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= 4 qts.,

3 += 44, just as 23 lbs. + 21 lbs. = 44 lbs.

Again. * + $ no more equals 4, or $, than 3 qts. + 1 pt. = or 4 pt.

ii tot no more equals 1., or &, than 11 3 + 53 = 16%, or 16 3.

Again. $ - 5 š, just as $8 - $3 = $5, or 8 lbs. — 3 lbs. = 5 lbs.

1 = 17, just as 15 rods 13 rods = 2 rods. 1-494 - 124, just as 358 apples 183 apples = 175 apples. Again. - $ no more equals 4, or , than 3 pints

1 gill

2 pints, or 2 gills.

- no more equals 7, or g, than 5 apples - 4 pears = 1 apple, or

1 pear.

24

Again. 5 times i = 1, just as 5 times 3 apples = 15 apples.
8 times = is, just as 8 times 11 oz. = 88 oz.
9 times 1:27

1729, just as 9 times 1387 cubic inches = 12483 cubic inches.

Again. 4 are contained 3 times in 6, just as 2 days are contained 3 times in 6 days.

are contained 7 times in 1, just as 3 grains are contained 7 times in 21 grains.

17+ are contained 182 times in 141, just as 7 apples are contained 184 times in 129 apples. Again. 4 of g

s, just as į of 8 apples = 2 apples. of 1 = 31°, just as } of 24 bushels 4 bushels.

1648 f of 1144=214-9, just as ģ of 1484 miles 1649 miles.

126. Reduction of Whole or Mixed Numbers to Improper

Fractions. The value of any whole or mixed number may be expressed by an improper fraction.

1. Reduce 8 to fifths. Solution. Since 1 §, 8 must equal 8 times , or 4 Therefore,

- 40 2. Reduce 99 to sevenths.

Solution. — Since 1 = 7, 9 must equal 9 times 7, or 62, and added are .

Therefore, 94 = 63 3. Reduce 4 to thirds. 1 4. Reduce 5ş to thirds.

8 =

5. Reduce 65 to eighths. 8. Reduce 9 to fourths. 6. Reduce 8 to eighths. 9. Reduce 4391 to fourths. 7. Reduce 93 to fifths.

First Solution. - Since 1 = 4 fourths, 439 must equal 439 times 4 fourths, which is equivalent to 4 times 439 fourths, or 1756 fourths, and 3 fourths added are 1759 fourths. Therefore, 4394 = 1759.

Second Solution. -- Since there are 4 fourths for each unit, there must be 4 times as many fourths in any number as there are units. Hence, 439 = 4 times 439 fourths, or 1756 fourths, and 3 fourths added are 1759 fourths. Hence, 439 = 1789.

Note. Compare these reasoning processes with those given on the

85th page.

10. Reduce 5783 to ninths. 11. Reduce 147625 to sevenths. 12. Reduce 74968 to seventeenths. 13. Reduce 64833to forty-sevenths. 14. Reduce 438613 to twenty-eighths. 15. Reduce 427] to two hundred and seventy-thirds. 16. 9 = how many times 1 ? *

First Solution. — Since 1 = 4, 9 must equal 9 times 4, or 36, and i fourth is contained in 36 fourths 36 times. Hence, 9 = 36 times 1.

Second Solution. Since 1 = 4 times ), or 4, 9 must equal 9 times 4 times ], or 36 times 1. Hence, 9 = 36 times 1.

17. 7 = how many times ] ?-
18. 11 = how many times ? ?
19. 49 = how many times ? ?
20. 487 = how many times as ?
21. 9 = how many times to ?
22. 7= how many times .1 ?
23. 13 = how many times .01 ?
24. 648 =

how
many

times .0001 ?

127. Reduction of Improper Fractions to Whole or Mixed

Numbers. The value of any improper fraction may be expressed by a whole or mixed number.

It will be seen by the solutions that this question is just equivalent to .9 = how many fourths ? "

1. Reduce 4 to ones.

Since : 1,43 must equal as many ones as there are times 8 in 43, which are 5š times. Hence, 42 = 5. NOTE. -- The preceding question is precisely like this : "43 qts. =

“ Since 8 qts. how many pecks?” as the following solution will show.

1 pk., 43 qts. will equal as many pecks as there are times 8 in 43, which are 5 times. Hence, 43 qts.

58 pks." 2. Reduce s to ones.

8. Reduce i to ones. 3. Reduce 4 to ones.

9. Reduce 1 to ones. 4. Reduce 189 to ones.

10. Reduce 109 to ones. 5. Reduce 427 to ones.

11. Reduce 82926 to ones. 6. Reduce Zeps to ones. 12. Reduce 27946 to ones. 7. Reduce 173 ; to ones. 13. Reduce 49844 to ones.

