No. of pieces Wood Dimensions Bd. ft. Price Cost 2OO 400 Chestnut Chestnut 2" X 4 M"Xi 9 " iK"X 3 %"X8" It- it- LESSON LXXIII ALLOYS An alloy is a mixture obtained by fusing metals with each other. When two or more metals are fused we obtain a new metal that often has properties exhibited by none of the metals in the combination nor by any single metal. For example, gold, which in its pure state is too soft and flexible for use as coins, jewelry, etc., is hardened by mixing with copper in the ratio of 9 parts gold to i part of copper. Brass is an alloy of copper and zinc, harder than copper and easier to work. Unless a small quantity of lead is added to brass it cannot be used in turning operations, since the tool will tear and not cut it. An alloy of 50% bismuth, 30% lead and 20% tin has the property of melting at a much lower temperature than any of the metals in the combination. In industrial work metals that have certain peculiar proper- ties, are needed for special purposes. Type metal must have the property of making sharp distinct lines; likewise pattern metal. This is accomplished by adding antimony to the alloy. Antimony when used as a part of the alloy causes the metal to expand on cooling; hence to fill out the corners of the molds making a pattern with sharp lines. Antimony also renders the alloy harder. The bearings for machinery must be antifrictional, that is, not easily heated by contact with the revolving machinery. They must be hard and strong particularly for heavy work; for high-speed machin- ery softer bearings may be used. A fairly good bearing for high-speed machinery contains 140 ALLOYS . 141 80% lead and 20% tin, but is not very hard. A metal widely used for heavy machinery bearings contains 80.5% lead, 11.5% tin, 7.5% antimony and .5% copper. If a metal is required that melts at a low temperature bismuth is used as part of the alloy. The plugs of automatic sprinklers and doors used for fire protection are made of such an alloy. Alloys containing sodium have the peculiar property of producing by oxidation, material which will saponify with the oil used in the bearing and thus assist lubrication. Thus by combining copper, lead, zinc, etc., in different ratios we often produce a new metal that answers the peculiar needs of various industrial problems. Exercises 1. In the foundry the formula for "yellow brass castings" contains 66%% copper and 33^% zinc. How many pounds of each must be used to make 1500 Ib. of yellow brass? 2. How many pounds of copper are used with 200 Ib. of zinc to make yellow brass? 3. Recently in your foundry 44 Ib. of copper, 3 Ib. zinc, 2 Ib. tin and i Ib. of lead were used to make a bronze casting. De- termine the per cent, of each element used. 4. How many pounds of each element must be used to make 1800 Ib. of bronze according to the formula in problem 3? 5. How many pounds of bronze will 100 Ib. of tin make? 6. In making pattern metal we use 50% tin, 45% zinc and 5% antimony. How much pattern metal will 90 Ib. of zinc make? 28 Ib. of antimony? 500 Ib. of tin? 7. A pattern that weighs 10 Ib. will contain how much of each of the elements of pattern metal? '8. Fire sand is about 98% silica. What is the weight of the silica in 5000 Ib. of fire sand? 9. A pile of fire sand contains i ton of silica. How many tons of sand in the pile? 142 INDUSTRIAL ARITHMETIC 10. A certain grade of molder's sand is 76% silica and 4% aluminum. How much of the sand will contain 758 Ib. of silica? How much aluminum in the sand? 11. Another grade of molder's sand is 86% silica and 8% aluminum. How much silica in a pile of the sand that contains 872 Ib. of aluminum? 12. One of the bronze castings made in your foundry weighed 28.5 Ib. How many pounds of tin in it? Of lead? 13. A certain alloy contains 50 Ib. of copper and 25 Ib. of zinc. Write the formula for this alloy in per cents. LESSON LXX1V ALLOYS 1. How many pounds of carbon in a ton of cast iron that contains 2.75% carbon? 2. If tool steel is 1.25% carbon how many pounds of tool steel will contain 25 Ib. of carbon? 