MC-NRLF MENTAL JPfcl I O iff £Lr JL *' %** «# |U #? • BOOK « COM B\ NY YORK- QNCINNATI-CHCCAGO I IJMtfljfiftit! COMPLIMENTS AMERICAN BOOK CO, A. P. aUNN, Gen'l Ag't PIN E& BATTERY SAN FRANCISCO. ^ MENTAL ARITHMETIC BY i. c. mcneill PRESIDENT SEVENTH WISCONSIN STATE NORMAL SCHOOL >*^c NEW YORK-:. CINCINNATI-:. CHICAGO AMERICAN BOOK COMPANY QlMc M \ ^ COPTEIGHT, 1902, BY i. c. McNeill. Entered at Stationers' Hall, London. MENTAL ARITH. EDUCATION £>&% PREFACE This work is designed for use in grammar grades. It may also be used with advantage for review work in high schools and normal schools. Mental or oral arithmetic when properly developed and taught not only does more than any other agency in giving pupils insight into the real nature of number and numerical relations, but it also adds keenness to the mind in general and strengthens its power of concentration. To be thus efficacious, however, any course in mental arithmetic must be based upon sound peda- gogic principles and proceed along sound pedagogic lines. The principles upon which this book is based may be briefly formulated. Observation must precede compari- son ; sense impressions must come before thought rela- tions can be established ; rational movement proceeds from the known to the related unknown, from the simple to the complex, and from the particular notion to the concept. The method of this book recognizes the impor- tant truth that first perceptions must be clear, distinct, and vivid in order to leave in the mind correct traces of ideas through which the later impressions may be apper- ceived. The value of clear images in bringing before the mind the material for reasoning and in leading gradually to the formation of accurate concepts is duly emphasized. The problems have been specially prepared to illustrate and call forth ideas. The first problem looks to the last through all that come between. The development of 541529 • 4 • : ,'<-"': ' . «" ' '../-''*"• 4 PREFACE each section furnishes a foundation upon which the new thoughts that immediately follow may most readily erect themselves. A constantly increasing demand upon the pupil's mental activity goes hand in hand with the pro- gressive gain in power. But there has been rigid exclu- sion of problems of so unnatural a complexity as to impede instead of furthering the free movement of the mind. The author acknowledges his indebtedness to Professor C. W. Smith of the Superior State Normal School for much valuable aid in the preparation of the problems in this book, and for testing in preparatory classes every step taken in the development of the subjects. SUGGESTIONS TO TEACHERS (a) How to prepare the lesson is the most important question for the pupils. They cannot work intelligently unless they understand the aim of each day's study. The great function of the teacher is to. guide activity. Learning is the pupils' act. Before an intelligent assignment of the lesson can be made by the teacher, he must con- sider carefully just what steps are new and how far the pupils are prepared by what they already know for the advanced work. In assigning the lesson, the teacher should impress upon his pupils the particular end in view, and should make sure that they understand just what they are to accomplish and by what plans and devices they may best succeed. The next day's recitation will test the pupils' understanding of such directions and their faithfulness in following them. In determining the pupils' preparation to begin a new line of work, the skilled teacher will approach the learners on their highest plane of old work, directly related to the new material to be considered. If students move from this position with ease and freedom, it is good evidence that the lower phases of the subject are well organized in their minds. If they do not show a mastery here, it is well to descend to the next lower phase or to a place where they are able to stand firm. The higher phases of the old work should be brought clearly within the field of conscious comprehension before attempting to present new ideas. (6) As a rule it is a good plan to leave the diagrams on the board for constant use until pupils can image them. When the diagrams can be held in the mind, it is not productive of attention or study to have them before the pupils. In their study it is of great value to have pupils picture or represent in diagrams, if possible, the condi- tions of new problems. Mensuration is not comprehended until pupils through practice and tests perceive the relations. (c) Good results may be reached in class by asking the pupils with 5 6 SUGGESTIONS TO TEACHERS books closed to work the problem given and rise at a signal from the teacher when they have finished. This reveals at a glance the back- ward students who need the most assistance. Analyses should be given in simple, direct sentences. It is often a good plan to have one pupil take up the analysis when another has partly given it and complete the explanation. This will stimulate attention, especially on the part of the unprepared pupils. When by questioning or by hearing the explanation, the backward pupils have learned how to dispose of the problem in hand, they should be called upon to show that they can take all the steps leading to the correct solution. A good test is to change one condition somewhat and then ask for a solution. Ques- tioning, in most cases, is better than telling. In questioning a pupil to bring him to an understanding of a problem or a principle, the teacher must always go back to what is known, as a starting point. It is sometimes a good plan to distribute small slips of paper, read a problem, have each pupil write the answer at a signal, after sufficient time has elapsed for its solution ; then give another in the same way until opportunity to test all with several problems has been given. Analyses, explanations, and modes of solution given by the pupils should follow. (d) Time will be saved and accuracy of results insured by encour- aging pupils to use the largest measures possible in finding ratios or making comparisons between like numbers. Take, for instance, What part of an 18|-ft. square is a 12i-ft. square? If the pupil finds that the 12|-ft. square is twice as long as a 6^-ft. square, and the 18|-ft. square is three times as long as a 6^-ft. square, he readily sees that the ratio of the length of the 12|-ft. square to the length of the 18§-ft. square is as 2 to 3, or f. In like manner he can compare widths. After this it is an easy step to bring the compound elements together, and find that the 12|-ft. square is $ as large as the 18f-ft. square. (e) The scientific teacher will find many other plans of developing ideas of subjects presented in the text. He should always feel at liberty to use his own methods and devices if they are consistent and will not confuse past or future lessons. It is, however, better to know one plan well than to have a superficial knowledge of many plans. CONTENTS SECTION' PAGE I. Introducing the Idea of Ratio 9 II. Introducing the Fraction One Half and developing Relations 12 III. Introducing the Fraction One Third and developing Relations • . . . .18 IV. Introducing the Fraction One Fifth and developing Relations . .23 V. Introducing the Fraction One Seventh and developing Relations 29 VI. Developing the Idea of Decimals and their Relation to Other Fractions 35 VII. Introducing the Idea of Per Cent and developing Rela- tions .48 VIII. Introducing General Compound Numbers and develop- ing Relations . . 56 IX. Introducing the Study of Surfaces and developing Rela- tions 65 X. Introducing the Study of Solids and developing Relations 79 XI. Review of Measurements 92 XII. Introducing the Ideas of Analysis and Mental Algebra in solving Problems and finding Relations . . . 101 XIII. Reviewing and extending the Idea of Percentage . . 122 XIV. Reviewing and extending Ideas previously presented and establishing New Relations i 134 7 SECTION I INTRODUCING THE IDEA OF RATIO 1. Rectangle a is what part of rectangle b ? 2. Rectangle b is how many times rectangle a ? 3. What is the ratio of rectangle a to rectangle b ? ^. 4. What is the ratio of rectangle b to rectangle a ? 2. 5. What is the ratio of a line 1 in. long to a line 2 in. long? 6. What is the ratio of a line 2 in. long to aline 1 in. long ? 7. What is the ratio of a line 4 in. long to a line 2 in. long ? 8. What is the ratio of a line 2 in. long to a line 4 in. long? 9. What is the ratio of 1 qt. to 2 qt. ? Of 2 qt. to 1 qt. ? 10. What is the ratio of 4 qt. to 2 qt. ? Of 2 qt. to 4 qt.? 9 10 THE IDEA OF RATIO 11. What is the ratio of 4 bu. to 2 bu. ? Of 2 bu. to 1 bu. ? 12. What is the ratio of 6 in. to 3 in. ? Of 6 in. to 2 in. ? Of 2 in. to 6 in. ? 13. What is the ratio of 3 gal. to 9 gal. ? Of 9 gal. to 3 gal. ? 14. James has 9 marbles and Henry has 6 marbles. Henry's marbles equal what part of James's marbles? What is the ratio of Henry's marbles to James's marbles ? Of James's to Henry's ? 15. A has $ 12 and B has $ 9. What is the ratio of B's money to A's ? Of A's to B's ? 16. What is the ratio of 1 ft. to 3 ft. ? Of 1 ft. to 1 yd. ? 17. * What is the ratio of 1 yd. to 1 ft. ? Of 1 yd. to 2 ft. ? 18. What is the ratio of 1 yd. to 6 in. ? Of 1^ yd. to 6 in. ? 19. What is the ratio of 6 in. to 2 yd. ? Of 2 yd. to 6 in. ? 20. A boy measured a stick with a 6-inch rule and found it to be 2 yd. long. How many times did he have to use the rule to measure the stick ? The ratio of 2 yd. to 6 in. is what ? 21. What is the ratio of 1 qt. to 1 gal ? Of 1 qt. to 5 gal. ? 22. A. man is dipping water from a barrel into a 5-gallon can, using a 2-quart dipper. How many times must he dip the dipper full to fill the can ? The ratio of 2 qt. to 5 gal. is — ? * The student must observe that ratios can be expressed only between quantities of like name. What must be done to quantities of unlike names ? THE IDEA OF RATIO 11 23. What part of a canf ill is each dipperful ? 24. At 20 $ a yard, how many yards of cloth can be bought for % 1.20 ? What is the ratio of $1.20 to 20^? 25. If 22 yd. of cloth cost % 15.50, what part of $ 15.50 will 2 yd. cost? 4 yd. ? 5 yd. ? 9 yd. ? 11 yd. ? 26. How many steps 2 ft. 6 in. long must a man take to walk a distance of 10 ft. ? What is the ratio of 120 in. to 30 in. ? Of 10 ft. to 1\ ft. ? 27. What is the ratio of 1 ft. 6 in. to 1 yd. 1 ft. ? 28. From a stick 1 yd. 2 ft. long, a piece 1 ft. 8 in. long was cut off. What part of the stick was cut off ? 29. From a can containing 3 gal. 3 qt. of milk, 1 gal. 1 qt. was sold. What part was sold ? 30. A boy carrying a package of sugar, weighing 5 lb. 8 oz., spilled 1 lb. 6 oz. of it. What part of the sugar did he spill ? What is the ratio of 1 lb. 6 oz. to 5 lb. 8 oz. ? 31. What is the ratio of 5 ft. to 5 in. ? Of 6 ft. to 6 in. ? Of 12 ft. to 12 in. ? 32. What is the ratio of 8 in. to 1 yd. 1 ft. ? Of 1 yd. 1 ft. to 8 in. ? 33. What part of a bushel is 1 pk. 4 qt. ? 34. What part of 1 bu. 2 pk. 2 qt. of cherries is sold, if 1 pk. 2 qt. is sold ? 