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OPERATION.

8) 3.0 (3 tenths.

24

8) 60 (7 hundredths.
56

8) 40 (5 thousandths.
40

Or thus:

Ans. .375.

8) 3.0 0 0

.37 5 Ans.

Since we cannot divide the numerator, 3, by 8, we reduce it to tenths by annexing a cipher, and then dividing, we obtain 3 tenths and a remainder of 6 tenths. Reducing this remainder to hundredths by annexing a cipher, and dividing, we obtain 7 hundredths and a remainder of 4 hundredths; which being reduced to thousandths by annexing a cipher, and then divided, gives a quotient of 5 thousandths. The sum of the several quotients, .375, is the answer.

To prove that .375 is equal to 8, we change it to the form of a common fraction, by writing its denomi1875% = 8.

nator, and reducing it to its lowest terms. Thus, .375

nator.

=

1000

RULE. Annex ciphers to the numerator, and divide by the denomiPoint off in the quotient as many decimal places as there have been ciphers annexed.

NOTE. It is not usually necessary that the decimals should be carried to more than six places. When a decimal does not terminate, the sign plus (+) is generally annexed. Thus, in the expression .333+, the sign annexed indicates that the division could be carried further.

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6. Reduce 19 to an equivalent decimal expression.
7. Reduce $3157 to an equivalent decimal expression.

Ans. $315.875.

8. Reduce $11633 to an equivalent decimal expression.

Ans. 1163.75.

NOTE. A decimal with a common fraction annexed constitutes what is called a complex decimal; as, .871, .314, and .182. In such expressions, instead of the common fraction, its equivalent decimal, with the decimal point omitted, may be substituted. Thus, .45

.404.

9. Reduce .62 to a simple decimal. 10. Reduce .37 to a simple decimal.

Ans. .625. Ans. .370625.

11. Reduce $ 4.314 to a simple decimal expression.

Ans. $4.3125.

12. Reduce $60.183 to a simple decimal expression.

Ans. $60.1875.

13. What decimal expression is equivalent to of 2

of 2.04?

14. What decimal expression is equivalent to 22, +0.374.

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279. To reduce a simple or compound number to a deci mal of a higher denomination.

Ex. 1. Reduce 15s. 9d. 3far. to the decimal of a £.

OPERATION.

4

3.0 0

12

far. 9.7 500 d.

2 0 1 5.8 1 250 s.

Ans. .790625.

We commence with the 3far., which we reduce to hundredths by annexing two ciphers; and then, to reduce these to the decimal of a penny, we divide by 4far., since there will be as many hundredths of a penny as of a farthing, and obtain .75d. Annexing this to the 9d., we divide by 12d., since there will be as many shillings as pence; and then, the 15s. and this quotient by 20s., since there will be as many pounds as shillings, and obtain .790625£. for the answer. Hence the following

.790625 £.

20

RULE. Divide the lowest denomination, annexing ciphers if neces sary, by that number which will reduce it to one of the next higher denomination. Then divide as before, and so continue dividing till the decimal is of the denomination required.

NOTE 1. - The given number may also be first reduced to a common fraction of the given denomination (Art. 256), and then the fraction changed to a decimal. Thus, if it be required to reduce 15s. 6d. to a decimal of a £.: 15s. 6d. 186d.; 1£. 280d.; 188 £. = 31 £. = .775 £. Answer.

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NOTE 2. Shillings, pence, and farthings may be readily reduced to a decimal of three places, by inspection, thus: Call half of the greatest even number of shillings TENTHS, and, if there be an odd shilling, call it 5 HUNDREDTHS; reduce the pence and farthings to farthings, and increase them by 1, if they amount to 24 or more, for THOUSANDTHS. Thus, if it be required to reduce, by inspection, 19s. 10d. 2far. to the decimal of a £.; half of 18s. 9s., which denote a value of .9£.; the 1s. denotes a value of .05£.; and 10d. 2far. 42far., which increased by 1far. = 43far., which denote a value of .043£.; .9£. + .05.£.+.043.£. .993 £. Answer.

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The reason for this process is, that 2s. equal a tenth of a £.; 1 shilling equals 5 hundredths of a £., and 1 farthing equals £., or so nearly a thousandth of a £. that 24 farthings exactly equal 25 thousandths of a £.; and therefore farthings require to be increased only by 1 when they amount to 24 or more, to denote with sufficient accuracy their value in thousandths of a £.

EXAMPLES.

2. Reduce 9s. to the fraction of a pound.

Ans. .45.

3. Reduce 15cwt. 3qr. 14lb. to the decimal of a ton. 4. Reduce 2qr. 21lb. 8oz. 12dr. to the decimal of a cwt. Ans. .71546875.

5. Reduce 1qr. 3na. to the decimal of a yard.

Ans. .4375.

6. Reduce 5fur. 35rd. 2yd. 2ft. 9in. to the decimal of a mile. Ans. .73603219+.

7. Reduce 3gal. 2qt. 1pt. of wine to the decimal of a hogshead. Ans. .0575396+.

8. Reduce 1pt. to the decimal of a bushel. Ans. .015625. 9. Reduce 2R. 16p. to the decimal of an acre. Ans. .6. 10. Reduce 175 cubic feet to the decimal of a ton of timber.

Ans. 4.375.

11. Reduce 3.755 pecks to the decimal of a bushel.

Ans. .93875.