3. Naval brass is 62% copper, i% tin and 37% zinc. Find the amount of each in 1200 Ib. of naval brass. U. S. Navy Department. 4. Hard bronze for piston rings is 22% tin and 78% cop- per. How many pounds of hard bronze will contain 2340 Ib. of carbon? U. S. Navy Department. 6. An alloy of 88% tin, 8% antimony, 3.5% copper and .5% bismuth is used for the bearings of high-speed dynamos. Cal- culate the amount of each in 2450 Ib. of such alloy. 6. The bearings for railway trucks contain 42% tin, 56% zinc and 2% copper. How much tin and copper must be used to make such an alloy that contains 672 Ib. of zinc? 7. Babbitt metal for high pressure bearings is 90% tin, 7% antimony and 3% copper. Find the amount of antimony and copper in such a composition that contains 1800 Ib. of tin. 8. One ton of babbitt metal adapted for low pressure and medium speed contains 160 Ib. tin, 400 Ib. antimony and 1440 Ib. lead. Find the per cent, of each for this composition. 9. One thousand pounds of plastic metal contain 800 Ib. tin, 100 Ib. lead, 10 Ib. antimony, 80 Ib. copper, and 10 Ib. bismuth. Find the per cent, of each in this alloy. 10. Find the amount of each element in 2500 Ib. of the alloys in problems 8 and 9. 11. The U. S. Navy Department uses brazing metal that 143 I 144 . INDUSTRIAL ARITHMETIC is 85% copper and 15% zinc. How many pounds of each will be required to make 3200 Ib. of this alloy? 12. At $13 per 100 Ib. for copper and $5 for zinc, find the cost of the alloy in problem n. 13. Use the following to find the cost of as many different alloys given above as possible: Cost per 100 Ib. lead $4; zinc $5; antimony $9; copper $13; and tin $30. LESSON LXXV THE PRINT SHOP The units of length in the print shop are the inch, the pica, the nonpareil and the point. The dimensions of cards, sheets of paper, etc., are expressed in inches. The pica, which is ^ in. is used to measure the lengths of printed matter, e.g., the dimensions of a piece of printed matter is usually expressed as 13 picas wide and 20 picas long, or 20 picas wide by 30 picas long, etc. Sometimes, however, the width is expressed in picas and the length in inches, e.g., a newspaper column 13 picas wide and 20 inches long. The nonpareil is one-half of a pica, and the point is one- twelfth of a pica. flonp. FIG. 85. The body of metal type is measured by the point, as i6-point type, 36-point type, etc. A 24-point type means the height of the type is 24 points or 2 ^f 2 m - Metal type is made in the following number of points: Common sizes: 6, 8, 10, 12, 14, 18, 24, 30, 36, 48, 60, 72, 84, 96, 1 20. Odd sizes, chiefly book and newspaper sizes: 3^3, 4, 4^, 5, 5>, 7, 9, ii. Sizes rarely used: 16, 20, 22. e-poiNT 10-POINT 12-POINT FIG. 86. 10 145 146 INDUSTRIAL ARITHMETIC The amount of type in any composition is measured by a square whose side is any number of points. This unit is called the EM. The number of ems in any body of printed matter corresponds to the area of a rectangle. When measuring printed matter set in 8-point type the side of the em is 8 points, set in lo-point type the side of the em is 10 points, etc. Exercises 1. How many picas in 3 in.? 4)-^ in.? % in.? 6 in.? 24 in.? How many nonpareils? How many points? 2. In the following number of points find the number of picas, inches, and nonpareils: 360; 24; 320; 144; 100; 168. 3. Find the number of square points in an 8-point em; a lo-point em, a 6-point em; an a-point em; a y-point em. 4. What is the value of each of the following ratios: 8-point em: 6-point em; lo-point em: 1 6-point em; 12 -point em: 3 6-point em? 5. How many 8-point ems in each of the following ems: 640 lo-point ems? 3200 i2-point ems? 1280 1 6-point ems? 6. At the same rate per M which will cost the most, a page set in 8-point type or lo-point type or i6-point type? Why? 7. An edition of a certain newspaper had seven columns 13 picas wide and 21 in. long on each page. How many i2-point ems per page? How many lo-point ems? 8. The body of the printed matter on the page of a certain book is 20 picas wide and 33 picas long. How many 4-point ems per page? Find the cost of setting the page at 50^. per M 8-point ems. 9. A double newspaper column is 26^ picas wide and 24 in. long. Find the cost of setting 2)^ such columns at 54^. per M 8-point ems. 10. How many ems, and what size on the page of this book? Answer the same questions about your text-book in English. PRINT SHOP 147 11. If 130 ems of composition contain 50 words, how many ems in an article of 2000 words? 