35. -How many strips of wall paper 18 in. wide will it take to cover a wall 4 yd. long ? SECTION II INTRODUCING THE FRACTION ONE HALF AND DEVELOPING RELATIONS . SUGGESTIONS a. Pupils should think problems through before attempting to give expression to ratios. b. Analysts should, in the main, be the oral expression of the observations the pupils make in thinking through their problems. Analyses should, if possible, be given in short, simple sentences in which the notion of what each step is should come in its natural order. c. Sure progress results if pupils think relations in diagrams. When the diagram of a class of problems can be carried in the mind, the pupils should image it. d. In presenting Section II, it is advisable to let the diagram of each new step remain on the blackboard during the first recitation. In subsequent lessons it should be reproduced if pupils cannot recall and image it. . c a b 1. What part of the square is a ? What part is b + c ? 2. a and b + c together equal the whole square. What is the value of \ + \ ? 12 THE FRACTION ONE HALF 13 3. What part of b + c is b ? What part is c ? . 4. What is the ratio of b to 6 + c ? Of 6 to a ? Of c to a? 5. What part of the whole square is b ? What part is * ¥to|- 32. Using the fractions ^ A' i' 2T> 2T' t> 2V' 2Y' h add each one to each one following, as ^ + fa 2V + 9' 2Y + 2T> etc - 33. Using the same fractions, subtract each one from each one that follows, as 27 — 2V 9 — fa e ^ c# 34. Add fa |,&, and 3 ' 35. What part of 9 yd. is 2 ft. ? THE FRACTION ONE THIRD 21 36. If ^ of a rectangle is divided into 2 equal parts, how many of those parts are there in the rectangle ? What is £ of f ? i of i = ? i of 1 = ? 37. If ^ of a rectangle is divided into 2 equal parts, how many of those parts are there in the rectangle ? iof| = ? iofi=? lofi=? $of*=? &ofl = ? 38. If Jg of a rectangle is divided into 2 equal parts, what fraction of the whole rectangle is each of them ? |of| = ? ^ofl = ? 39. If fa ol a rectangle is divided into 2 equal parts, what fraction of the rectangle is each of them ? 1 of ^ = ? 40. State the ways in which 1 of anything may be found. Ans. | of |, | of ±, | of 1. 41. State the ways in which fa of anything may be found. fa of anything, fa of anything. 42. If 1 of a rectangle is divided into 2 equal parts, what fraction of the whole is each of them ? \ of -^ = ? lof| = ? lofi = ? lofi = ? T Vofl = ? 43. ^ofJ Y =? _ ? 1_5 ? 12. What is the ratio of fa to | ? To ^ ? 13. What is the ratio of f to fa ? f to ^ ? f to fa ? 14. What is the ratio of f to fa ? f to -^ ? Of any number of 5ths to the same number of lOths ? 15. What part of the rectangle is one of the 6 equal parts of a ? One of the 8 equal parts of a ? One of the 9 equal parts? 10? 12? 15? 16? 18? 20? 16. Give the ratio of each of the following fractions to each of the others: \, fa, fa fa fa, fa fa ^ 17. When the numerators of two fractions are alike, what simple way do you see of telling the ratio of one to the other ? The ratio of f to f is f ; of f to fa is ^ or 4. 18. Give the ratio of : t to A |to^ tw to 2 % 4* to if *to^ A- to -& A to ifo A to A fto «& tV to gV ■is to ^y TOT to T V ¥*°H «to& A to & * to £ I to ^ A to a <& to ^jr &t°A 19. Give the ratio of each of the following fractions to each of the others: fa fa fa fa fa fa fa £|. THE FRACTION ONE FIFTH 25 20. What is the ratio of f to f ? Of T \ to T % ? Of f to |? 21. When the denominators are the same, what simple way do you see of rinding the ratio of one fraction to another ? The ratio of f to f is -| ; of -fa to ^ is $. 22. What must be done before finding the ratio of fractions whose denominators are unlike ? 23. What is the ratio of | to £? Of | to f ? 24. What is the ratio of ^ to -f? Of f to f ? 25. If f of a yard of cloth costs $1, how much will a yard cost ? Suggestion. The ratio of 1 yd. to f yd. is f. Therefore 1 yd. will cost f of what £ of a yard costs, or £ of $ 1, or $ If. 26. If | of a yard of cloth costs $ 1, what will be the cost of |- of a yard ? 27. If |- of a quantity of sugar costs $-$-, how much will | of the quantity cost ? 28. If 5 lb. of sugar cost $ J, find the cost of 7 lb. 29. A can mow \ of an acre in half a day, and B \ an acre in the same time. How many acres could both together mow in half a day ? In a day ? In 2 days ? 30. A can do f of a certain piece of work in one day, B | of it in the same time. How many 12ths of the work can both do in a day ? In Jy of a day ? If they can do -^2- of the work in Jy of a day, how many 17ths of a day will it take them to do ^|, or the whole work ? 31. A can break ^ of an acre of new land in a day, B \ of an acre. How long would it take both of them to break one acre ? 26 THE FRACTION ONE FIFTH 32. James has -§ as m'any marbles as John. If John has 30 marbles, how many has James ? 33. v James has |- as many marbles as John. If James has 18, how many has John ? 34. Four boys found a sum of money, and divided it in such a way that the first got f of it, the second fa, the third ^, and the fourth the remainder. What part of the money did the fourth get ? 35. A owned ^ of a mill, and B ^1. Which owned the greater amount, and how much ? 36. A man set out on a four days' journey. The first day he traveled Jl of the distance, the second day | of it, the third day -fa of it. What part of the distance did he have to travel the fourth day ? 37. A owned -fa of a mill and sold f of what he owned. What part of the mill did he sell, and what part did he still own ? 38. A owned fa of a mill and sold ^ of the mill to B. What part of the mill did A still own ? What part of his share did he sell ? What is the ratio of A's share, after selling, to B's ? 39. A has 60 A. of land and B 50 A. A gives half of his land to B, receiving in return half of B's. How many acres has each after the exchange ? 40. Five men together own a factory. The first owns | of it, the second -fa of it, the third jfc of it, and the fourth -fa of it. What part does the fifth own ? 41. If the share of the fifth was $ 2000, what was the value of fa of the factory? Of£f? Of fa? Of if? Of fa? What was the share of each of the first four men ? THE FRACTION ONE FIFTH 27 42. If the yearly profits of the factory were $ 9600, what should the first man receive as his share of the profits ? | of $9600= ? What should each of the others receive? 43. A, B, and C engaged in business. A furnished 1 of the capital, B \ of it, and C the remainder. What part of the capital did C furnish ? 44. If the whole capital was $ 2400, how much money did A invest? B? C? 45. If their profits were $600, how much should A receive? B ? C ? 46. A, B, and C rent a pasture. A puts in 25 sheep, B 30 sheep, and C 17 sheep. What part of the rent should each pay ? If the whole rent is $ 36, how much should each pay ? What is the cost for 1 sheep if the cost for 72 sheep is $ 36 ? 47. If gV °f a quantity of sugar cost $ |, how much will {^ of the same quantity cost ? What is the ratio of 48. I of a yard of cloth cost $ f . How much will -^ of a yard cost ? 49. At $f a yard, how many yards of cloth can be bought for $ I ? What is the ratio of $ £ to $ f ? 50. At $ 21 each, how many hats can be bought for $ 7 J ? What is the ratio of $ - 1 /- to $ f ? Of 1 7J to $ 21 ? 51. At $3J each how many hats can be bought for $131? What is the ratio of $4f to $-!?•? Of $131 to $31? 52. How many yards of ribbon at $£-% each can be bought for $-^? What is the ratio of $-fo to $£%? 28 THE FRACTION ONE FIFTH 53. If you know the cost of f of a quantity, how can you find the cost of |-| of the quantity? What is the ratio of JJ to f ? 54. Given the cost of ■£$ of a quantity, how can you find the cost of ^ of the quantity ? Ans. By taking |^ of the cost of ^ of it. 55. Give the ratio of : \ to &. *M- | to A- itof &tof ftof. &t°tt- ftof. if to if- ftof. 56. Given the cost of \ of a quantity, how can you find the cost of T 5 2 of it ? Given cost of : |, to find cost of 2 7 o* ^51 to find cost of \\. ^, to find cost of \. f , to find cost of f . ^, to find cost of \. J$, to find cost of ||. J, to find cost of |. f , to find cost of J. 57. If 3 yd. 1 ft. of rubber hose cost $ .90, how much will 1 yd. cost ? 58. A merchant sold 1 bu. 1 pk. of timothy seed for $1.20. At that rate how much should he charge for 1 pk. 2 qt. ? 59. 8 is the ratio of what to 1 gal. 1 qt. ? 8 is the ratio of 10 gal. to what ? 60. The cost of one chair is J of the cost of another chair. If the first chair cost $10, how much did the second cost ? If the second cost $ 12, how much did the first cost ? What is the ratio of 10 to 8 ? Of 15 to 12 ? 61. What is the ratio of If to 1J? To j ? To |? Tof? To 2? To5f? SECTION V INTRODUCING THE FRACTION ONE SEVENTH AND DEVELOPING RELATIONS a i i ! c ! 1. What part of the rectangle is a? b? c? 2. What is the ratio of a to the whole rectangle ? Of b ? Qf el 3. What is the ratio of \ to | ? ftSOf? fto|? f? t? f? £? 4. If a is divided into 2 equal parts, what part of the rectangle is each part ? What part are both parts ? 3 such parts? 4? 5? 6? 8? 10? 13? 14? 5. T \ = how many 7ths ? T \? T \? T \? £f? if? jf? 6. Using the fractions fo f ^, f, T \, f, J, f T 9 4 , f, T?' f » il' I"* gi ye ^ ne ra ^io of each to each of the others. 7. If ^ is divided into 3 equal parts, how many such parts are there in £ ? What is the name of each part ? Of 2 such parts ? 3? 5? 19? 21? 8. ^ T = how many 7ths? &? & ? if? ft? |f? 2 1 ? 21 ' 30 THE FRACTION ONE SEVENTH ». iof*=? |off=? Joff=? I off-? *of* = ? *of$ = ? Joff = ? 10. If ^ is divided into 4 equal parts, how many such parts are there in £ ? In 1 ? If -^ is divided into 2 equal parts, how many such parts are there in ^| ? In 1 ? ii. }of} = ? iof T i 3 = ? fof| = ? Jof T % = ? What is the ratio of J of any number of 7ths to \ of the same number of 14ths ? 12. What must be done to \ to produce -^? How many times ^ = |? f? f? f? £? f? J? 13. What must be done to | to produce -£%? What must be done to ^ to produce ^ ? ^ of ^ = ? J of ^ = ? 14. | of ^ = ? Is ^g larger or smaller than jfc ? Than ^3 ? How many 7ths = f f ? 15 . i of i = ? i f^ = ? Jof^ = ? ioff-T lof^ T = ? T Vof|=? iof T V = ? |bf.*-? 16. How many 7ths in §£ ? In |f ? In ff ? 17. How many llths are there in anything ? What is the ratio of 2 to 22 ? \ of ^ = ? \ of ^ = ? \1 \t 1 ? l ? i ? l ? _i_ of l = ? - Ar of I = ? - 1 - of I = ? 1? 1? 1? 1? 1? z - 6 • y • 8 • 9 • 18. If a rectangle is divided into 13 equal parts, what name is given to one of the parts ? To 2 of them ? To 3 of them? 4? 7? 12? 13? 19 1 of - 1 - = ? 1 of -X- = ? -1 ? 4 ? l ? i ? 20. What name is given to one of the 17 equal parts of a rectangle ? To one of the 19 equal parts ? To one of TBE FRACTION ONE SEVENTH 31 the 23 equal parts ? Of the 29 equal parts ? 31? 37? 41? 43? 47? 53? 59? 61? 67? 71? 73? 79? 83? 89? 97? 21. We may think of ^ as being £ of -|-, or | of J, or J of ^, or \ of J. Why is it not possible to think of any of the fractions indicated in problem 20 in a similar way ? 22. i of T L = ? 1 of Jg = ? I of Jg = ? What are the values of the similar parts of T x y , ^, and ^9 ? 23. If T 3 y of a quantity costs $ 25, how much will ^ 9 T of it cost? 24. What is the ratio of ^ to ^ ? Of ^ to & ? 0f A to ,V Of A to W? Of .A to &? Of.fr to &? Of A to A? 25. y of a certain distance is 21 miles. What is ^ of the distance ? 26. A owned T 5 T of a mill, B fo C ^, and D the rest. What is the ratio of D's share to that of each of the others ? 27. If D's share is $1000, what is the value of A's share ? B's ? C's ? Of the whole mill ? 28. A piece of property is divided into 63 equal shares. A owns y of it, B ^, C £, and D the remainder. How many shares does each own? What is the ratio of A's part to each of the others ? Of B's ? Of C's ? Of D's ? 