12. What decimal part of a degree is 25′ 34′′.6? 13. Reduce 12T. 3cwt. 2qr. 20lb. to hundred-weight and the decimal of a hundred-weight. Ans. 243.7.

14. Reduce 2hhd. 30gal. 2qt. 14pt. to gallons and the decimal

of a gallon.

15. Reduce to the decimal of a pound, and 17s. 51⁄2d., and find their sum.

Ans. 156.6875.

19s. 11 d., 16s. 94d., Ans. 2.710416+.

280. To find the value of a decimal in whole numbers of lower denominations.

Ex. 1. What is the value of .790625 £.?

OPERATION.

.790 625£.
20

1 5.8 1 2 5 0 0s.
12

9.7 5 0 0 0 Od.
4

3.0 0 0 0 0 Ofar. Ans. 15s. 9d. 3far.

Ans. 15s. 9d. 3far.

There will be 20 times as many mil lionths of a shilling as of a pound; therefore, we multiply the decimal, .790625, by 20, and reduce the improper fraction to a mixed number by pointing off six figures on the right, which is dividing by its denominator, 1000000. The figures on the left of the point are shillings, and those on the right, the decimal of a shilling. The decimal .812500 we multiply by 12, and, pointing off as before, obtain 9d., and a decimal of a penny. The decimal

.750000 we multiply by 4, and pointing off have 3 farthings, which, taken with the other denominations obtained, gives 15s. 9d. 3far. for the answer.

RULE. Multiply the decimal by that number which will reduce it to the next lower denomination, and point off as in multiplication of decimals.

Then, multiply the decimal part of the product, and point off as before. So continue till the decimal is reduced to the denominations required.

The several whole numbers of the successive products will be the

answer.

NOTE.

When there is a decimal in the last product, it may be changed to a common fraction.

EXAMPLES.

2. What is the value of .625 of a shilling? 3. What is the value of .6725 of a cwt.?

Ans. 7 d.

Ans. 2qr. 171b. 4oz.

4. What is the value of .9375 of a yard? 5. What is the value of .7895 of a mile?

Ans. 6fur. 12rd. 10ft. 61ğin.

6. What is the value of .9378 of an acre?

Ans. 3R. 30p. 13ft. 9 in.

7. Reduce .5615 of a hogshead of wine to its value in gallons, &c. Ans. 35gal. 1qt. Opt. 3133gi. 8. Reduce .367 of a year to its value in days, &c.

Ans. 134d. 1h. 7m. 19 sec.

9. What is the value of .6923828125 of a cwt.?

Ans. 2qr. 191b. 3oz. 13d.

10. What is the value of .015625 of a bushel? 11. What is the value of .55 of an ell English?

12. What is the value of .6 of an acre?

Ans. 2qr. 3na.
Ans. 2R. 16p.

MISCELLANEOUS EXAMPLES.

1. What is the value of 7cwt. 2qr. 18lb. of sugar, at $11.75 per cwt.? Ans. $90.24. 2. What cost 19cwt. 3qr. 14lb. of iron, at $9.25 per cwt.?

3. What cost 39A. 2R. 15p. of land, at $87.375 per acre? Ans. $3459.50332.

4. What would be the expense of making a turnpike 87m. 3fur. 15rd., at $578.75 per mile? Ans. $50595.41. 5. What is the cost of a board 18ft. 9in. long, and 2ft. 34in. wide, at $.053 Ans. $2.277. foot? per

6. Goliath of Gath was 64 cubits high; what was his height in feet, the cubit being 1ft. 7.168in. ? Ans. 10ft. 4.592in.

7. If a man travel 4.316 miles in an hour, how long would he be in travelling from Bradford to Boston, the distance being 29 miles? Ans. 6h. 50m. 6sec. + 8. What is the cost of 5yd. 1qr. 2na. of broadcloth, at $5.62 per yard? Ans. $30.234ğ. 9. Bought 17 bags of hops, each weighing 4cwt. 3qr. 7lb., at $5.87 per cwt.; what was the cost?

10. Purchased a farm, containing 176A. 3R. 25rd., at $75.37 per acre; what did it cost? Ans. $13334.30813. 11. What cost 17625 feet of boards, at $12.75 per thousand? Ans. $224.7183. 12. How many square feet in a floor 19ft. 3in. long, and 15ft. 9in. wide? Ans. 303ft. 27in. 13. How many square yards of paper will it take to cover a room 14ft. 6in. long, 12ft. 6in. wide, and 8ft. 9in. high? 14. How many solid feet in a pile of wood 10ft. 7in. long, 4ft. wide, and 5ft. 10in. high? Ans. 24617ft.

15. How many garments, each containing 4yd. 2qr. 3na., can be made from 112yd. 2qr. of cloth?

16. Bought 1gal. 2qt. 1pt. of wine for $1.82; what would be the price of a hogshead? Ans. $70.56.

17. Bought 125yd. of lace for $15.06; what was the price of 1 yard? Ans. $0.12.

18. What cost 17cwt. 3qr. of wool, at $35.75 per hundredweight? Ans. $634.5624. 19. What cost 7hhd. 47gal. of wine, at $87.25 per hogs. head? Ans. $675.843. 20. How many solid feet in a stick of timber 34ft. 9in. long, 1ft. 3in. wide, and 1ft. 6in. deep? Ans. 65.15625ft. 21. If 18yd. 1qr. of cloth cost $36.50, what is the price of 1 Ans. $2.00.

yard?

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