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4. If 5 yards of cloth cost $21, what will 20 yards cost at the same rate ?

5. If 6 tons of coal are bought for $19}, what will 30 tons cost at the same rate ?

6. A has $20 and B has as much, plus $3; what part of A's money equals B’s money?

WRITTEN EXERCISES. What part is 2. 25 of 75 ?

Ans. s., 6. 105 of 225 ? Ans. As. 3. 18 of 54 ? Ans. $. 7. 420 of 588 ?

Ans. 4. f. 76 of 24? Ans. 4.8. 540 of 432?

Ans. I. 5. 84 of 91? Ans. If. I 9. 1899 of 2743 ?

Ans. is 10. If 16 apples cost 13 cents, what will 48 apples cost at the same rate ?

SOLUTION.-If 16 apples cost 13 cents, OPERATION. 48 apples, which are it, or 3. times 16 16 A. cost 13 cts, apples, cost 3 times 13 cents, or 39 48 A. cost 3X13 cts. cents.

= 39 cts, 11. If 15 oranges cost 14 cents, what will 60 oranges cost at the same rate ?

Ans. 56 cents. 12. What cost 125 sheep, if 25 sheep cost 137 dollars ?

Ans. $685. 13. If 56 cows cost $1000, what will 42 cows cost at the same rate ?

Ans. $750. 14. If 105 pens cost 77 cents, what will 225 pens cost at the same rate?

Ans. $1.65. NOTR.--Classes nooding a shorter course may omit the other three cases unless otherwise directed.

CASE II. 186. To And the relation of a fraction to a nurse ner.

1. The fraction f is what part of 6? SOLUTION.-One is 4 of 6, and if 1 is 3 of 6, 4 is of t, which is the of B, and f is 3 times , which is 3 of 6.

MENTAL EXERCISES. 1. What part of 3 is $? Of 4 is f? Of 7 is f? Of 9 is f? 2. What part of 5 is 1 of ? Of 7 is of ? Of 5 is of $?

3. If of a ton of coal cost $27, what will 6 tons cost at the same rate?

4. A hat was bought for $2 and sold at a gain of $f; what part of the cost equals the gain?

5. Sarah bought 5 yards of silk and used 81 yards ; what part of the silk then remained ?

6. A bought 6 oranges, and B bought 4 oranges ; what did B ay for his oranges, if A paid 26 cents for his?

WRITTEN EXERCISES.

What is the relation of 2. to 92 Ans. 1. 6. f to 64? 3. $ to 12 ? Ans. Zg. 7. to 35 ? 4. 7 to 147 Ans. zo. 8. 9 to 14 ? 6. H to 28? Ans. 3o. 19. to 35? 10. A having $25, gave $5 to B and of the remainder to C. what part of $25 then remained ?

Ans. Ans. Ans. Ans.

Ans.

CASE III. 187. To find the relation of a number to a fraction.

1. 6 is how many times ? SOLUTION.—is of , and since } is } of , or 1 is 4 times ], or $ of t, and 6 is 6 times $, or 44, which equals 8.

MENTAL EXERCISES. 1. 8 is how many times $? 14 is how many times 3? 24 is how many times ?

2. 96 is how many times 19? 115 is how many times it? 144 is how many times It?

3. If a yard of cloth cost $$, how many yards will $9 buy? 4. A can walk a certain distance in 24 hours; how many timee that distance can be walk in 6 hours?

5. A book cost $4 and a slate $$ ; how many times the cost of the slate equals the cost of the book?

WRITTEN EXERCISER, What is the relation of -. 288 to 10? Ans. 414. 5. 462 to 85? Ans. 54. 3. 495 to 43? Ans. 1124. 6. 588 to 3i? Ans. 154 4. 1896 to 44? Ans. 2133. 7. 63504 to 50%? Ans. 1260

8. Mary had 12 quarts of nuts and Sarah bad as many, minds of a quart; how many times Sarah's number equals Mary's?

Ans. 176.

CASE IV. 188. To find the relation of a fraction to a fracHon.

1. What part of His ? SOLUTION.I's is of 15, and t}, or one, iş 15 times ty, which equals tf of 1. If 1 equals if of tf, I equals ts of 1, and it equals 1 times it of 14, which equals ils of it, or f.

MENTAL EXERCISES. 1. What part of fis *? Of is ? Of is *? Of is f?

2. What part of is 21? Of is of $? Of is of 1? Of to is of 3?

3. A had $f and spent $; what part of his money did he spend?

4. If 24 yards of silk cost $7}, what will 71 yards cost at the same rate?

5. A merchant bought f of a cargo of oranges and sold of them : what part of the cargo did he sell?

6. Edwin and Emma each had $2; Edwin spent of his, and Emma spent } of hers; what part of Emma's money then equaled Edwin's ?

WRITTEN EXERCISES. What is the relation of 2. $ to V? Ans. 1. 16. 91} to 113? Ans. 1148 3. H to 7? Ans. f. 7. 111ł to 7?

