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REDUCTION ASCENDING.

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, OPERATION. therefore of the number of farthings equals the 3x1x1=1

xix1=74 number of pence: there are 12 pence in one shilling, therefore iz of the number of pence equals the number of shillings; hence À far. equals exèx12= 4 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. Ans. 238. 3. 315 sq. in. to the fraction of a sq. rd. Ans. Ta12 4. 8 in. to the fraction of a mile. Ans. 30788. 6. of em to the fraction of a Cong. Ans. 1228806. 4}x 64 cu. in. to the fraction of a cord. Ans. 7313304 7. What part of a cable length is 1 link? Ans. 12000.

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.

OPERATION. 6 pwt. 16 gr. equals 4480 gr.; and also 1 lb.

oz. pwt. gr. equals 5760 gr.; 1 gr. equals 5767 of a 9 6 16=4480 gr. pound, and 4480 gr.equals 4480 times 376,

1 lb.=5760 gr. which equals $488, which reduced to its

1486=, Ans. lowest terms, equals. Therefore, etc.

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. 21 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. 112 oz. ? Ans. 4. 6. What part of 9 feet square is 9 sq. ft. ? Ans. j. 7. What part of 10 ft. square is 4 ft. square and 4 sq. ft. ?

Ans. } - 8. Reduce 114 P. 8 sq. yd. 5 sq. ft. 1134 sq. in. to the fraction of an acre.

dns. 5. . 9. What part of a bhd. does a box 7 in. long, 51 in. wide, and 3 in. deep contain ?

Ans. The

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 OPERATION. the number of gills equals the number of pints ;

412.00 of 2 equals .5, which with the 1 pt., equals 1.5 pt.:

211.50 there are 2 pt. in 1 qt., hence } of the number of pints equals the number of quarts ; 1 of 1.5 equals 4 3.75 .75, which with 3 qt. equals 3.75 qt.: there are .9375, Ans. 4 qt. in 1 gal., hence of the number of quarts equals the number of gallons ; 1 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 £.

Ans. £.675. 3. Reduce 2 cwt. 59} lb. to tons. Ans. .1296 ton.

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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 £j, and of a shilling.

OPERATION SOLUTION.-£3, we find by Art. 452, equals 13 s. £=13 s. 4d. 4 d.; and is. equals 9 d. Taking the sum of 13 s.

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 4 bu. and į bu. Ans. 1 bu. 1 pk. 6 qt. 113 pt. 3. Of oz. and { lb.

Ans. 11 oz. 6 pwt. 16 gr. 4. Of s mi. and frd. Ans. 214 rd. 1 yd. 54 in. 5. Of .325 gal. and į gal.

Ans. 3 qt. 114 pt. 6. Of $}, £j, and į of a franc.

Ans. $4.039z. 7. Of .678 A. and s rood.

Ans. 144 P. 1 sq. yd. 97.92 sq. in. 8. Of į wk. į 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 i of a shilling from of a pound.

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

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

is.= 9 d. 7d. Therefore, etc.

12 s. 7 d.

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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 £f by 7. SOLUTION.-7 times £} equals £21, which

OPERATION. by reduction we find equals £5 5 s. There-. £3X7= £21=£5 5s. 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 1b. 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 Ans. 34 P. 17 yd. 8 ft. 12644 in. 8. 12.16 C. by 7.01.

Ans. 85 C. 10 S. 3° 20'. 9. If a man dig 6 yd. of ditch in 1 day, how much would be dig in 7 days?

Ans. 9 rd. 2 yd. 2 ft. 54 in.

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

1. Divide {bu. by 5. SOLUTION.- of of a bushel

OPERATION. equals o of a bushel, which, by bu. x1=7 bu.=5gt. 14 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 3.434 da. by .18%.

Ans. 18 da. 8 h. 5. Divide 6.3 mi. by 44. Ans. 1 mi. 106 rd. 3 yd. 2 ft. 6. Divide 2.7 bu. by .436. Ans. 6 bu. 1 pk. 40909 pt. 7. Divide 12.16 wk. by . 3. 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 denominate integer or fraction. 1. Divide 13} cwt. by 16 lb.

OPERATION. SOLUTION.-1}{cwt. equals 144 lb.; 111 cwt.= 144 lb. dividing 144 lb. by 16 lb. we have for a xà lb. - 16 lb. = 9. Ans. quotient 9. Hence the following

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