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reduce the decimal as before, and thus continue as far as necessary.

What is the value
2. Of.9375 of a gal.?

Ans. 3 qt. 1 pt. 2 gi. 3. Of .1296 of a ton ?

Ans. 2 cwt. 59 lb. 3.2 oz. 4. Of.6783 lb. Ap.?

Ans. 83 13 9.6 gr. 5. Of 1.426 of a day? Ans. 1 da. 10 h. 13 min. 26.4 sec. 6. Of .2845 of a year? Ans. 103 da. 20 h. 13 min. 12 sec. 7. Of.8469 deg. ? Ans. 58 mi. 182 rd. 5 yd. .82944 in.

REDUCTION ASCENDING.

OPERATION.

CASE I. 454. To reduce a common denominate fraction to a common fraction of a higher denomination.

1. Reduce of a farthing to the fraction of a shilling.

SOLUTION.—There are 4 farthings in a penny, therefore of the number of farthings equals the

xix1=74 number of pence: there are 12 pence in one shilling, therefore 1 of the number of pence equals the number of shillings; hence i far. equals èx&x1=of a shilling.

Rule.-Express the division by the required divisors, and reduce by cancellation.

Reduce
2. of a gill to the fraction of a gallon.
3. 315 sq. in. to the fraction of a sq. rd.

Ans. 04152 4. 8 in. to the fraction of a mile.

Ans. 70688 6. of im to the fraction of a Cong.

Ans. 122880 6. 4}x 63 cu. in, to the fraction of a cord. Ans. 9813304 7. Wbat part of a cable length is 1 link ?

Ans. 12000

Ans. 238

CASE II. 455. To reduce a compound number to a common fraction of a higher denomination.

1. Reduce 9 oz. 6 pwt. 16 gr. to the fraction of a pound. SOLUTION.- By reduction we find 9 oz. 6 pwt. 16 gr. equals 4480 gr.; and also 1 lb.

oz. pwt. gr. equals 5760 gr.; 1 gr. equals 5760 of a

9 6

16=4480 gr. pound, and 4480 gr. equals 4480 times 3760,

1 lb. which equals 488, which reduced to its

$448= , Ans. lowest terms, equals. Therefore, etc.

OPERATION.

5760 gr.

Rule.- Reduce the number to its lowest denomination, and write under it the number of units of this denomination which make a unit of the required denomination; and then reduce the resulting fraction to its lowest terms.

Reduce 2. 213 rd. 1 yd. 22 ft. to the fraction of a mile. Ans. . 3. 13 quires 8 sheets to the fraction of a ream. Ans. . 4. What part of a bu. of wheat is 52 lb. 8 oz.?

Ans. 5. What part of a quintal is 85 lb. 11% oz.?

Ans. 4. 6. What part of 9 feet square is 9 sq. ft. ?

Ans. 7. What part of 10 ft. square is 4 ft. square and 4 sq. ft. ?

Ans. }

acre.

8. Reduce 114 P. 8 sq. yd. 5 sq. ft. 1134 sq. in. to the fraction of

dns.. 9. What part of a bhd. does a box 7 in. long, 54 in. wide, and 3 in. deep contain ?

Ans. 6

OPERATION.

CASE III. 456. To reduce a compound number to a decimal of a higher denomination.

1. Reduce 3 qt. 1 pt. 2 gi. to the decimal of a gallon.

SOLUTION.—There are 4 gi. in 1 pt., hence of the number of gills equals the number of pints ; $

412.00 of 2 equals .5, which with the 1 pt., equals 1.5 pt.: there are 2 pt. in 1 qt., hence } of the number of

21.50 pints equals the number of quarts ; } of 1.5 equals 4 3.75 .75, which with 3 qt. equals 3.75 qt.: there are

.9375, Ans. 4qt. in 1 gal., hence & of the number of quarts equals the number of gallons ; ; of 3.75 equals .9375; hence 3 qt. 1 pt. 2 gi. equals .9375 gal.

Rule.-1. Divide the lowest term by the number of units which equals one of the next higher, and annex the decimal quotient to the integer of the next higher denomination.

II. Proceed in a similar manner until the whole is reduced to the required denomination.

NOTE. It may also be done by reducing to a common fraction, and the common fraction to a decimal.

2. Reduce 13 s. 6 d. to £.
3. Reduce 2 cwt. 597 lb. to tons.

Ans. £.675.
Ans. .1296 ton.

4. Reduce 93 13 29 8 gr. to lb.

Ans. .768 lb. 6. Reduce 12 cwt. 87 lb. 3} oz. to tons. Ans. .6436 ton. 6. 58 mi. 182 rd. 5 yd. .82944 in. to deg. Ans. .8469 deg. 7. 1 da. 10 h. 13 min. 26.4 sec. to days. Ans. 1.426 da. 8. 103 da. 20 h. 13 min. 12 sec. to yr.

Ans. .2845 yr.

ADDITION OF DENOMINATE FRACTIONS.

457. Addition of Denominate Fractions is the process of finding the sum of two or more denominate fractions. 1. Find the sum of £}, and of a shilling.

OPERATION. SOLUTION.-£}, we find by Art. 452, equals 13 s.

£}=13 s. 4d. 4 d.; and {s. equals 9 d. Taking the sum of 13 s.

9 d. 4 d. and 9d. we have 14 s, 1 d. Therefore, etc.