2500 words? 1600 words? 12. How many lines set solid of i8-point type in i in.? In 4 in.? In 12 in.? In 5^ in.? In 10% in.? In a in.? In b in.? In an ordinary newspaper column? 13. How many lines per inch set solid of the sizes of type in common use? 14. What size type is used to set solid 9 lines per in.? 7 lines per in.? 12 lines per in.? 2 lines per in.? 15. Find the width of a printed sheet of seven columns 12^ picas wide with a margin of % in. on each side if the columns are spaced with 6-point rule. 16. Measure several of your text-books to determine the size of the type used. LESSON LXXVI PRINT SHOP 1. A sheet of cardboard 22^ in. X 28^2 in. is to be cut into tickets 2^ in. X 4 in. Find the greatest number of tickets that can be cut from the piece and tell how you will cut it. 2. How many sheets of cardboard 22^ in. X 28^ in. will be required to cut 1500 tickets 2^ in. X 4 in.? If the stock costs $2.50 per 100 sheets, how much will the tickets cost? 3. Your schedule cards are 4 in. X 5% in. and were cut from cardboard 22^ in. X 28^ in. How many sheets were required to make 5000 of these cards? Find their cost at $2.40 per 100 sheets. 4. How many cards 2% in. X 4% in. can be cut from 50 sheets of the cardboard used in problem 3 ? 6. A card 2 in. X 7 in. was cut from cardboard 22 in. X 28 in. If the cardboard costs $2.40 per 100 sheets, find the cost of the cards per 1000. 6. Answer each of the above questions allowing 10% for press waste for each. 7. Letter paper sheets 16 in. X 21 in. are cut into letter heads 8 in. X 10^ in. How many sheets will be required for 2M such letter heads allowing 10% for press waste? What will the paper cost at $2.25 per ream of 500 sheets? 8. Which cuts to the better advantage sheets 17 in. X 22 in. or sheets 16 in. X 21 in., if letter heads 8 in. X 10^ in. are wanted? If letter heads 8^ in. X n/4 in- are wanted? 9. Use the size sheets that will be more economical for cutting half-size letter heads 8% in. X $% in. and find the number of sheets you must use to print $M such letter heads allowing 10% for press waste. 148 PRINT SHOP 149 10. An order for 2oM slips 3^ in. X 2 in. was sent to your print shop. How shall these be set up in order that they may be printed with 2M impressions of the press? What will be the dimensions of the sheet on which they are printed? How many sheets will be required allowing 10% for press waste? 11. How many sheets of cardboard 22 in. X 28 in. will be required for backs used in padding the slips of problem 10, if 50 slips are put in each pad? 12. The cardboard back of a certain calendar is 8% in. X 6% in. If they were cut from sheets 22 in. X 28 in., how many sheets were required for 10,000 calendars allowing 10% for waste in printing? 13. How many cards n in. X 14 in. can be cut from 1000 sheets 22 in. X 28 in. If the sheets cost $3 per 100, find the cost of the cards per 1000, adding 10% for waste in printing. LESSON LXXVII PRINT SHOP 1. Find the number of lines of i6-point type spaced with 2- point leads that can be set in 10 in.; in 15 in.; in 12 in.; in 53^ in.; in 24 in. 2. The page of a certain book contains 33 lines. If the type is set solid and the composition is 33 picas long what is the size of the type used? 3. It is required to set solid 72 lines in a i2-in. space. What size type must be used? What size type must be used if the lines are spaced with 2-point leads? 4. Find the cost of setting the page in problem 2 at 55^. per M ems. 5. If 65 ems of composition contain 25 words, how many ems in an article of 4000 words? 