29. A pole 77 ft. long stands ^ in the mud, y in the water, and the rest in the air. How many feet of the pole are in the air ? 30. A miner found 5 nuggets of gold. The first weighed y oz., the second § oz., the third § oz., the fourth ^j oz., and the fifth 1 oz. What was the weight of all? 31. A man owning ^ of a piece of property sold -^ of the property. What part of the property had he left ? 32 THE FRACTION ONE SEVENTH 32. A man owning f- of a mill sold | of what he owned. What part of the mill did he sell ? What part did he still own? 33. A can mow f of an acre in half a day, and B | of an acre in the same time. How many acres can both mow in half a day ? In 10 days ? In 16 days ? 34. A can plow \ acres in one day, and B \ acres. How long would it take both of them working together to plow 1 acre ? 9 acres ? 21 acres ? 35. A can do a certain piece of work in 7 days, B in 5 days. What part of the work can A do in 1 day ? B ? Both ? How many days will it take both to do the work if they can do ^-| in 1 day ? 36. A can plow a certain field in 4 days, B in 5 days. How many days would it take both ? 37. A can saw seven cords of wood in 3 days, B in 4 days, and C in 5 days. How many days would it take all three to saw seven cords ? 38. James has T 6 T as many marbles as John. If John has 55 marbles, how many has James ? 39. James has ^ T as many marbles as John. How many has John, if James has 42 ? 40. Four boys found a sum of money, and divided it in such a way that the first got f of it, the second T 3 T , the third f£, and the fourth the remainder. What part of the money did the fourth get ? What is the ratio of the share of the first to each of the others ? Of the second ? Third? Fourth? THE FRACTION ONE SEVENTH 33 41. If the sum of money found was $2.31, how many cents did each receive ? What is the ratio of the share of each to the share of each of the others ? 42. A owned T 5 T of a mill, and B \ of it. Which owned the greater amount, and how much ? 43. A owned ^ of a mill and sold \ of his share. What part of the mill did he still own ? - What is the ratio of the part he owned after selling to the part he owned at first ? 44. A owned -Jf of a mill and sold \ of the mill to B. What part of the mill did A still own ? 45. A has 60 acres of land, and B 50 acres ? A exchanges 1 of his land for \ of B's. How many acres has each after the exchange ? 46. The ratio of the circumference of a circle to its diameter is about 3 r 2 -.* What is the circumference of a circle whose diameter is 7 inches ? 47. A wagon wheel is 35 inches in diameter. What is its circumference ? 48. How many times will a wheelbarrow wheel 14 in. in diameter turn in going 12 ft. ? 49. A bicycle wheel turns 3 times in going 22 ft. What is its diameter ? 50. What is the ratio of 1\ ft. to 28 in. ? 51. A horse is tied to a stake with a rope 14 yd. long. If he walks around the stake in as large a circle as the rope will permit, how many yards does he travel in going around once? * This ratio is a little too large, but is close enough for small circles. BtCN. MENT. AR. 3 34 THE FRACTION ONE SEVENTH 52. What is the ratio of 5 to 7 ? Of £ of 5 to } of 7 ? Of J of 5 to J of 7 ? Of 3 x 5 to 3 x 7 ? Of \ 2 - x 5 to -^ x 7 ? 53. What is the ratio of the circumferences of two circles whose diameters are 5 and 7 ? 54. What is the ratio of the circumferences of two circles whose radii are 2 J and 3 J ? 55. A, B, and C rent a pasture for $23-|-. A puts in 20 sheep, B 21, and C 6. How much should each pay ? What is the ratio of what A pays to what B pays? To what C pays ? Of B's share to A's ? To C's ? Of C's share to B's? Of the number of sheep of each to the number of each of the others ? Of the fraction of the whole number of sheep each has to the fraction of the whole number each of the others has ? 56. A, B, C, and D engage in business. A puts in $1000, B $2000, C $3000, and D $4000. At the end of a year they find they have gained $ 5000. What is each one's share of the gain ? 57. What is the ratio of the gain of each to the gain of each of the others ? Of the investment of each to the investment of each of the others? 58. If you know the cost of -^ of a quantity, how can you find the cost of -j 1 ^ of the same quantity ? 59. If -^ of a quantity of sugar is worth $ 35, what is the value of ^ of the quantity ? 60. The cost of a horse was -£ of the cost of a cow. If the horse cost $140, how much did the cow cost? If the cow had cost $ 39, how much would the horse have cost? SECTION VI DEVELOPING THE IDEA OF DECIMALS AND THEIR RELATION TO OTHER FRACTIONS 1. What part of the whole square is each of the small squares? How many small squares are there iij the whole ? 2. What part of the whole square are fifty of the small squares ? -j 5 ^- = what ? ^ = how many hundredths ? .5?) of the square is the same as what part of the square ? (Note the decimal point and learn its use.) 3. James has 24 marbles; John has .50 as many. How many has John ? 4. How many small squares are there in ^ of the large square? In f ? |? f? £? 35 36 THE IDEA OF DECIMALS 5. Give in fourths the values of .75, .25, .50, 1.00. 6. If \ — .25, how many hundredths are there in J? How many small squares = \ of the large square ? \ ? f ? 3? 1? 4? 5? 3? 6? 1? 8. ? 4 ? 2 ? 8 * 2 ' 8" • 8 * 4 ' 8 * 8 ' 8 * 4 ' 2 ' 7. Give in eighths the values of .25; .371; .121 ; .621; .87J; .50; .75; 1.00. 8. If .12| = 1, how many hundredths =3^? ^'? -|? a ? _4_ ? 1 ? X ? - 6 - ? & ? A ? - 8 - ? i ? - 9 - ? 44 ? 4 ? 44 ? 16- 16' ^* 16- 16* 8* 16* 16" 2' 16' 16' tf" 16' 12? 3.? 13? 14? 1? 15? 16? 8? 4? 16 * 4 * 16 • 16 • 8 * 16 ' 16 " 8 ' 4 " 9. Give the values in sixteenths of .31 J; .50; .121 ; .87|; .43|; .18|; .93|; '.56J; .68|; .75; .06J; .37 J; .621; .811; .25. 10. Give the ratio of : .121 to .871 .871 to .25 .061 to .621 .50 to.43| .93fto.75 .25 to .371 .18| to .561 .93| to .68| .25 to .81| .75 to .371 .75 to .061 .43f to .18f .811 to .25 .621- to .81| .433 to .371 .50 to .25 .871 to .93| .81 J to .93f .75 to .25 .68f to .75 .68} to .56% « 11. If you know the cost of .81 \ of a quantity, how can you find the cost of .871 of it ? Of .93f of it ? Of .25 ? Of .68f ? Of .621 ? of .121 ? Of .18| ? 12. If you know the weight of .184 of a bag of grain, how can you find the weight of .56^ of it ? Of .43 J of it ? Of .871- of it? Of .50? Of .621? Of .371? 13. If .25 of a quantity of coal weighs 3 T., how much does .75 of it weigh? .50? .18$? .621? THE IDEA OF DECIMALS 37 14. At the rate of $ .30 for .371 yd. of cloth, how much will .871 of a yd. cost? .62* ? .121? 15. At the rate of $.21 for .43| of a bushel of wheat, what is the value of ft of a bushel? T %? 11? If? J? 4 • 2 * 16. T 5 g of a piece of land is worth $ 5000. What is the value of .43f of it ? .75 ? .871 ? .68f ? >QQ ? 17. A farmer sold .18| of a load of corn. What was the ratio of what remained to what was sold ? Of what remained to the whole load ? 18. .37 -| of a certain distance is 24 miles. What is .87| of it ? .43f of it ? .561 of {t ? The whole distance ? 19. James has .31^ as many marbles as John. If James has 25, how many has John ? If John has 32, how many has James? What is the ratio of James's marbles to John's ? 20. The ratio of a certain number to another is .68|. The larger number is 96. What is the other ? If the smaller number had been 44, what would the larger have been ? 21. A owned f of a mill, B .18f, C ft, D .12J, and E the remainder. What part of the mill did E own? What was the ratio of A's share to each of the others ? Of B's ? OfC's? OfD's? OfE's? 22. At $.121 a yard, how many yards of cloth can be bought for $.15? For $ 1 ? For $ J- ? For $ 11 ? 23. At $6J each, how many plows can be bought for | of $ 100 ? For T % of $ 100 ? For $ 43| ? 38 THE IDEA OF DECIMALS 24. At $ . 31 J each, how many books can be bought for 15? For $10? 25. How many quarts are there in .43| of a bushel? What is the ratio of .18f pk. to .31£ bu. ? 26. How many hundredths are there in 1 ? In 1 of 1 ? Inf? In* of*? In J? In^ofJ? Inl?|?J?i? 3? 2? 4? 5? &? 6 * 3 ' 6 ' 6 ' 6 * 27. Give in sixths the values of .331 .50, .66}, .831 .16|, 1.00. Which of these can be expressed exactly in thirds ? 28. Give the ratio of : .16} to . 50 . 331 to . 66| . 66| to . 83J .50 to .331 .83£ to .16| 1.00 to .831 .66} to .50 .331 to .50 .83J to .66f .33J to 1.00 29. If you know the cost of . 33 J of a quantity, how can you find the cost of .66f of it? Of .50 of it? Of all of it? 30. Given the weight of .83^ of a load of corn, how can you find the weight of .16f of it? Of .66f ? Of .331? Of .50? 31. How many hundredths are there in ^ ? In 1 of \ ? In JW ? In A ? 1 ? - 3 - ? l ? - 4 - ? 1 ? -5- ? A ? l ? J- ? xii 12 . ±ii 12 . 6 . 12 . 1 . 12 . 3 . 12 . 12 . 2 . 12 . _a^ ? 2 ? _9_ ? 3 ? 10? 6 ? 11? 12? How manv new 12 * 3 • 12 • If • 12 * 6 * 12 * 12 ' X1UW many «ew decimals are there here ? What are the four ? 32. Give in twelfths the values of .91f, .41f, .581 .081 Give in twelfths and also in lower terms the equivalents of .831, .66}, .50, .331 .75, .25, .16}, 1.00. THE IDEA OF DECIMALS 39 33. Give the ratio of : .91| to .41f .66f to .75 .16§ to f .41|to.58| .831 to 1.00 ll to .581 .081 to .50 .75 to .41| .41| to 1 .50 to .081 .66| to .581 1.00 to .16| .331 to .75 .25 to .75 f to .41f .16fto.08i .331 to .25 .25 to .91§ .331 to .50 .121 to .16| 34. If you know the cost of .41| hundredths of a quan- tity, how can you find the cost of 1J of it ? Of .75 of it ? Of | of it? Of .621 of it? 35. If you know the weight of .58^ of a bag of sugar, how can you find the weight of .41| of it ? Of .66% of it? Of ll of it? 36. At the rate of * .22 for .91f yd. of cloth, how much will 1.831 yd. cost ? .50 yd. ? .331 v d. ? .58J yd. ? 37. ^ of a farm is worth $ 2100. Wha*t is the value of .581 of it? Of .50 of it? Of .621 of it? Of .66|- of it ? 38. If | of a piece of property is worth $62.50, how much is the whole worth ' ? .31 r i? .66|? .43|? .581? .831? .93|? J_? 3? ll? 16 ' ll? a? 12 • 6 * 39. A owns y 1 ^ of a factory. , B i G 1 D i, and E i. What part of the whole do all own ? 40. Give the ratio of : .50 to^ .331 to | i to .75 T V to - 62 i .81^to^ .831 to 1 I to .91| i to .66f .93ftof .50 tof f to .50 if to .68| .43f tof .66|to T ^ H to .93f I to .91| .68|to T 5 e .75 to-f s | to .43| | to .581 40 THE IDEA OF DECIMALS 41. How many hundredths are there in 1 ? In ^ of 1 ? | ? | ? f ? f? Give the value in 5ths of .20, .60, .40, .80, 1.00. 42. How many hundredths are there in ^ ? In ^ of \ ? T n JL ? _2_ ? 1 ? _3__ ? _4_ ? 2. ? _5_ ? 1 ? _fi_ ? 3 ? , in 10 • 10- 5" 10- 10* 5* 10' 2' 10' 5' JLV JL? 4? _9_? 10.9. 10 ' 10 ' 5 * 10 • 10 ' 43. How many hundredths are there in |- of -j 1 ^-? In ^? T n __2_ y l ? _s_ ? 1 ? I ? _3_ v J7_ y 2 ? _9_ y i ? ia 20' 10- 20- 5- 4* 10- 20- 5' 20' 2* 11 y 1? 13 y jL_y 3y i? ixy jl? loy 20 y 20* 5' 20* 10" 4" 5' 20" 10' 20* 20 ' - 44. Give in hundredths the values of -§, ^, J^, ^, 2%, 1_8_ 16 5 15 J3_ 14 12. 20' 20' 20> 20' 20' 20' 20* 45. Give the ratio of : .65 to .25 .50 to .55 £ to .85 if to .35 .40 to .65 .75 to .85 f to .55 -§- to .25 .70 to .35 .75 to .80 | to .65 ^ to .95 .45 to .20 .90 to .15 -^ to .45 ^ to .50 .85 to .45 :95 to .60 |# to .75 J to .75 46. If you know the cost of ^ of a quantity, how can you find the cost of .25 of it ? Given cost of -|, to find cost of .65. Given cost of .85, to find cost of -^ Given cost of .90, to find cost of -^-q. Given cost of J-J, to find cost of -|. Given cost of .75, to find cost of .85. 