Ans. 413. 4. 47 to ? Ans. t. 8. fx 1 to 4x4? Ans. 184. 5. 3 to 4 ? Ans. 9. 19. #xif to 1fx 347 Ans. ¥. 10. If a merchant sold of g of his stock, how many fifths of f of his stock remained ?

Ans. REMARK.-He sold f, and hence there remained }, and l equals trou fifths of $ of his stock; hence the answer is g. 11. Harry lost of f of a certain sum of money;

bow many sevenths of f of the sum remained ?

Ans. 12. Two boys had each to of a bushel of walnuts, when he first sold ß of his to the second; what part of what the second has equals what the first has ?

Ans. 13. James had f of a ton of coal and Thomas had g of a ton; if James burns } of his coal and Thomas 4 of his how many ninths of a ton will remain ? Ans. 67% ninths.

NOTE.-For Greatest Common Divisor and Least Common Multipla

Partinn a Manical.

ARITHMETICAL ANALYSIS. 189. Analysis is the process of solving problems by . comparison of their elements. In comparing, we reason to the unit and from the unit; the unit being the basis of the reasoning process.

CASE I. 190. To pass from one integer to another integer.

1. If 5 oranges cost 12 cents, what will 8 oranges cos at the same rate ?

SOLUTION.-If 5 oranges cost 124 or y cents, 1 orange cost š of %, which is

5 cost a cents; and if 1 orange cost of cents, 8 oranges will cost 8 times į cents, or 20 cents.

8 “ **8=20, Ans.

OPERATION.

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MENTAL EXERCISES. 1. If 4 apples cost 5} cents, what will 5 apples cost at the same rate?

2. If 3 girls earn $24 in a day, how much could 5 girls earn at the same rate ?

3. How long will it take a boy to plow 8 furrows, if it takes him 13} minutes to plow 5 furrows ?

4. How much will 9 barrels of four cost, if 2 barrels cost $101? 5. If 5 hats can be bought for $11}, how much will 8 hats cost ? 6. What cost 5 pounds of butter at the rate of $3f for 10 pounds ?

WRITTEN EXERCISES.

2. If 6 hens cost $3.54, what will 7 hens cost at the same rate?

Ans. $4.13. 3. If 8 turkeys cost $19}, what will 10 turkey cost at the Bame rate ?

Ans. $24. 4. How much must I pay for 22 yards of muslin, at the rate of 5 yards for $H?

Ans. $4.18. 5. How much will 21 books cost at the rate of 9 books for

Ans. $36. 6. What must I pay 8 arithmetics, at the rate of 5 arithmetics for $4.75?

Ans. $7.60. 7. If 3 tons of hay cost $47}, how much will 12 tons cost at the same rate ?

Ans. $1891 8. Required the cost of 21 harrels of four, at the rate of $17.50 for 3 barrels

Ans $122.50

154 ?

9. A gave $43.50 for pigs, at the rate of $7.25 for 3 pigs ; how many did he buy?

Ans. 18 pigs. 10. B gave $54 for flour, at the rate of $6 for 4f bushels; bow many bushels did he buy?

Ans. 39 bushels.

CASE U. 191. To pass from a fraction to an integer.

1. If of an acre of land cost $108, what will one acre cost?

OPERATION. SOLUTION.-If & of an acre cost $108, 4 of an

* cost $108 acre cost of $108, which is $36; and if of an acre cost $36, # of an acre, or one acre, will cost 4

3 $36 times $36, which are $144.

+ “ $144, Ans.

MENTAL EXERCISES.

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1. If of a yard of cloth is worth 24 dimes, what are 8 yards of the cloth worth?

2. If f of & barrel of sugar cost $12, what will 7 barrels cost ?

3. I bought 5 yards of summer silk at the rate of $10 for Bf yards · what was the cost ?!

4. Mary bought 9 apples, at the rate of of an apple for of a cent; what did they cost ?

5. Henry bought 25 peaches, at the rate of of a peach for of a cent; required the cost.

6. Three-Afths of what I paid for a cow was $24; what did I pay foi a horse whose value was 5 times that of the cow?

WRITTEN EXERCISES, 2. If į of the cost of A's farm is $2480, what did the farm cost ?

Ans. $3720. 3. If 4 of a barrel of flour cost $2.50, wbat will 5 barrels cost at the same rate ?

Ants. $21.871. 4. If & of a yard of cloth cost $2.10, what will enough for suit containing 9 yards cost?

Ans. $22.05. 6. If of a cwt. of sugar cost $12.45, wbat will be the cost of 20 cwt, or 1 ton of sugar ?

Ans. $298.80. 6. What will 21 tons of hay cost, if [ of a ton of bay cost $12.85 ?

Ans. $308.40. 7. If $467 is so of the price I pay for a borse, what would be the cost of 4 pairs of horses at the same rate ?

Ans. $1200

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