6. If 10,000 cards 4 in. X 6 in. are to be printed with 2500 impressions of the press, assuming no waste, how shall they be set up? What size sheets must be used? How long will it take to print them if the press averages 25 impressions per min.? 7. How long will it take to print the cards 19)^ in. X 14 in. that can be cut from 2000 sheets 22 in. X 28 in. if the press averages 1200 impressions per hr.? 8. The page of a certain book averages 34 words for each 84 ems. How many words in three pages of this book if the page is 21 picas X 33 picas? 9. Advertising announcements 83^ in. X 1 2 in. are cut from sheets 22'^ in. X 28^ in. Find the number of sheets required to print 150 of the "ads" if 10 are allowed for press waste. 10. Find the time required to print the letter heads 8 in. X 150 PRINT SHOP 151 10^2 in- that can be cut from 800 sheets 16 in. X 21 in. when the press averages 18 impressions per min. 11. A certain job can be completed in i hr. 45 min. if the press averages 1200 impressions per hr. What must be the average number of impressions per hour to complete the job in 2 hr.? In i hr. 30 min.? 12. A piece of printed matter set solid 16 picas X 32 picas contains 1152 ems. Find the number of points for each em. 13. A certain page 20 picas X 30 picas contains 864 ems. What size type is used for the page if the work is set solid? LESSON LXXVIII PRINT SHOP 1. Common type is 60% lead, 30% antimony and 10% tin. How much of each in 400 Ib. of type? 620 lb.? 157 lb.? 235 lb.? 2. The best type is 50% lead, 25% tin and 25% antimony. How many pounds of lead and of antimony must be melted with 75 lb. of tin to make this grade of type? 3. In a 2oo-lb. font of best type, how many pounds of each of the metals? 4. Type metal is 77.5% lead, 6.5% tin and 16% antimony. How many pounds of type metal can be made from 64 lb. of antimony? How many pounds of lead and of tin must be used? 6. Type is also made by melting 5 lb. of tin, 9 lb. of anti- mony, 35 lb. of lead and i lb. of copper. Each metal is what per cent, of the alloy? 6. A 2-in. pulley is belted to a 2o-in. pulley on a* counter- shaft. The counter contains a i5-tooth gear that meshes with a Qo-tooth gear. A revolution of the go-tooth gear makes one impression of the press. If the driving pulley is making 500 R.P.M., how many impressions does the press make? 1000 R.P.M.? 800 R.P.M.? 1250 R.P.M? 850 R.P.M.? 7. In order that the number of impressions of the press for any given time may be the same, what change would have to be made in the driven pulley if a 3-in. pulley is substituted for the 2-in. pulley? 8. A label for a can 6 in. high and 6 in. in diameter is required. The label is to extend three-fourths of the distance 152 PRINT SHOP 153 around the can and within i in. of the top and % in. of the bottom. How many sheets 25 in. X 38 in. will be required for 6000 of these labels allowing 8% for waste? 9. The page of a certain magazine contains three columns each 1 8 picas wide; between each column is i pica. The margin on the right-hand side of the page is 5^ picas and there is an equal margin at the left. Find the width of the page in inches. 10. The printed matter of the page of problem 9 is set solid and has 9 lines per in. What size type is used? How many 8-point ems are there per inch? LESSON LXXIX FRACTIONAL REVIEW 5% X iH? - 1 simplify p y 2. Simplify ^*-ia7ji ~ JTT ~ ?M 3. isH X 12^3 X 5 M X 6M,= ? % X 7K X 3% = ? X 7% X 8> X 25 = ? 18% X 9^7 X 3^ = ? 4. Simplify 1X 7 . , X H of - i>^ of % ^ ioH 13 5 Simplify M + KsXH-KXfr mpMy MXWi+XXM 6. What is the value of 3 ff 7 f** 4/13 Of 2M.6 7 . Simplify '-Hrf 9. 4^8 + 2 x $% - 3 x % + y 2 = ? 10. % + K 2 + KG + Ko - % - Mo 11. Find the sum of ^ + % + %Q + Mo correct to four decimal places. 12. 12^ X 4 M X 5 K 2 = ? 1748 + (6M X 7^2) = ? X i 5 K2 X % = ? ^2 X K2 X 18% = ? x ISM x $y 2 = ? 1890 -=- ( 7 y 3 x ioH) = ? X 9% X H = ? ^2 X sM X % = ? 154 UC SOUTHERN REG10IWL LIBRARY FACILITY A 000933218 ANGELES STATE NORMAL SCHOOL