128. Miscellaneous Questions involving Fractions. 1. How much will 227 yards of cloth cost at of a dollar per yard ?

Solution. Since 1 yard costs of a dollar, 227 yards will cost 227 times it of a dollar, equivalent to 3 times 247 of a dollar, which, found by multiplying 227 by 3, is 61 of a dollar, equal (by 127) to 1701 dollars.

NOTE. — Compare the preceding example and solution with the following:

How many bushels of grain in 227 bags, each holding 3 pecks ?

Solution. - Since 1 bag holds 3 pecks, 227 bags will hold 227 times 3 pecks, equivalent to 3 times 227 pecks, which, found by multiplying 227 by 3, is 681 pecks, equal to 170 bushels, 1 peck. 2. What will 493 lbs. of tea cost at of a dollar

per

lb. ? 3. What will 1286 bu. of apples cost at at of a dollar

per bu. ?

4. What will 347 gal. of burning fluid cost at få of a dollar per gal.?

5. How many acres in 169 house lots, each containing it of an acre ?

6. How many acres in 3 lots, each containing 43627 acres ?

7. How many quarts in 9 boxes, each holding is of a quart?

8. How many pounds of silver in 28 bars, each weigling $473 of a pound ?

9. How many tons of hay in 37 loads, each containing 1278 of a ton ?

10. How much will 7 acres of land cost at $275, per

acre ?

NOTE. — Compare the last example with this : “ How much corn in 7 bins, each holding 275 bushels, 3 pecks ?

11. How much will 6 acres of land cost at $4937 per acre ?

12. If a ship sails uniformly at the rate of 17911 miles per day, how far will she sail in 5 days ?

13. If a steamship sails uniformly at the rate of 4179347 rods per hour, how many rods will she sail in 85 hours ?

14. How many baskets, each holding of a perk, can be filled from 82% pecks of corn ?

Solution. — Since } of a peck will fill one basket, 82% pecks will fill as many baskets as there are times ] in 82g. But 823 639, which contains & as many times as 659 contains 7, which is 944 times. Hence, 944 baskets can be filled.

Note. — Compare the above question and solution with the following: “How many baskets, each holding 7 quarts, can be filled from 82 pk. 3 qt. of corn?”

Solution. — Since 7 quarts will fill one basket, 82 pk. 3 qt. will fill as many baskets as there are times 7 qt. in 82 pk. 3 qt. 82 pk. 3 qt. 659 quarts, which contains 7 quarts as many times as 659 contains 7, which is 944 times. Hence, 944 baskets can be filled.

15. How many vest patterns, each containing i of a yard, can be cut from a piece of vesting containing 254 yards ?

16. How many books, at of a dollar a piece, can be bought for 2324 dollars ?

17. How many baskets, each holding 33 of a bushel, can be filled from 4537 bushels of apples?

18. How many barrels, holding 24 bushels each, can be filled from 594 bushels of apples ?

NOTE. Compare the last question with, “How many jugs, hold. ing 2 gal. 1 qt. each, can be filled from 59 gal. 3 qt. of Auid ? ”

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19. How many spoons, each weighing 75 dwt., can be made from 3287 dwt. of silver ?

20. How many yards of cloth, at 23dollars per yard, can be bought for 893dollars ?

21. How many days, at $235 per day, must a man labor to earn $23713?

22. A man, who owned 247dwt. of silver, had it made into spoons, each weighing 414 dwt. How many spoons did it make ?

23. What is the sum of H+ totetut 1a + ?* Ans. 14 = 317

Note. Compare the above example with 11 in. + 7 in. + 9 in. + 4 in. + 7 in. + 5 in. = 43 in. = 3 ft. 7 in.

What is the sum
24. Of +$+++?
25. Of it + 1} + 15 + + + 15 + 13?
26. Of if + f + 3 + 4 + 4 + 2 + 3?
27. Of 133 + 13 + 24 + 224 + 434 ?

NOTE. Compare the 27th example with the following: 3 da. t 13 w. 5 da. + 2 w. 4 da. + 22 w. 6 da. + 43 w. 2 da. = 95 w. 6 da. To make the resemblance to compound addition still more apparent, we give two forms of writing the work of the 27th example, and place opposite them the work of the example in this note.

Ones. Sovenths.
134 = 13 3
134 = 13 5
24 2

4
229 = 22 6
434

43 2
95$ = 95
6 = Ans.

6= Ans. What is the value 28. Of 175 + 1841 + 234 + 4211 + 2711 ? 29. Of 24} +963 + 145 + 26} + 557?

13 w.

da.

W.
13
13

3

5

2

4

22

6

43

2

95

* These, like the compound numbers of 56, can best be reduced as they are added ; thus, 11 + is (by adding one of the 3 to 13) = 1 and 112 + fa (by adding 3 of the to the 2) = 2; &c.

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