14s. 1 d. Rule.- Reduce the fraction to integers, and then add as in addition of compound numbers.

Find the sum 2. Of bu. and i bu. Ans. 1 bu. 1 pk. 6 qt. 113 pt. 3. Of oz. and } lb.

Ans. 11 oz. 6 pwt. 16 gr. 4. Of mi. and frd.

Ans. 214 rd. 1 yd. 51 in. 5. Of .325 gal. and í gal.

Ans. 3 qt. 114 pt. 6. Of $3, £s, and sof a franc.

Ans. $4.0393. 7. Of.678 A. and s rood.

Ans. 144 P. 1 sq. yd. 97.92 sq. in. 8. Of { wk. } da. 1 h. and min.

Ans. 2 da. 2 h. 30 min. 45 sec.

SUBTRACTION OF DENOMINATE FRACTIONS.

458. Subtraction of Denominate Fractions is the process of finding the difference between two denominate fractions. 1. Subtract of a shilling from of a pound.

OPERATION. SOLUTION.-£3 equals 13 s.4d.; of a shilling

£}=13s. 4d. equals 9 d.; 9 d. subtracted from 13 s. 4 d. leaves 12 s.

9 d. 7d. Therefore, etc.

12 s. 7 d.

Rule.-Reduce the fractions to integers and then subtract as in subtraction of compound numbers.

2. Subtract Ő yd. from { of a rd. Ans. 3 yd. 2 ft. 114 in. 3. j cwt. from .35 ton. Ans. 6 cwt. 33 lb. 5} oz. 4. _ 3 from 1.45 lb.

Ans. 1 lb. 43 43 13 12 gr. 5. 0. from 18 of a Cong. Ans. 5 0. 10 f3 5 f3 20 m. 6. á cu. ft. from { cd. ft. Ans. 13 cu. ft. 288 cu. in. 7. § mo. from .36 of a leap year.

Ans. 115 da. 2 h. 10 min. 5414 sec. 8..83 square

rod from .27 acre.

Ans. 42 P. 24 sq. yd. 2 sq. ft. 90 sq. in.

OPERATION.

MULTIPLICATION OF DENOMINATE

FRACTIONS. 459. Multiplication of Denominate Fractions is the process of finding the product when the multiplicand is a denominate fraction.

1. Multiply £} by 7. SOLUTION.—7 times £{equals £4, which by reduction we find equals £5 5s. There

£1x7= £4=£55 s. fore, etc.

Rule.- Multiply the denominate fraction by the given multiplier and reduce the product to integers.

NOTE.-It may also be done by first reducing the fraction to integers and multiplying as in multiplication of denominate numbers. 2. Multiply lb. Troy by 11.

Ans. 7 lb. 4 oz. 3. Multiply .82 yd. by 13. Ans. 10 yd. 1 ft. 11.76 in. 4. Multiply 2 lb. by 3.25.

Ans. 9 Ib. 43 13. 5. .978 mi. by 3.45. Ans. 3 mi, 29 ch. 92 li. 6.336 in. 6. .083 A. by .35. Ans. 4 P. 20 sq. yd. 1 sq. ft. 72 sq. in. 7. .324 A. by 2

Ans. 34 P. 17 yd. 8 ft. 12634 in.

31 8. 12.16 C. by 7.01.

Ans. 85 C. 10 S. 3° 20'. 9. If a man dig 64 yd. of ditch in 1 day, how much would be dig in 7 days?

Ans. 9 rd. 2 yd. 2 ft. 51 in.

DIVISION OF DENOMINATE FRACTIONS. 460. Division of Denominate Fractions is the process of finding the quotient when the dividend is a denominate fraction.

461. There are Two Cases :
1st. To divide a denominate fraction into equal parts.

2d. To divide a denominate fraction by a denominate integer or fraction.

CASE I.

OPERATION.

462. To divide a denominate fraction into a number of equal parts.

1. Divide bu. by 5.

SOLUTION. of } of a bushel equals o of a bushel, which, by

Zbu. X}=% bu.=5qt. 1} pt. reduction, we find equals 5 qt. 1}pt. Therefore, etc.

Rule.-Divide the denominate fraction by the given divisor, and reduce to integers. 2. Divide { rd. by 3.

Ans. 2 ft. 9 in. 3. Divide .325 yr. by 8. Ans. 14 da. 19 b. 52 min. 30 sec. 4. Divide 3.43da. by .184

Ans. 18 da. 8 h. 5. Divide 6.3 mi. by 44. Ans. 1 mi. 106 rd. 3 yd. 2 ft. 6. Divide 2. 1 bu. by .436. Ans. 6 bu. 1 pk. 11099 pt. 7. Divide 12.16 wk. by .1. Ans. 3 mo. 3 wk. 4 da. 12 h. 8. Divide 4.35 P. by 14.

Ans. 2 P. 12 sq. yd. 5 sq. ft. 63 sq. in. 9. If a bird fly 20.36 mi. in 1.33} hours, how far does it fly in one hour?

Ans. 15 mi. 86 rd. 2 yd. 7.2 in.

CASE II.

463. To divide a denominate fraction by a denomi. nate integer or fraction. 1. Divide 11} cwt. by 16 lb.

OPERATION. SOLUTION.–13} cwt. equals 144 lb.;

1}} cwt. = 144 lb. dividing 144 lb. by 16 lb. we have for a

144 lb. ; 16 lb. = 9, Ans. quotient 9. Hence the following

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