47. How many hundredths ; are there in ^ ? Iniofi? jl v jl y j_ y jl y j§ ? 25 * 25 • 25 • 2 5 * 2Z ' ft* etc. 48. Give the ratio of : .24 to .36 .20 to .64 .32 to .52 .68 to .40 .76 to .16 .72 to .48 .84 to .92 .88 to .44 .56 to .88 .60 to .64 .88 to .76 .36 to .80 THE IDEA OF DECIMALS 41 49. How many hundredths are there in J? ^? \1 ^? 1 ? i ? JL? JL ? - 1 - ? J- 9 - 1 -? 6 ' 8 • 10 • 12 • 16 * 20 ' 25 * 50. Give the number of hundredths in f , §, |, |, |, |, _2_ 2 2_ _2_ _2_ 10' 12' 16' 20' 25* 51. Give the number of hundredths in f, |, f, |, |, 10' 12' 16' 27' 2 5* 52. Give the number of hundredths in |, -|, |, |, ^, _4_ 4 _4_ 4 T n 5 5 5 JL __5_ _5_ JL _5_ 12' 16' 20' 25* 1U 5' 6' 8' 10' 12' 16' 20' 25* 53. Give the number of hundredths in |, |, t 6 q, y 6 ^, ^, _6_ JL Tn 1 JL _7_ JL JL _7_ Tn .§- _8_ __8_ _8__ _8 8 20' 25* - Ln 8' 10' 12' 16' 20' 25* "* "8"' 10' 12' 16' 20' 25* 54. Give the number of hundredths in -^, -j 9 ^, T 9 g, ^, _9 Tn l£ 44 iD. JLO io in i-l 44 44 44 In 4i 25* A11 10' 12' 16' 20' 25* " L11 12' 16' 20' 25* - 111 12"' 12 12 12 T n 13 la 13 I n 14 14 14 I n 15 15 16' 20' 25* Xil 16' 20' 25* ±1L 16' 20' 25* AU 16' 20' 15. I n 15. l£ 16. Tn 11 11 JL8 11 19 19 2 _2 111 16' 2 0' 2 5* in 2 0' 2 5' 2 0' 2 5* X11 16' 2 0' 2 5* Aii 2 0' 2 5' 2 0' 2 5' 2 0' 2 5' 2 0' 2 5* 7n 21 22 23 24 25 111 25' 2 5' 2 5' 2 5' 2^' 55.* Give the ratio of : .25 to .16| .80 to .66| .061 to .081 .60 to .371 .04 to .121 i.oo to .18f .081 to .10 .621 to .41| .05 to .331 .go to .75 .121 to .06^ .50 to .15 .25 to .331 .68f to M .66| to .50 .68f to .91f .50 to .40 .621 to .831 .121 to .16f .40 to .66| 56. A certain piece of property is divided into 120 shares. A owns \ of it, B /■§• of it, C \ of it, D 1 of it, and * In solving such questions as these the pupils will generally find it easiest to change the decimal form to the corresponding common fraction. But if by inspection, as in the ratio of .12^ to .06^, the pupils see that .12| is twice .06£, it is advisable that they shall go directly to the point and state the fact. See also problem 17, page 24. 42 THE IDEA OF DECIMALS E the remainder. How many shares does each own ? How many hundredths of the property does each own ? 57. .18| of a certain distance is 7 J miles. What is .50 of the distance ? The whole distance ? 58. .55 of a certain distance is 22 miles. What is .68| of the distance ? .25 of it ? The whole distance ? 59. If .33J of a ton of coal costs $ 2J, what is the cost of .80 of a ton ? What is the ratio of £ to T y? What is the value of \ 2 - of $ §? 60. Into how many pieces ,12| yd. long can 2.37| yd. of ribbon be cut ? 61. At 12|^ per yard, how many yards of cloth can you buy for $2. 37 f? 62. A can mow .66| of an acre in half a day ; B .75 of an acre in the same time. How many acres can both mow in half a day ? In a day ? In 12 days ? 63. John can do a piece of work in 1.5 days ; James in 1.33 J days. How long will it take them both to do it ? 64. Charles has .68| as many marbles as Henry. If Henry has 64, how many has Charles ? 65. Charles has .68| as many marbles as Henry. If Charles has 55, how many has Henry ? 66. .41| is the ratio of the cost of one chair to the cost of another. If the first cost $10, how much did the second cost ? If the second cost $ 6, how much did the first cost ? 67. A, B, and C engaged in business. A furnished .37 J of the capital, B .31^ of it, and C the remainder. What part did C furnish ? THE IDEA OF DECIMALS 43 68. If the whole capital was $3200, how much money did A invest? B? C? 69. If their profits were $1600, how much should A receive ? B ? C ? 70. What is the ratio of A's share of the profits to B's share? A's to C's ? B's to C's? B's to A's? C's to A's? C's to B's ? 71. What is the ratio of A's share of the capital to B's share ? A's to C's ? B's to C's ? B's to A's ? C's to A's ? C's to B's ? 72. Given the cost of .41| of a quantity, how can you find the cost of .58^ of it ? Given the cost of .16§, to find cost of .12|. Given the cost of .25, to find cost of .91§. Given the cost of .75, to find cost of .25. Given the cost of .43|, to find cost of .50. Given the cost of .87^, to find cost of .93|. Given the cost of .18|, to find cost of .43|. Given the cost of .81^, to find cost of .93 J. Given the cost of .85, to find cost of .45. Given the cost of .45, to find cost of .20. 73. .18| is the ratio of 12 gal. to what ? .66§ is the ratio of 5 ft. to what ? 1.12J is the ratio of 18 bu. to what? .91| is the ratio of 27 j lb. to what ? 1.93| is the ratio of 62 mi. to what ? 74. A pole 34 ft. long stood .31^ in the water, .33^ in the mud, and the rest in the air. How many feet were in the air ? 44 THE IDEA OF DECIMALS 75. A pole stood .25 in the mud, .41f in the water, and 15J ft. in the air. How long was the pole ? How many feet were in the mud ? In the water ? 76. A man, owning .80 of an acre of land, sold .18 J of his share for $6. What, was the price per acre ? 77. The value of a certain man's property was $ 6000. .81^ of it was real estate, .02^ was cash, and the remainder live stock. What was his live stock worth ? 78. A man owning .40 of a piece of property sold .37^ of his share. How many hundredths did he still own ? 79. A man chops 2J cd. of wood in 1.121 days. How long should it take him to chop 20 cd. ? Suggestion. It will take him 20 x § of f days. 80. What is the cost of 6 lb. of butter, at the rate of $.371 for fib.? 81. .41f of 72 is .311 of what number? 82. 50 is .121 of .66f of what number? 83. If . 83| of a quantity of sugar is worth $ 14, what is the value of .621 f the quantity ? Why is it worth | of $14? 84. At $ .16f each, how much will 33 note books cost ? What is the cost of 32 note books at $.18f each ? 85. How many pounds of butter can you buy for $1.25 if | lb. cost$.18f ? 86. How many cents are there in$f? In$|? In $ f ? In$ T 5_? In $^? In$|? In$f? 87. What part of a dollar is 25 cents? 331^? 20/? 18f^? 41f^? 93f^? THE IDEA OF DECIMALS 45 88. A owned .66% of a mill and sold .37 J of his share. How many hundredths of the mill did he still own ? 89. A man bought a horse and carriage, paying for the latter .68f of what he paid for the former. If the carriage cost $ 132, how much did the horse cost ? 90. A man bought a horse and carriage, paying $192 for the latter. If the ratio of the cost of the carriage to the cost of the horse was .68f, how much did the carriage cost? 91. One chair cost .66% as much as another. If both cost $ 50, what was the cost of each ? 92. Ho w. many rods are there in the circumference of a circle .56 rd. in diameter? 93. What is the ratio of .08 J ft. to .56% in. ? 94. Give the ratio of : .16| to .25 .25 to .66% .621- to .831 .12J to .081 .75 to .18} .60 to .371 .25 to .331 .371 to .25 .43| to .70 .20 to .50 .30 to .18f .75 to .56% .04 to .16% .80 to .66% .68f to .91| .16f to .66$ .331 to .50 .66f to .831 .40 to .12£ .41f to.31£ .84 to .561 95. -^ of a piece of land is worth $800. What is the- value of .43f of it? 96. A farmer sold .564 of a load of corn. What was the ratio of what remained to what was sold? 97. .25 of a certain distance is 60 miles. What is .58J of it ? .18| of it ? .371 of it ? .91f of it ? 46 THE IDEA OF DECIMALS 98. James has .83^ as many marbles as John. If James has 50, how many has John ? 99. The ratio of a certain number to another is f . The smaller number is .621. What is the other ? 100. At $ .18 J a yard, what part of a yard can be bought for |.16 ? What is the ratio of 9& to '••£? 101. At $ .41 J per yard, how many yards can be bought for $ . 621 ? For $ . 31| ? For $ 1. 25 ? What is the ratio offto&? Of^to^? Offto^? 102. At $.16$ per yard, how many yards of cloth can be bought for $.25? For*. 33$? For J. 12 J? For $.40? 103. At $.56;|- per pound, how many pounds can be bought for $.75? For $ 1.12J? For $21 ? 104. What is the ratio of .87$ pk. to .24 bu. ? 105. How many quarts are there in .93 J bu. ? In .68| bu. ? 106. If -y- of a lot is worth $ 91$, what is J of the lot worth ? f of it ? T % of it ? 107. .41 1 of a certain distance is 80 miles. What is . 311 of the distance ? . 91f of it ? . 58£ of it ? 108. .68| of a certain distance is 99 miles. What is .91f of the distance ? .75 of it ? .93f of it ? All of it ? 109. If ,66| of a ton of coal is worth $2, what is the value of .50 of a ton? Of .25 of a ton? Of .12J of a ton? 110. Into how many pieces .25 yd. long can 10^ ft. of rope be cut ? 101 yd. of rope ? ill. At $.06^- each, how many pencils can be bought for $1.25? For $1.50? For $.75? For$l? THE IDEA OF DECIMALS 47 112. At $ .08 J each, how many blank books can you buy for $.75? For $.25? For $1.50? 113. A can mow .75 of an acre in .33^ of a day ; B can mow .66% of .an acre in the same time. How many acres can both mow in 5 days ? In 8 days ? 114. Give in hundredths the values of : |, T 8 g, ^, f, -^g-, 11 11 5 4 14 18 4 11 JL _i>_ 12? 16' 8"' 5' 16' 2 5' ?' 2 0' 12' 16* 115. The ratio of the wheat in one bin to that in another is .831. If there are 60 bu. in the second bin, how many are there in the first ? 116. A, B, and C engage in business. A invests $ 2000, B $3000, and C $1000. How many hundredths of the profits should each receive ? 117. A farmer sold a cow for $ 45, which was f of what he paid for her. How many hundredths of the cost did he gain ? 118. If a merchant sells goods costing $48, so as to gain .16 J of the cost, how much does he gain ? 119. .83| of A's money equals .75 of B's. What is the ratio of A's money to B's ? 120. .311 of A's property is land, .41| of it is buildings, and the remainder is cash. If all his property is valued at $48,000, what is the value of his land? Of his build- ings ? How much cash has he ? 121. A man bought a horse and carriage for $ 282. If .18| of the cost of the horse equaled .40 of the cost of the carriage, what was the cost of each ? SECTION VII INTRODUCING THE IDEA OF PER CENT AND DEVELOP- ING RELATIONS 1. What is the ratio of 1 to 2 ? Of 2 to 4 ? Of 50 to 100 ? Another name for this relation is 50 per cent, writ- ten 50%, which means the same as .50. 2. What is 50% of 2 ? Of 4? Of 100? Of $4? Of $5? Of 18 bu.? Of 60 min.? 3. |2 is 50% of what? $6 is 50% of what? $10? 40 lb. ? 100 bu. ? 4. What part of anything is 25% of it? (.25 or J.) What is 25% of $12? Of 4 bu. ? Of 100 lb.? 5. How many per cent are there in the whole of any- thing? In 1 of it? ^ofit? | of it? 6. What is 75% of $20? 50% of 7 lb.? 25% of 16 bu.? 100% of $5? 7. Give the values in per cent of the following: l, ^, fi 8 > • J -^2' 8' ,0, 2> ¥' ,u ^2' 8' ' ni 2' 8* 8. What is 121% f 24? 371% of 16? 62-|% of 32? 87|-% of 100 ? 75% of $1 ? 100% of $1 ? 9. The amount of grain in a certain bin was 80 bu. 621% f it W as sold. How many bushels were sold? What per cent was left ? 30 bu. is what per cent of 80 bu. ? 50 bu. is what per cent of 80 bu. ? 48 THE IDEA OF PER CENT 49 10. A grocer sold 50 lb. of sugar, which was 12^-% of what he had in stock. How much did he have in stock ? What per cent did he have left after selling ? How many- pounds of sugar had he left ? 350 is what per cent of 400 ? 11. Give the values in per cent of the following: ^ Also Of 3, ^, g, 6 , T 2> 12' JL JL JL 9 11 13 15 16' 16 16' 16' 16' 16' 16' •L 11 12' 12* 12. How many pounds are there in .16f of 100 lbs. ? What is 16|% of 100 lbs. ? What is .41f of 144 bu. of corn ? What is 41|% of 144 bu. of corn ? 