+ no more equals 12, or 4, than 11 3 +53

28+, just as 23 lbs. + 21 lbs.

=

= 44 lbs.

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

or 4 pt.

16 3.

=175 apples.

5 lbs.

19/

$58

1, just as 15 rods

=

159 — 187–114, just as 358 apples - 183 apples

Again. — no more equals, or, than 3 pints pints, or 2 gills.

1 pear.

no more equals, or, than 5 apples 4 pears 1 apple, or

12483

1728, just as 9 times 1387 cubic inches = 12483

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

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

Tare contained 18 times in 12, just as 7 apples are contained 18 times in 129 apples.

=2147, just as of 1484 miles 1648 miles.

Reduction of Whole or Mixed Numbers to Improper
Fractions.

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

Solution. Since 1 = 7, 9 must equal 9 times 7, or 3, and added are. Therefore, 94 = 5.

3. Reduce 4 to thirds.

4. Reduce 5 to thirds.

5. Reduce 6 to eighths. 6. Reduce 8 to eighths.

7. Reduce 9g to fifths.

8. Reduce 93 to fourths.

9. Reduce 439 to fourths.

First Solution. - Since 14 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, 439 = 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, 4391759.

NOTE.

85th page.

Compare these reasoning processes with those given on the

10. Reduce 5785 to ninths.

11. Reduce 147625 to sevenths.

12. Reduce 74968 to seventeenths.

13. Reduce 64832 to forty-sevenths.

14. Reduce 43861 to twenty-eighths.

15. Reduce 427 to two hundred and seventy-thirds. 16. 9 how many times ? *

First Solution. Since 1 =

,9 must equal 9 times 4, or 36, and 1 fourth is contained in 36 fourths 36 times. Hence, 9 = 36 times 4. Second Solution. Since 14 times, or 4, 9 must equal 9 times 4 times, or 36 times. Hence, 9 = 36 times .

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

=

to ones.

Since 1,43 must equal as many ones as there are times 8 in 43, which are 5 times. Hence, 3 = 58.

NOTE. The preceding question is precisely like this: "43 qts. = how many pecks?" as the following solution will show. "Since 8 qts. = = 1 pk., 43 qts. will equal as many pecks as there are times 8 in 43, which are 5 times. Hence, 43 qts. = 53 pks."

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 of a dollar, equivalent to 3 times 227 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 of a dollar per bu.?

4. What will 347 gal. of burning fluid cost at 48 of a dollar per gal.?

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

6. How many acres in 3 lots, each containing
7. How many quarts in 9 boxes, each holding

7 acres? of a

quart?

8. How many pounds of silver in 28 bars, each weighing

용골궁 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 4179337 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 823 pecks of corn?

Solution. Since of a peck will fill one basket, 829 pecks will fill as many baskets as there are times in 828. But 82 659, which contains as many times as 659 contains 7, which is 944 times. Hence, 94 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 94 times. Hence, 944 baskets can be filled.

15. How many vest patterns, each containing can be cut from a piece of vesting containing 25

of a yard, yards?

16. How many books, at of a dollar a piece, can be bought for 232 dollars?

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

18. How many barrels, holding 21 bushels each, can be filled from 593 bushels of apples?

ing 2 gal. 1 qt. each, can be filled from 59 gal. 3 qt. of fluid?"

19. How many spoons, each weighing 73 dwt., can be made from 3287 dwt. of silver?

20. How many yards of cloth, at 21 dollars per yard, can be bought for 892 dollars?

21. How many days, at $2 per day, must a man labor to earn $2371?

22. A man, who owned 247 into spoons, each weighing 41 it make?

dwt. of silver, had it made

dwt.

How many spoons did

23. What is the sum of ++&+&+¿1⁄2 + $?* ? Ans. 13 = 372.

NOTE. Compare the above example with 11 in. + 7 in. + 9 in. + 4 in. +7 in. + 5 in. 43 in. =

3 ft. 7 in.

24. Of ++¿+&+§?

25. Of++ √3 + 1s + ♬ +19 + 13 ? 26. Of +8 + 38 + 13 +33 +33 + ¿§ 27. Of 1313§ + 24 + 229 + 434 ?

NOTE. Compare the 27th example with the following:-13 w. 3 da.+ 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.

28. Of 17+18+231 + 42,77 + 2710?

29. Of 247 +963 + 14§ + 261 + 557?

* These, like the compound numbers of 56, can best be reduced as they are added; thus, 12+12 (by adding one of the to) = 1; and 1 (by adding 3 of the to the 1) = 2131⁄2; &c.

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