1. How many sheets in 10 bundles of paper?

2. If paper is $6 a ream, what does it cost a quire?

3. A bookseller bought 10 reams of paper, at $21 a ream; he retailed it at 1 cent a sheet. What was his gain?

Ans. $23.

4. How many reams of paper will be needed for 1000 books, if each book requires a dozen sheets? Ans. 25 reams.

5. If a score of boys have each 5 boxes of pens, containing a gross apiece, how many pens have they in all?

6. A tailor uses 13 dozen buttons out of a great gross; how many buttons has he left?

7. If a stationer manufactures 48 dozen copy-books a day, excluding Sundays, how many great gross will he make in fiftytwo weeks? Ans. 104 great gross.

Reduction of Denominate Fractions,

Common and Decimal.

278. A Common Fraction or Decimal is called Denominate when it is used in connection with a denomination; as, £, .25 oz.

279. Denominate Fractions, whether common or decimal, are reduced, like integers, to lower denominations by multiplication, to higher denominations by division.

277. Recite the Table relating to collections of nnits.-278. When is a common fraction or decimal called denominate?-279. How are denominate fractions reduced to lower denominations? To higher denominations?

280. CASE I.-To reduce one denominate fraction to another of a lower denomination.

EXAMPLE.-Reduce gall. to the fraction of a gill.

RULE.-Multiply the given fraction by the number or numbers that connect its denomination with that of the required fraction.

1. Reduce

of a bushel? 68000 acre?

too ton to the fraction of an oz. 2. Reduce £1 to the fraction of a penny. 3. What fraction of a pint is 4. What part of a sq. foot is 5. What part of an inch is go 6. What part of a second is

7. What part of a quire is

40000 of a week?

of a mile?

of a bundle of paper?

8. Reduce of a pound to the fraction of a scruple.

1944

281. CASE II.-To reduce a denominate fraction to whole numbers of lower denominations.

EXAMPLE.-Reduce

to pecks, &c.

of a bushel

2

4

To reduce bushels to pecks, multiply by 4. Multiplying the numerator of the fraction by 4 and dividing the product by its denominator, we get 2 pk. Reduce the fraction, pk., to quarts. Multiplying its numerator by 8 and dividing by its denominator, we get 5 qt. Reduce the fraction, qt., to pints. Multiplying its numerator by 2 and dividing by its denominator, we get 3 pt. Collect the integers in the several quotients, and the last fraction, for the answer.

3)8

2 pk. | 2 rem.
8

3)16

5 qt. 1 rem.

2

3)2

0% pt.

Ans. 2 pk. 5 qt. pt.

280. What is the first Case of the reduction of denominate fractions? Solve the given example. Recite the rule.-281. What is Case II.? Go through the given example.

RULE.-Multiply the numerator of the given fraction by the number that will reduce it to the next lower denomination, and divide by its denominator. If there is a remainder, multiply and divide it in the same way; and proceed thus to the lowest denomination. Collect the integers and the last fraction, if any, for the answer.

EXAMPLES FOR PRACTICE.

Reduce the following to integers of lower denominations:

11. How many acres, &c., in a piece of land mile long and of a mile wide?

Ans. 142 A. 35 sq. rd.

Area = } × ¦ = ? sq. mi. Reduce 3 sq. mi. to acres, &c.
12. Required the solid

1 yd. wide, yd. thick.

contents of a block of stone, 24 yd. long,

Ans. 1 cu. yd. 21 cu. ft. 10364 cu. in.

282. CASE III.-To reduce one denominate fraction to another of a higher denomination.

EXAMPLE. Reduce of a gill to the fraction of a gallon.

This is a case of Reduction Ascending. Divide the fraction: that is, mul tiply its denominator by 4 (since 4 gi. = 1 pt.); by 2 (2 pt. 1qt.); by 4 (4 qt. =1 gall.). Cancel 2; multiply the remaining factors.

2

=

1

7 × 4 × 2 × 4 112 Ans. The gall.

Under Case I. we reduced a gall. to gill.

Here we have reduced

Recite the rule for reducing a denominato fraction to whole numbers of lower denominations.-292. What is Case III.? Solve the given example. How may it be

proved?

gill to The gall.

each other.

Hence the operations in Case I. and Case III. prove

RULE.-Divide the given fraction by the number or numbers that connect its denomination with that of the required fraction.

EXAMPLES FOR PRACTICE.

1. Reduce of a rod to the fraction of a league. Ans. lea. 2. Reduce pt. to the fraction of a puncheon. Ans. 1440 pun.

3. Reducefathom to the fraction of a mile.

Ans. 24 mi.

8. What part of a circle is § of a second?

9. What part of a piece of 40 yards is a nail of cloth?

1 nail = yd. 16×40 = 6 Ans.

10. What part of 20 gallons is 1 of a pint?

Ans. 176.

Ans.

11. What part of a five-acre lot is of a perch? 12. What part of the month of Aug. is 7 min.? Ans. 580320.

283. CASE IV.-To reduce one denominate number to the fraction of another.

EXAMPLE I.-Reduce 16s. 6d. 2 far. to the fraction of a pound.

Reduce 16s. 6d. 2 far. to farthings, the

lowest denomination mentioned:

Reduce £1 to the same denomination: 794 far. = of 960 far.

16s. 6d. 2 far.

794 far.

960 far.

Reduce this fraction to its lowest terms.

£ t = £87 Ans.

EXAMPLE II.-Reduce 20 rods 24 yards to the fraction of a mile.

If the lowest denomination given contains, we must reduce both numbers to halves of that denomination; if it contains thirds, to thirds,

Give the rule for reducing a denominate fraction to a higher denomination.— 253. What is Case IV.? Solve Example I. If the lowest denomination given contains, what must we do? If it contains thirds, wha's must we do? Illustrate this with Example II.

REDUCTION OF DENOMINATE DECIMALS.

163

&c. In this example, for instance, we must reduce both numbers to halfyards.

20 rd. 24 yd. = 225 half-yards.

1 mile 3520 half-yards.

mile Ans.

RULE.-Reduce the given numbers to the lowest denomination in either. Of the numbers thus reduced, take the one of which the fraction is required for the denomi nator, and the other for the numerator.

EXAMPLES FOR PRACTICE.

Reduce the following; give the fraction in its lowest terms:

7. 1 English ell to the fraction of 1 French ell.

Reduce both to the common denomination, quarters.

8. What part of 1 ch. 501. is 44 inches?

9. What part of 6s. 84d. is 3s. 5d.?

Ans. § ell Fr.

10. Reduce 5 hours to the fraction of a leap year.

Ans.
Ans. 19.

284. CASE V.-To reduce a denominate decimal to whole numbers of lower denominations.

EXAMPLE.-Reduce .471875 lb., apothecaries' weight, to ounces, &c.

This is a case of Reduction Descending. Multiply by 12, to reduce to ounces, pointing off the product as in multiplication of decimals. Reserve the integer, and reduce the decimal to drams by multiplying by 8. Again reserve the integer, and reduce the decimal to scruples by multiplying by 3. There being no integer, multiply this product by 20 to reduce it to grains. Finally, collect the integers in the several products for the answer.

.471875 lb. 12

oz. 51.662500

8

3

sc. .900000

20

dr. 5.300000

gr. 18.000000

Ans. 5 oz. 5 dr. 18 gr.

Recite the rule for reducing one denominate number to the fraction of another.284. What is Case V.? Go through the given example, explaining the steps.