1 13. In a tank there were 132 gal. of water. 58 J % of it was drawn off. How many gallons were drawn off? What per cent of the whole amount remained ? 14. A farmer sold at a gain of 8|% a cow that cost him $36. How much did he gain? What was his selling price ? 15. A farmer sold a cow at a gain of $ 3, which was 8|% of what she cost. How much did she cost? If $3 is 8^%, or ^2 of the cost, how many dollars equal 100%, or 1| of the cost ? 16. A farmer sold a cow for $ 39, thereby gaining 8^%. How much did she cost ? If $ 39 is Jf of the cost, what is the cost ? 17. A farmer sold for $39 a cow that cost him $36. What per cent of the cost did he gain ? $ 3 is what part of $ 36 ? What per cent of $ 36 ? 18. A dealer sold a watch at a loss of 12 J%. If it cost him $48, how much did he lose? What was his selling price ? MCN. MENT. AR. — 4 50 THE IDEA OF PER CENT 19. A dealer sold a watch for $42, which was 121% less than it cost him. What was the cost ? The loss ? 20. A dealer sold a watch at a loss of 12|%. If his loss was $ 6, what was the cost ? The selling price ? 21. A watch that cost $48 was sold for $42. What was the loss per cent ? 22. A man having $4000 invested 20% of it in corn, 30% of it in wheat, and the remainder in oats. How much did he invest in each kind of grain ? What per cent of his money did he invest in oats ? 23. A 50-gallon barrel is 60% full of oil. How many gallons of oil are in it ? 24. A has $1000 invested in a factory, and B $1400. At the end of a year they find they have gained 16|%. What is each one's share of the gain ? 25. A owned § of a mill and sold 75% of his share. What part of the mill did he sell ? What per cent of the mill did he sell ? What per cent did he still own ? 26. What part of 66 is 22 ? What per cent of 66 is 22? What per cent of 18 is 9 ? 27. What per cent of : 24 is 3 ? 39 is 13 ? 25 is 24? 100 is 80? 12 is 6? 30 is 20? 25 is 25? 100 is 125? 18 is 3? 25 is 20? 25 is 26 ? 75 is 125? 48 is 8? 41f is 16|? 25 is 30 ? 32 is 28 ? 50 is 12J? 28 is 21? 10 is 10? 48 is 44? 28. A has 50 A. of land, and B 60 A. What per cent of A's land is B's ? Of B's is A's ? THE IDEA OF PER CENT 51 29. A has 50 A., and that is 83 J % of what B has. How many acres has B ? B has 60 A., which is 120% of what A has. How many has A ? 30. After selling some cattle a dealer found that he had gained $ 30, which was 18f % of the cost. How much did they cost ? What was his selling price ? 31. If A has 33^% more money than B, what per cent of A's money is B's ? What per cent of B's is A's ? 32. A dealer sold some cattle for $ 190, thereby gaining 18f %. How much did they cost him ? 33. Goods were sold at auction for $30, at a loss of 25%. How much did they cost ? 34. 16f% of a certain number is 2 greater than 12 J % of it. What is the number ? 35. The difference between two numbers is 3, and one of them is 25% greater than the other. What are the numbers ? 36. One number is 3 less than another. One is 20% less than the other. What are the numbers ? 37. A bicycle was sold for $ 50«at a loss of 50%. How much did it cost ? What was the loss ? 38. A newsboy buys papers at the rate of 2 for 3 cents and sells them for 2^ apiece. What per cent does he gain ? 39. If a merchant sells \ a pound of butter for what | of a pound cost, what per cent does he gain ? 40: By selling apples at the rate of 3 for 5 cents, a dealer gains 25%. What per cent would he have lost had he sold them at the rate of 5 for 6 cents ? 52 THE IDEA OF PER CENT 41. A farmer sold 2 cows for $24 each. On one he gained 20%, and on the other he lost 20%. How much did each cost ? 42. 3% of a certain number is 2 less than 5% of the number. What is the number ? 43. A clothing dealer sells a suit of clothes marked $ 20, at 25% less than the marked price and still gains 50% on the cost. What is the cost ? 44. 8% of a certain sum is $30 less than 14^% of it. What is the sum ? 45. A boy buys oranges at the rate of 4 for 10 cts., and sells them at the rate of 3 for 10 cts. What is his per cent of gain ? 46. If 3 oranges are sold for what 4 oranges cost, what is the gain per cent ? 47. What is the loss per cent if 4 oranges are sold for what 3 cost ? 5 for what 4 cost ? 48. What is the gain per cent if 3 newspapers are sold for what 5 cost ? If 3 are sold for ^ of what 5 cost? 49. A grocer buys strawberries at the rate of 5 qt. for 12 cts., and sells them at the rate of 3 qt. for 12 cts. What is his gain per cent ? How much does he make on 30 qts. of berries ? 50. A bicycle is marked $ 48, but in order to make a sale the dealer has to allow a discount of 12 J%. How much did the bicycle cost him, if he made 16|% on the transaction ? 51. If a coal dealer sold f of a ton of coal for what f of a ton cost, what per cent did he gain ? THE IDEA OF PER CENT 53 52. An overcoat is sold for $ 24 at a gain of 20% . How much did it cost ? 53. If a horse that cost $ 174 was sold for $ 203, what per cent was gained ? 54. What is 12|% f 16 |^ f 95? 55. 20% of 50% of a certain number is 78. What is the number ? 56. 5 times 25% of a certain number is 375. What is the number ? 57. 24 is 12 times 18|% of a certain number. What is the number ? 58. 12 1 % of a certain number plus 16|% of it equals 21. What is the number ? 59. What number is it of which 25% exceeds 18f % by 20 ? 60. 31|% of one number equals 41|% of another. If the first is 48, what is the second ? 61. 62 J % of one number equals 83^% of another. The sum of the two numbers is 56. What are the numbers ? If f of the first number equals § of the second, what is the ratio of f of the first number to the second ? Since the first number is § of f, or f of the second, both numbers are how many thirds of the second number ? If 56 = | of the second number, what is the second number ? The first? 62. 37^-% of James's money equals 30% of John's. If both have 81 cts., how much has each ? 63. A certain part of a pole 37 ft. long is in the ground, and the rest in the air. 18 j % of the part in the air equals in length 58J% of the part in the ground. What is the length of each part ? 54 THE ILEA OF PER CENT 64. A watch that cost $65 was sold for $52. What was the loss per cent ? 65. A man having $ 60,000, invested 20 of it in molasses, and the remainder in corn. How much corn did he buy, if he paid $.30 per bushel? 66. A has $2000 invested in a factory, B $2400, and C $1600. At the end of a year they find they have gained 25% on the whole investment. What is each one's share of the gain ? 67. A owned f of a mill and sold 50% of his share for $ 3000. What was the value of the mill at that rate, and what was the value of A's share, both before and after selling ? 68. A has | as much land as B. What per cent of A's land is B's ? 69. If A has 25% more money than B, what per cent of A's -money is B's ? What per cent of B's is A's ? 70. Goods were sold at auction for $ 62J, which was at a loss of 37^ %. How much did they cost ? 71. The difference between two numbers is 5, and one of them is 33^% greater than the other. What are the numbers ? 72. By selling at the rate of 4 for 5 cents what was bought at the rate of 5 for 6 cents, what per cent is gained ? 73. A farmer sold two horses for $ 75 each. On one he gained 50%, and on the other he lost 50%. What was the cost of each ? THE IDEA OF PER CENT 55 74. A couch is marked $35, but in order to make a sale, the dealer gives a discount of 20%. If he still gains 12%, what is the cost? 75. If a grocer sells | of a dozen eggs for what % of a dozen cost, what per cent does he gain ? 76. What is 18f % of 133£% of 84 ? 77. 27 is 75% of 75% of what number? 78. 60 is 6 x 31|% f w hat number ? 79. 28 is 331% + 25% of what number? 80. What number is it of which 66f % exceeds 5^% by 65? 81. A horse and carriage are worth $ 246. If -| of the value of the horse equals | of the value of the carriage, what is the value of each ? 82. A and B invest $8800 in business. 62^% of A's investment equals 75% of B's. How much does each invest ? 83. What per cent of the circumference of a circle 2 inches in diameter is the circumference of a circle 3 inches in diameter ? (See problem 46, page 33.) 84. What per cent of A's money is B's, if A has $ 50, and B has 10 fo more ? What per cent of B's is A's ? 85. What per cent of A's money is B's if A has 33 J J6 more than B ? What per cent of B's is A's ? SECTION VIII INTRODUCING GENERAL COMPOUND NUMBERS AND DEVELOPING RELATIONS 1. What is the ratio of 8 in. to 12 in. ? Of 2 ft. to 3 ft. ? Of 3 yd. to 5£ yd. ? Of 32 rd. to 320 rd. ? 2. What is the ratio of 8 in. to 1 ft. ? Of 2 ft. to 1 yd. ? Of 3 yd. to 1 rd ? Of 32 rd. to 1 mi. ? 3. What is the ratio of 1 yd. to 6 in. ? Of 1£ yd. to 6 in. ? Of 2 yd. to 6 in. ? 4. A boy measures a stick with a 6-inch rule, and finds that he has to apply the rule exactly 12 times. How many yards long is the stick ? 5. How many steps 2 ft. 4 in. long must a man take to walk a distance of 28 ft. ? What is the ratio of 28 ft. to 21 ft. ? 6. How many inches are there in 1 yd. 2 ft. 6 in. ? In 1 yd. 1 ft. 3 in. ? In 1 rd. ? In 1 rd. 2 in. ? In 1 rd. 1 yd. 1 ft. 1 in. ? 7. Change 66 in. to integers of higher denominations. (^66 in. = 1 yd. 2 ft. 6 in.) Change 200 in. to integers of higher denominations. 198 in. 51 in. 247 in. 56 GENERAL COMPOUND NUMBERS 57 8. How many feet are there in 1 rd. ? In 2 rd. ? In 4 rd. ? In 6 rd. ? In 1 rd, 3 yd. ? In 1 rd. 5 yd. ? In 6 rd. 2 ft. ? 9. Reduce to integers of higher denominations: 33 ft.; 99 ft.; 66 ft.; 25$ ft.; 100 ft.; 31 J ft. 10. Five planks, 5 ft. 6 in., 7 ft. 3 in., 6 ft. 3 in., 5 ft. 7 in., and 8 ft. 5 in. long, respectively, are laid end to end. How many yards along the ground will they reach ? 11. A certain fence is made of boards 8 in. wide stand- ing upright. How many boards are there in 6 yd. 2 ft. of the fence ? 12. How many feet will a boy walk in taking 10 steps of 26 in. each ? 13. What is the ratio of 1 pt. (dry measure) to 2 pt.? Of 3 qt. to 8 qt.? Of 1 pk. to 4 pk.? 14. What is the ratio of 1 pt. to 1 qt.? Of 3 qt. to 1 pk.? Of 1 pk. to 1 bu.? 15. W T hat is the ratio of 1 bu. to 1 qt.? Of 1 bu. to 1 pt.? Of 1 bu. to 2 pt.? Of 1 bu. to 2 qt.? To 3 qt.? To 8 qt.? To 1 pk.? To 1 pk. 1 qt.? To 2 pk. 3 qt.? To 1 pk. 1 qt. 1 pt.? 16. How many times must a half -peck measure be filled in measuring 2^ bu. of beans ? 17. Reduce to quarts : 4 pk. 2 qt. ; 1 bu. 1 pk. ; 2 bu. 2pk. 2qt.; 1 bu. 3 pk. 5 qt. 18. Reduce to integers of higher denominations : 48 qt. ; 25 pt. ; 57 qt. ; 69 qt. 19. What is the ratio of 1 gi. (liquid measure) to 4 gi.? Of 1 pt. to 2 pt.? Of 1 qt. to 4 qt.? Of 1 gal. to 3£ gal.? 58 GENERAL COMPOUND NUMBERS 20. What is the ratio of 1 gi. to 1 pt.? Of 1 pt. to 1 qt.? Of 1 qt. to 1 gal.? Of 1 gal. to 1 bbl.? 21. Give the ratio of 1 gal. to 1 qt. To 1 gi. To 1 pt. Of lqt. tolpt. Tolgi. 22. Give the ratio of 1 gal. to 1 gal. 2 qt. To 1 gal. 1 qt. 1 pt. To 2 gal. 2 gi. Of 1 qt. to 2 qt. 1 pt. To 1 gal. 1 qt. 1 pt. To 1 pt. 3 gi. Of 1 pt. to 1 gal. 3 qt. To 2 qt. 1 pt. To 1 pt. 2 gi. 23. How many times can a dipper holding 1 pt. 1 gi. be filled from a can containing 3 gal. 1 pt.? 24. A milkman has in his wagon 5 cans. The first con- tains 10 gal. of milk; the second 4 gal. 2 qt.; the third 1 gal. 2 qt. ; the fourth 4 gal. 1 pt.; and the fifth 1 gal. 3 qt. 1 pt. How many gallons of milk has he ? 25. How many bottles holding 1 pt. 3 gi. can be filled from a cask containing 1 gal. of wine ? 26. How many inch squares does a twelve-inch square contain ? What is the ratio of 1 sq. ft. to 1 sq. in.? Of 1 sq. in. to 1 sq. ft.? Of 36 sq. in. to 1 sq. ft.? Of 72 sq. in. to 1 sq. ft.? Of 108 sq. in. to 1 sq. ft.? 27. What is the ratio of 1 sq. ft. to 1 sq. yd.? Of 1 sq. yd. to 1 sq. ft.? 28. How many square inches are there in a 2-inch square ? In a 3-inch square ? A 4-inch square ? A 5-inch square? A 10-inch square? 29. What is the ratio of a 2-inch square to a 3-inch square ? To a 4-inch square ? To a 6-inch square ? GENERAL COMPOUND NUMBERS 59 30. Give the ratio of : A 3-inch square to a 2-inch square ; a 4-inch square to a 2-inch square ; a 3-inch square to a 5-inch square ; a 5-inch square to a 10-inch square ; a 12-inch square to a 36-inch square. 31. Rectangle a is 3 in. long and 2 in. wide; rectangle b is 5 in. long and 3 in. wide. What is the ratio of a to b ? Oibtoa ? Of 1 of b to -i of a ? Of 1 of a to 1 of b ? 32. At the rate of 10 shingles to the square foot, how many shingles are necessary for a roof 20 ft. wide and 50 ft. long ? 33. A cube is a solid having six equal square faces. A 1-inch cube (or cubic inch) has each edge 1 inch in length. How many 1-inch cubes are there in a 2-inch cube ? How many cubic inches are there in a 2-inch cube ? In a 3-inch cube ? A 4-inch cube ? A 5-inch cube ? A 10- inch cube ? A 12-inch cube ? 34. What is the ratio of : A 2-inch cube to a 3-inch cube ? To a 4-inch cube ? To a 5-inch cube ? To a 10-inch cube ? 35. What is the ratio of a 3-inch cube to a 4-inch cube? To a 5-inch cube ? To a 6-inch cube ? To a 10-inch cube ? Of a 4-inch cube to a 5-inch cube ? To a 10-inch cube ? 36. How many cubic inches are there in a block of wood 4 in. long, 3 in. wide, and 1 in. thick ? In a block 4 in. long, 3 in. wide, and 2 in. thick ? In a block 4 in. long, 3 in. wide, and 5 in. thick ? 37. How many cubic inches are there in a cubic foot ? How many cubic feet are there in a 3-foot cube ? In a cubic yard ? 60 GENERAL COMPOUND NUMBERS 38. In digging a ditch 20 ft. long, 2 ft. wide, and 1 ft. deep, how many cubic feet of dirt are handled? How many, if the ditch is 2 ft. deep? 3 ft. deep? 4 ft.? 5 ft.? 39. A wheat bin is 10 ft. long, 8 ft. wide, and 5 ft. deep. How many cubic feet of wheat will it hold ? 40. Allowing 1J cu. ft. of wheat to the bushel, how many bushels will the bin hold ? 41. How many bushels of oats can be put into a box 4 ft. long, 2 ft. wide, and 18 in. deep ? 42. There are about 7J gal. in a cubic foot. How many gallons will a tank 2 ft. x 3 ft. x 4 ft. hold ? 43. An aquarium measures 4 ft. long, 2J ft. wide, and 18 in. deep. How many gallons of water are necessary to fill it half full ? 44. A gallon contains 231 cubic inches. How deep must a box be to hold exactly a gallon, if it is 11 in. long and 7 in. wide ? 45. What is the ratio of 3 bu. of wheat to 3| cu. ft. of wheat ? 46. Give the ratio of a 5-inch square to a 6-inch square. Of a 4-inch square to a 5-inch square. Of a 4-inch cube to a 5-inch cube. Of a 10-inch cube to a 5-inch cube. 47. How many cubic inches are there in a block of wood 12 in. long, 5J in. wide, and 1 J in. thick ? 48. What is the cost of digging a cellar 10 ft. deep, 10 ft. wide, and 27 ft. long at $ 3.50 per cubic yard ? 49. How many 2-bushel grain sacks can be filled from a bin 3 ft. x 4 ft. x 5 ft. full of wheat ? GENERAL COMPOUND NUMBERS 61 50. About how many gallons of milk will a cheese vat 10 ft. x 4 ft. x 3 ft. hold ? 51. If the bottom of a jar has an area of 21 sq. in., how deep must the jar be to hold exactly a gallon ? 52. A steel plate 20 in. long and 1 ft. wide weighs 8 oz. to the square inch of surface. How many pounds does the plate weigh ? 53. How many pieces 1 ft. square and 1 in. thick are there in an inch board 1 ft. wide and 10 ft. long ? In a 2-inch plank of the same width and length ? 54. How many units 1 in. thick, having a surface of 1 sq. ft., are there in a board 1 in. thick, 6 in. wide, and 12 ft. long ? Such units are called board feet. The board foot is used in meas- uring lumber. 55. How many board feet are there in a plank 8 in. wide, 15 ft. long, and 2 in. thick ? The surface of the plank = 10 sq. ft. Since it is 2 in. thick, for each square foot there are 2 bd. ft. Therefore there are 20 bd. ft. in the plank. 56. How many board feet are there in a 2-inch plank 6 in. wide .and 16 ft. long ? In 10 such planks ? 57. How many board feet are there in seven 2 by 4's (i.e. plank 2 in. thick and 4 in. wide) 12 ft. long? In twenty 2 by 4's 12 ft. long ? In ten 2 by 6's 20 ft. long ? 58. How many feet (board feet) are there in ten 8 by 8's 18 ft. long ? In ten 4 by 4's ? 59. How many feet of lumber are necessary to plank the floor of a stall 8 ft. x 10 ft. with 2 by 6's ? How much will it cost at $ 20 per M. ? 62 GENERAL COMPOUND NUMBERS 60. Boards less than an inch thick are reckoned as though they were an inch thick. What is the value of ten |-inch black walnut boards, 10 in. wide and 12 ft. long, at $ 100 per M. ? 61. At the rate of 10 shingles to the square foot, how much will shingles cost for a roof 20 ft. by 50 ft. at $ 3 per M. ? 62. At $2.50 per M., how much will the shingles cost for a roof of two parts, each 40 ft. by 20 ft. ? 63. How much would it cost to cover the same roof with inch boards at $ 10 per M. ? 64. What is the ratio of 2 bu. 3 pk. to 1 pk. 3 qt. ? Of 1 bu. 1 pk. 1 qt. to 2 bu. 2 pk. 2 qt. ? Of 4 gal. 1 pt. to 3 qt. ? Of 4 rd. 2J ft. to 8 rd. 1 yd. 2 ft. ? 65. Give the ratio of : 2 lb. Avoir, to 4J lb. Avoir. 6 oz. Avoir, to 2 lb. Avoir. 1 lb. 4 oz. Avoir to 3 lb. Avoir. 2 lb. 12 oz. Avoir, to 3 lb. 8 oz. Avoir. 7 oz. Avoir, to 2 lb. 3 oz. Avoir. 2 lb. Troy to 4J lb. Troy. 6 oz. Troy to 2 lb. Troy. 1 lb. 4 oz. Troy to 3 lb. Troy. 2 lb. 6 oz. Troy to 3 lb. 4 oz. Troy. 7 oz. Troy to 2 lb. 4 oz. Troy. Suggestion. If the teacher has time, Liquid Measure and the Metric System can be easily treated in a manner similar to that in 65 above. GENERAL COMPOUND NUMBERS 63 €6. How many bushels of wheat are there in a bin 5 J ft. x 6 ft., and 5 ft. deep, when it is f full? (See problem 40.) 67. What is the cost of digging a cellar 6 ft. deep, 10 ft. long, and 9 ft. wide, at $2.70 per cu. yd. ? 68. About how many gallons of water will a tank hold, whose inside measurements are 4 ft. x 3| ft. x 1J ft.? (See problem 42.) 69. At $10 per M., what is the cost of eight 2xl0's 12 ft. long, and ten 8 x 8's 18 ft. long ? 70. At the rate of 1000 shingles to the 100 sq. ft., how many shingles are necessary for a roof 40 ft. x 80 ft. ? 71. The area of the bottom of a jar is 77 sq. in. How deep must it be to hold exactly 2 gal. ? 72. How many times can a cask holding 3| gal. be filled from a tank 4 ft. long, 3 ft. wide, and 2 ft. deep ? 73. How many cubic feet are there in a block of stone 30 in. x 24 in. x 18 in.? 74. At 40 lb. to the cubic foot, what is the weight of a block of wood 6 ft. long, 18 in. wide, and 1J ft. thick ? 75. How many feet of lumber are necessary to cover with 2-in. plank a surface 20 ft. x 80 ft. ? How much will the lumber cost at $20 per M.? 76. How many revolutions does a wagon wheel 3J ft. in diameter make in going 12 rd. ? (See problem 46, page 33.) 77. At $ 3 per cubic yard, how much will it cost to dig a cellar 30 ft. x 33 ft., and 12 ft. deep ? 64 GENERAL COMPOUND NUMBERS 78. How many bushels of grain can be put into a bin 10 ft. x 8 ft., and 6 ft. deep ? How many bushels can be put into a bin 20 ft. x 16 ft. x 12 ft. ? 79. If wall paper is 18 in. wide, how many strips are needed to cover a wall 30 ft. long ? How many strips are needed for the four walls of a room 30 ft. long and 24 ft. wide, making no allowance for doors and windows ? 80. How many more steps must be taken in walking a mile if the steps are 2 ft. long, than if they are 3 ft. long ? 81. At the rate of 8 mi. per hour, how far will a horse travel in 4 hr. 35 min. ? 82. At the rate of 37| mi. in 90 min., how far will a train run in 3^ hr. ? 83. Reduce 208 in. to integers of higher denomination. 92 ft.; 125 in.; 144 in. 84. At $10 per M., find the cost of ten 4 x 4's 18 ft. long, and thirty 2 x 6's 12 ft. long. SECTION IX INTRODUCING THE STUDY OF SURFACES AND DEVELOPING RELATIONS 1. If the above rectangle is 8 units long, how many square units are there along one side ? If it is 4 units wide, how many square units are there along one end ? How many rows are there . of 8 square units each ? How many square units are there in the whole rectangle ? How many rows are there of 4 square units each ? How many square units are there in the rectangle ? 2. Does 4x8 equal the number of square units in a rectangle 8 units long and 4 units wide ? How many square units are there in a rectangle 10 units long and 7 units wide ? 3. The number of square units in the area of a rectangle is equal to the product of the numbers of the like units in the length and width. What is the area of a rectangle 6 in. long and 4 in. wide ? 1 ft. long and 5 in. wide ? MON. MENT. AR. 5 65 66 THE STUDY OF SURFACES 4. A parallelogram is a four-sided figure whose opposite sides are equal and parallel. Is a rectangle a parallelo- gram ? The base is the side on which the parallelogram seems to stand. The altitude is the perpendicular dis- tance from the base to the side opposite. In figure a what is the base ? The altitude ? In figure b what is the base ? What two lines represent the altitude ? 5. If in figure b we cut off a three-sided portion at the right, and add on an equal three-sided portion at the left, what kind of a figure have we ? Compare the area of the rectangle with the area of parallelogram b. 6. If the rectangle is 8 units long and 4 units wide, how many square units are there in its area? If the parallelogram has a base of 8 units, and an altitude of 4 units, how many square units are there in its area ? 7. The area of a parallelogram is represented by the product of the numbers of like units in the base and in the altitude. What is the area of a parallelogram whose base is 7 in. and altitude 4 in. ? Base 10 in. and altitude 6 in. ? Base 20 in. and altitude 7 in. ? B b ' b 8. A triangle is a three-sided figure. Its base is usu- ally considered to be the side it seems to stand on, although THE STUDY OF SURFACES ' 67 any side may be considered the base. Its altitude, AB, is its height, measured from the highest point to the base. If a line is drawn from corner to corner of a parallelo- gram, into what two figures is the parallelogram divided ? Compare triangle a with triangle b. Compare a with the parallelogram. Compare b with the parallelogram. 9. Is the altitude of triangles a and b the same as that of the parallelogram ? Are the bases- the same ? If the base of the parallelogram is 7 and its altitude 3, what is its area? What, then, is the area of triangle a ? Of triangle b ? 10. The area of a triangle is represented by one half the product of the numbers of like units in the base and in the altitude. The base of a certain triangle is 7 units, its altitude 3 units. What is its area ? 11. Give the area of a triangle whose base is 7 in. and altitude 3 in. Base 10 in. and altitude 4 in. Base 6 ft. and altitude 3 ft. Base 4 rd. and altitude 3 rd. 12. What is the area of a rectangle 12 ft. long and 7 ft. wide? Of a parallelogram whose base is 12 ft. and alti- tude 7 ft. ? Of a triangle whose base is 12 ft. and alti- tude 7 ft. ? What is the ratio of the rectangle to the parallelogram? Of the parallelogram to the triangle? Of the rectangle to the triangle? 13. A rectangular piece of land is divided into two parts by a diagonal line (one running from corner to corner). If the base of the rectangle is 40 rd. and its altitude 20 rd., what is the area of each of the parts? 14. A trapezoid is a four-sided figure having two and only two sides parallel. These sides are called the upper 68 THE STUDY OF SURFACES base and the lower base. The altitude is the perpen- dicular distance between the parallel sides. If a diagonal is drawn, into what does it divide the trapezoid? If the upper base is 4 and the altitude is 2, what is the area of triangle a? If the lower base is 3, what is the area of triangle b? Of both triangles? The area of the trape- zoid? Will you get the same area of the trapezoid if you take half the sum of the two bases times the altitude ? 15. The area of a trapezoid is represented by the product of the number of like units in half the sum of the two bases and in the altitude. What is the area of a trapezoid whose bases are 4 units and 3 units, and altitude 2 units ? Of one whose bases are 4 in. and 3 in., and altitude 2 in.? 16. Find the area of a trapezoid having : Bases 6 in. and 8 in., and altitude 4 in. Bases 4 ft. and 5 ft., and altitude 3 ft. Bases 2 ft. and 18 in., and altitude 15 in. 17. A certain rectangle has an area of 72 sq. in. If its length is 1 ft., how wide is it? 18. A parallelogram having an altitude of 7 ft. has an area of 81 sq. ft. How long is the base ? THE STUDY OF SURFACES 69 19. The area of a certain triangle is 40 sq. in. If the base is 10 in., what is the altitude? 20. The area of a certain trapezoid is 85 sq. ft. If one base is 20 ft., and the other 14 ft., what is the altitude of the trapezoid ? 21. Think of a circle divided into a number of equal triangles as in Fig. 1. Is the area of all the triangles greater or less than the area of the circle? Are the alti- tudes of the triangles equal? Is the altitude a greater Fig. i. Fig. 2. or less than the radius of the circle ? Is the sum of the bases (the sides next to the circumference) greater or less than the circumference? If the number of triangles be doubled, is their area nearer to that of the circle ? Is the altitude a more nearly equal to the radius? Is the sum of the bases more nearly equal to the circumference ? If you think of the number of triangles indefinitely increased, what will be true of their area, altitude, and the sum of their bases, as compared with the area, radius, and circum- ference of the circle respectively? Can they be made as nearly equal as we please? 22. Think of the circle in Fig. 2 above as unrolled. The sum of the bases of the triangles may be represented 70 THE STUDY OF SURFACES as below by a straight line (nearly) equal to the circum- ference. How is the area of each triangle found ? Is the area of all of them equal to ^ the product of the sum of the bases by the altitude? Since this is true, and since Circumference i Circumference the sum of the bases is nearly equal to the circumference, and the altitude to the radius, is the area of the circle nearly equal to ^ the product of the circumference by the radius, or the product of \ the circumference by the radius? By increasing the number of triangles, is the re- sult more nearly correct? (Geometry shows that the area of the circle is exactly equal to J the product of the circum- ference by the radius.^ In the above figure is represented a parallelogram nearly equal in area to the circle. How can you find its area ? 23. Find the area of a circle whose circumference is 44 in. and radius 7 in. Of one whose circumference is 22 in. and radius 3^ in. State the dimensions of the rectangle equal in area to each circle. A rectangle 7 in. x 22 in. is equal to the first circle; that is, one whose width is the radius, and whose length is \ the circumference. THE STUDY OF SURFACES 71 24. What is the ratio of the circumference to the diame- ter? (Nearly - 2 Y 2 -.) How many times the radius is the diameter? How many times the radius is the circumfer- ence ? What is the circumference of a circle whose radius is 7 in. ? Radius 8J in. ? Radius 14 ft.? Radius 17£ ft. ? Radius 21 ft. ? How long must a rectangle 7 in. wide be to be equal to the first circle ? Make a similar state- ment regarding each of the other circles. it x r \ Circumference = n x r 25. Find the area of a circle whose radius is 7 in. What rectangle is it equal to ? Into how many squares 7 in. on each side can this rectangle be divided ? (Each square may be thought of as the square of the radius rep- resented above by r 2 .) Is the area of the rectangle equal to 3^ x the square of 7 in. ? Using the symbol ir (read pi) for 34-, is the area of the rectangle equal to nr x 49 sq. in., or it x r 2 , or 77T 2 ? Is the area of the circle also equal to irr 2 ? (This way of finding the area is easier than find- ing ^ the product of the circumference by the radius, since the radius, not the circumference, is usually given.) 26. Find the area of a circle whose radius is 10^ in. 11 3 ( Area = 7rr 2 , or, ^ x ^ x — = 346£, number of sq. in. J 72 THE STUDY OF SURFACES Of one whose radius is 3^ in rd. yd ; 14 in.; 7ft.; 31 yd.; 14' Of one whose diameter is 7 in. ; 28 in. ; 14 ft. ; 7 28 rd. : 7 ft. 27. What is the ratio of the length of rectangle a to the length of rectangle b ? Of the width of rectangle a to the width of rectangle b ? Of the length x the width of rec- tangle a to the length x the width of rectangle b ? Of rectangle a to rectangle b ? Of b to a ? 28. If rectangle a is 36 in. long and 25 in. wide, and rectangle b is 27 in. long and 20 in. wide, what is the ratio of the length of a to the length of b ? Of the width of a to the width of b ? Of the area of a to the area of b ? Of b to a ? 29. Give the ratio of : A rectangle 12 in. by 9 in. to a rectangle 8 in. by 6 in. A rectangle 5 in. by 7 in. to a rectangle 7 J in. by 10 J in. A rectangle 18| in. by 25 in. to a rectangle 25 in. by 16| in. A rectangle 12J ft. by 8^ ft. to a rectangle 37^ ft. by 16| ft. A rectangle 31^ rd. by 20 rd. to a rectangle 37J rd. by 30 rd. THE STUDY OF SURFACES 73 A rectangle 83^ yd. by 33^ yd. to a rectangle 33J yd. by 8* yd. A rectangle 56^ ft. by 41| ft. to a rectangle 62^ ft. by 58^ ft. 30. How many rectangles 8 in. x 5 in., will a rectangle 32 in. x 15 in. contain ? 31. What is the area of : A parallelogram whose base is 8 ft. and altitude 7 ft. ? A triangle whose base is 8 ft. and altitude 7 ft. ? » A trapezoid whose bases are 6 ft. and 4 ft., and altitude 10 ft. ? A triangle whose base is 3 ft. and altitude 16 in. ? A rectangle 4 ft. by 4 yd. ? A circle whose radius is 7 in. ? A circle whose diameter is 7 in. ? A trapezoid whose bases are 8 ft. and 8 yd., and altitude 10 ft. ? n 32. What is the ratio of the base of a to the base of b ? Of the altitude of a to the altitude of b ? If the ratio of the bases is f , and the ratio of the altitudes is 2, what is the ratio of the base x altitude of a, to the base x altitude of b ? Or, what is the ratio of the area of a to the area of b ? Of b to a ? (f x i - ?) 74 THE STUDY OF SUBFACES 33. Give the ratio of the area of parallelogram a to parallelogram b in the following. Also of b to a. Base .8 in., alt. 6 in. Base 12 in., alt. 8 in. Base 18f ft., alt. 16$ ft. Base 911 r d., alt. 56{ rd. Base 66% yd., alt. 25 yd. Base 50 ft., alt. 62£ ft. Base 100 ft., alt. 300 ft. Base 41$ in., alt. 50 in. Base 4 in., alt. 3 in. Base 9 in., alt. 5 in. Base 25 ft., alt. 33£ ft. Base 75 rd., alt. 31Jrd. Base 100 yd., alt. 37£ yd. Base 66% ft., alt. 37J ft. Base 200 ft., alt. 150 ft. Base 33£ in., alt. 125 in. 34. What is the ratio of the area of parallelogram a to the area of parallelogram b ? If a is 3 x 6, what is the ratio of J of a to ^ of b ? Triangle a is what part of parallelogram a? Triangle b is what part of parallelo- gram b ? What, then, is the ratio of triangle a to triangle b ? 35. What is the ratio of a triangle whose base is 6, and altitude 3, to one whose base is 4 and altitude 1J? ct * ! = ?)• Why is it not necessary to consider the \ in comparing two triangles ? (See problem 10.) THE STUDY OF SUBFACES 75 36. What is the ratio of triangle a to triangle b in the following ? Of b to a ? a b Base 8 in., alt. 6 in. Base 4 in., alt. 3 in. Base 12 in., alt. 8 in. Base 9 in., alt. 5 in. Base 18f ft., alt. 16$ ft. Base 25 ft., alt.' 83} ft. Base 91 J- rd., alt. 56^ rd. Base 75 rd., alt. 81} rd. Base 34 in., alt. 2^ in. Base 5 in., alt. 5 in. Base 18f in., alt. 25 in. Base 25 in., alt. 16$ in. Base 121 f t ., a lt. 8} ft. Base 37* ft., alt. 16$ ft. Base 561 ft., alt. 41$ ft. Base 621 ft., alt. 581 ft. 37. Find the ratio of a 7-in. circle to a 14-in. circle. Find the dimensions of the rectangles equal to the two circles. (See problems 24 and 25.) If the altitudes are 7 in. and 14 in., and the bases are 3f x 7 in., and 3f x 14 in., what is the ratio of the alti- tudes and of the bases ? The ratio of the rectangles ? Of the circles ? Since the ratio of the altitudes is equal to the ratio of the bases, can the ratio of the areas be found by squaring the ratio of the altitudes, or their equal, the ratio of the two radii ? 38. What is the ratio of a circle of radius 3^ ft. to one of radius 7. ft. ? Of one of radius 6 ft. to one of radius 12 ft. ? Of one of radius 12J ft. to one of radius 18| ft. ? What is the ratio of the radii? If the ratio of the radii is §, what is the ratio of the circles ? 39. If the radius of one circle is. twice as long as the radius of another, what is the ratio of the area of the first to that of the second ? Of the second to the first ? 76 THE STUDY OF SURFACES 40. If the area of one circle is 4 times that of another, what is the ratio of the radius of the first to that of the second ? Of the diameter of the first to the diameter of the second ? Of the circumference of the first to the circumference of the second ? 41. What is the ratio of the area of a circle of radius 1 ft. to the area of one of radius 2 ft. ? Of one of radius 2 ft. to one of radius 4 ft. ? Of one of radius 3 ft. to one of radius 6 ft. ? Of one of diameter 1 ft. to one of diameter 2 ft. ? Of one of diameter 2 ft. to one of diameter 4 ft. ? Of one of diameter 3 ft. to one of diameter 6 ft. ? Of one of circumference 1 ft. to one of circumference 2 ft. ? Of one of circumference 2 ft. to one of circumference 4 ft. ? Of one of circumference 3 ft. to one of circumference 6 ft. ? 42. A pipe 1 inch in diameter carries 3 gal. of water per minute. At that rate how much per minute will a 2-inch pipe carry ? A 4-inch pipe ? An 8-inch pipe ? A ^-inch pipe ? A ^-inch pipe ? 43. A building is supplied with water by a 2-inch pipe. How many J-inch pipes would it take to carry the same amount of water running at the same rate ? 44." What is the ratio of the area of a circle of 1 in. ra- dius to that of a circle of 3 in. radius ? (See problem 27.) Of a circle of 2 in. radius to one of 6 in. radius ? Of a circle of 3 in. radius to one of 9 in. radius ? Of a circle of any radius to a circle of 3 times that radius ? Of a circle of any diameter to a circle of 3 times that diameter ? Of a circle of any circumference to a circle of 3 times that circumference ? THE STUDY OF SURFACES 77 45. What is the ratio of the area of a circle of radius 1 in. to that of a circle of radius 4 in. ? Of a circle of radius 2 in., to one of radius 8 in. ? Of a circle of radius 3 in., to one of radius 12 in. ? Of a circle of any radius, to One of 4 times that radius ? Of a circle of any diameter, to one of 4 times that diameter ? Of a circle of any cir- cumference, to one of 4 times that circumference ? 46. The square of a number is the product obtained by multiplying the number by itself. What is the square of 6 ? Of 5 ? Of 8 ? Of | ? Off? The square of 6 may be represented thus : 6 2 ; of -|, (|) 2 , etc. 7 2 = ? 9 2 = ? (t) 2 =? (t) 2 =? d) 2 =? 47. The ratio of the area of one circle to that of another is the square of the ratio of the radius of the first to the radius of the second, or the square of the ratio of the diameter of the first to the diameter of the second, or the square of the ratio of the circumference of the first to the circumference of the second. Or, The ratio of one circle to another equals the square of the ratio of two homol- ogous (like) lines. Give the ratio of : A circle of radius 1 ft. to one of radius 2 ft. A circle of diameter 1 ft. to one of diameter 2 ft. A circle of circumference 1 ft. to one of circumference 2 ft. A circle of radius 2 ft. to one of radius 8 ft. The ratio of the radii is \. Then the ratio of the circles is (i) 2 > or r V 78 THE STUDY OF SUBFACES A circle of radius 2 ft. to one of radius 10 ft. A circle of radius 2 ft. to one of radius 12 ft. A circle of diameter 5 ft. to one of diameter 30 ft. The ratio of the diameters is }. Then the ratio of the circles is G) 2 , or *V A circle of circumference »4 ft. to one of circumference 10 ft. The ratio of the circumferences is f. Then the ratio of the circles is (f ) 2 , or ^. A circle of circumference 6^ ft. to one of circumference 18| ft. A circle of diameter 2^ ft. to one of diameter 12J ft. A circle of radius 3^ ft. to one of radius 20 ft. A circle of radius 2^ ft. to one of radius 18 ft. A circle of circumference 2\ ft. to one of circumference 18 ft. A circle of radius 3 ft. to one of diameter 3 ft. A circle of radius 3J ft. to one of circumference 22 ft. A circle of radius 100 ft. to one of diameter 200 ft. SECTION X INTRODUCING THE STUDY OF SOLIDS AND DEVELOPING RELATIONS 1. Think of a square block of wood, 6 in. wide, 6 in. long, and 6 in. high. What is a solid of this shape called ? How many surfaces has it ? What shape is each ? How many square inches are there in the lower face of the 6-in. cube ? How many cubic inches are there in a section of it 1 in. thick ? How many such sections are there ? How many cubic inches are there, then, in a 6-in. cube ? 2. How many cubic inches are there in a 10-in. cube ? In a 5-in. cube ? In a 12-in. cube ? In a 1-in. cube ? In a 2-in. cube ? In a 4-in. cube ? In a 3-in. cube ? 79 80 THE STUDY OF SOLIDS 3. What is the ratio of the length of the base of cube a to the length of the base of cube b? Of the width of a to the width of b? . Of the height of a to the height of b? What is the ratio of the volume of a to the volume of b? 4. What is the ratio of a cube whose length, breadth, and height are 4 in., to one whose dimensions are 2 in.? Of an 8-in. cube to a 4-in. cube? 5. Give the ratio of : A 1-in. cube to a 2-in. cube. A 2-in. cube to a 4-in. cube. 25-in. cube to a 75-in. cube. 1-in. cube to a 4-in. cube. lj-in. cube to a 3-in. cube. A A A A A A 6j~in. cube to a 13-in. cube. 37fin. cube to a 12^-in. cube. 121 £-m. A 56i-in. cube to a 25-in. cube, cube to an 18|-in. cube. A 25-in. cube to a 5-in. cube. 6. A certain box measures 2 ft. each way, inside. How many 6-in. cubes of wood can be put into the box? 12 < > c 12 > b 10 4^ ^4 8\ ""4 7. A solid which has two bases (or ends) which are alike and parallel, and whose sides are rectangles (or par- THE STUDY OF SOLIDS 81 allelograms) is called a prism, a and b are square prisms. Is c a square prism? Is a cube a prism? Is a cube a square prism? A solid like c is called a parallelepiped. 8. How many square units are in the base of a? Of b? Of c? How many cubic units are there in a? In b? Inc? What is the ratio of a to c ? Of a to b? Gib toe? 9. What is the ratio of the length of the base of a to the length of the base of c? Of the width of the base of a to the width of the base of c? Of the base of a to the base of c? Of the height of a to the height of c? If the ratio of the bases of a and c is f, and the ratio of the heights is 1, what is the ratio of the volume of a to the volume of c? Find in a similar way the ratio of a to b. Of b to c. 10. Give the ratio of a to b : a b Base Height Base Height 12 in. by 8 in., 15 in. 9 in. by 4 in., 10 in. (f x f ) 6 in. by 7 in., 16 in. 4 in. by 14 in., 32 in. 121 i n . by §\ in., 16| in. 18| in. by 121 in., 25 in. 31 J ft. by 331 ft., 100 ft. 371 ft. by 50 ft., 1331 ft. 8 ft. by 9 ft., 20 ft. 12 ft. by 18 ft., 30 ft. 3 in. by 3 in., 6 in. 3 in. by 6 in., 12 in. 18 in. by 2 ft., 5 yd. 12 in. by 3 ft., 10 yd. 18f in. by 31 ft., 6* yd. 31* in. by 7 ft., 18f yd. 40 ft. by 40 yd., 200 ft. 20 ft. by 30 yd., 66f ft. 8 ft. by 8 ft., 8 ft. 4 ft. by 4 ft., 4 ft. 11. If a chalk box measures 6 in. x 4 in. x 3 in. outside, how many can be packed in a box 4 ft. x 3 ft. x 2 ft. ? MCN. MENT. AR. 6 82 THE STUDY OF SOLIDS 12. How many times as much air is there in a room 30 ft. x 20 ft. x 15 ft., as in a room 20 ft. x 25 ft. x 18 ft.? 13. How many cubic feet of water are there in a tank 16 ft. long, 4 ft. wide, and 2 ft. deep, when it is half full? In a tank 32 ft. long, 4 ft. wide, and 3 ft. deep, half full? 14. How many 3-in. cubes of wood can be put into a box 14 in. long, 11 in. wide, and 10 in. deep? Can the box be filled? Why? a 15. Prisms whose bases are triangles are called triangular prisms. The base of the base of prism a is 5 units, and its altitude 4 units. What is its area? If its area is 10 square units, how many cubic units are there in a section 1 unit thick, cut off the end of the prism ? How many such sec- tions are there in a, if the altitude is 8 units ? How many cubic units are there in a? How many cubic units are there in a triangular prism 8 units high, the triangle of the base having a base of 5 units, and an altitude of 4 units? 16. What is the volume of prism 5, which is 8 units high and whose base has a base of 10 units, and an altitude of 3 units ? THE STUDY OF SOLIDS 83 17. Give the volume of the following triangular prisms : Height 18 in., base of base 5 in., altitude of base 4 in. Height* 20 in., base of base 12 in., altitude of base 6 in. Height 15 in., base of base 20 in., altitude of base 10 in. Height 100 in., base of base 50 in., altitude of base 4 in. Height 30 ft., base of base 121 ft., altitude of base 8 ft. 18. In the above figures, what is the ratio of the base of the base of prism a, to the base of the base of prism b? What is the ratio of the altitude of the base of a to the altitude of the base of b ? Of the area of the base of a to the area of the base of b ? What is the ratio of the height of a to the height of b ? If the ratio of the bases is J, and the ratio of the heights is 1, what is the ratio of the volume of a to the volume of b ? 19. Give the ratio of prism a to prism b ; of c to d; of e to/; of g to h; of i to j. a. Base of base 12 in., altitude 8 in., height 15 in. b. Base of base 9 in., altitude 4 in., height 10 in. c. Base of base 6 in., altitude 7 in., height 16 in. d. Base of base 4 in., altitude 14 in., height 16 in. e. Base of base 3 in., altitude 6 in., height 6 in. /. Base of base 3 in., altitude 6 in., height 12 in. g. Base of base 18| in., altitude 3-| ft., height 6 \ yd. h. Base of base 31^ in., altitude 7 ft., height 18| yd.- i. Base of base 31^ in., altitude 33 J in., height 100 in. y. Base of base 37 J in., altitude 50 in., height 133 J in. 84 THE STUDY OF SOLIDS 20. A circular solid that has length is called a cylinder. A round pencil, or a stove pipe is a cylinder. What is 32 24 24 the shape of the end of a cylinder if cut off square ? How can you find the area of the end or base of a cylinder if you know its radius ? If you know its diameter ? 21. If the diameter of the base of cylinder a is 14 units, what is the area of the base ? If there are 154 square units in the base, how many cubic units are there in a sec- tion of the cylinder 1 unit thick ? How many such sec- tions are there if the height of the cylinder is 32 units ? Are there, then, 32 x 154 cubic units in cylinder a ? 22. The volume of a cylinder is represented by the number of units in the height times the number of square THE STUDY OF SOLIDS 85 units in the area of the base. What is the volume of a cylinder of radius 7 in. and height 10 in. ? (1540 cu. in.) Of a cylinder of radius 3^ in. and height 20 in. ? Of a cylinder.of radius 1 ft. and height 1 ft. ? Of a cylinder of radius 1 yd. and height 2 yd. ? 23. In the above figures, what is the ratio of the diameter of a to the diameter of c ? Of the area of the base of a to the area of the base of c ? (See problem 37, page 74.) Of the height of a to the height of c ? If the ratio of the bases is 4, and of the heights is |, what is the ratio of the volume of cylinder a to the volume of cylinder c ? The ratio of one cylinder to another equals the product of the ratio of the heights by the square of the ratio of the diameters or radii. 24. Give the ratio of cylinder a to cylinder b: a Diameter 14 in. height 32 in. u Diameter 7 in. height 24 in. Diameter 18 ft. height 50 ft. Diameter 24 ft. height 75 ft. Diameter 12J ft. height 83± ft. Diameter 37| ft. height 16| ft. Diameter 56£ ft. height 41§yd. Diameter 31£ ft. height 75 yd. Diameter 16| yd. , height 66f ft. Diameter 66| yd.. , height 83$ ft. Diameter 18f rd., height 20 rd. Diameter 12| rd. height 25 rd. Radius 6 iii. height 2 ft. Diameter 3 in. height 4 ft. Radius 12 in. height 4 ft. Diameter 2 ft. height 3 yd. Radius 2£ ft. height 100 in. Diameter 12| ft. height 20 in. Radius 2i in., height 40 ft. Diameter 18 in. height 5 ft. Radius 5 ft. height 72 yd. Diameter 30 ft. height 2 yd, Radius 32 in. height 45 in. Diameter 40 in.. , height 72 in. 25. How many cubic feet of water will a cylindrical tank 2 ft. in diameter and 7 ft. high hold ? 86 THE STUDY OF SOLIDS 26. A cylindrical pail 8 in. in diameter and 12 in. deep holds how many times as much as a pail 6 in. in diameter and 8 in. deep ? 27. How many gallons will a cylindrical cistern 15 ft. deep hold if the bottom has an area of 30 sq. ft., and it is half full ? (See problem 42, page 60.) 28. A solid that has a base of any number of sides, and whose other surfaces are triangles meeting at a common point at the top (the apex of the pyramid) is a pyramid. The altitude is the height, or usually the shortest distance from the apex to the base. Figure a represents a pyramid with a rectangular base. Figures b and e represent rect- angular pyramids inside of prisms of base and height equal to those of the pyramids. Figure d represents a triangular pyramid inside a prism of equal base and height. The volume of a pyramid is represented by \ the product of the number of square units in its base by the number of units in its altitude. What is the volume of pyramid a, THE STUDY OF SOLIDS 87 if its base is 3 units by 2 units, and its altitude is 8 units? Of pyramid b, if its base is 4 units by 3 units, and its altitude is 8 units ? Of pyramid