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16. How many times 6,25 dollars can I have in 1235 dollars ?

Ans. 197,6. Note. To divide by 10, 100, 1000, &c., remove the decimal point in the dividend as many places towards the left hand as there are ciphers in the divisor. Thus, 425, divided by 10, the quotient is 42,5 42,5 10,

4,25
425,4
100,

4,254

425.

6

425,

1000,

REDUCTION OF DECIMALS.

PROBLEM I.

To reduce a Vulgar Fraction to its equivalent decimal.

RULE.

Annex ciphers to the numerator of the given fraction, and divide by the denominator, the quotient will be the decimal required, which must contain as many decimal places as there are ciphers annexed to the numerator. If there are not so many figures in the quotient, make up

the deficiency by placing ciphers on the left.

EXAMPLES,

1. Reduce s to its equivalent decimal.

Operation. 8)7,000

,87 5 Answer. 2. Reduce to a decimal fraction. Operation.

We might continue annexing 9)5,00000 ciphers to this remainder, and car

ry on the quotient still lower; but ,55555

were it carried to an infinite number of figures, we should never arrive at a complete quotient.* 3. Reduce } to a decimal fraction.

Ans. ,5. 4. Reduce to a decimal.

Ans. ,75. 5. What decimal is equal to }?

Ans. ,2. 6. Reduce 11 to a decimal.

Ans. ,91666+ 7. Reduce

23
to a decimal.

Ans. ,12.
* Such fractions are called circulating, or repeating decimals.

3

9

8. Reduce to a decimal. 9. Reduce to a decimal. 10. Reduce o to its equal decimal. 11. Reduce

14286

to a decimal. 12. Reduce to its equivalent decimal.

Ans. ,04166+
Ans. ,33333+.
Ans. ,03846+.
Ans. ,00631+.
Ans. ,875.

PROBLEM II.

To reduce quantities of different denominations to a decimal of the highest denomination.

RULE.

1. Reduce the given denominations to a Vulgar Fraction, as taught in Problem 7, page 93 ; then reduce the Vulgar Fraction to its equivalent decimal.

EXAMPLES.

414 qrs.

1. Reduce 8s. 70. 2qrs. to the decimal of a pound. 960) 414,00000(,43125 Ans. 8s. 7d. 2qrs. 3840

12 3000

414 2880

103 £1=960 qrs. 1200

4
960
2400

Nole. This Rule will give 1920

the true decimal; but the fol

lowing Rule is much easier, 4800

and therefore the best for 4800

practice. See this example perform

ed by Rule 2. Rule 2. Write the denominations below each other, plàcing the lowest name at the top ; then divide each denomination (beginning at the top,) by that number which makes one of the next higher, and the last quotient will be the de- · cimal sought.

Perform the preceding example by this Rule.
Thus 42,

In dividing by the several divi

sors, we annex as many ciphers to 127,5

each dividend as are necessary.

Thus we annex 1 cipher to the 2018,625 farthings, and divide by the number

of qrs. in a penny, and so on with ,43125 each denomination.

2. Reduce 7s. 60. to the decimal of a pound. Ans. ,375. 3. Reduce 14s. 9d. 3qrs. to the decimal of a pound.

Ans. ,740625. 4. Reduce 6d. to the decimal of a shilling. Ans. ,5 5. Reduce 15s. 9d. 3qrs. to the decimal of a pound.

Ans. ,790625. 6. Reduce 5s. 3d. to the decimal of a dollar, New England currency, (=6s.)

Ans. ,875. 7. Reduce 14s. to the decimal of a pound. Ans. ,7.

Note. If the shillings be an even number, half that number, with a point prefixed, is their decimal expression ; but if the number be odd, annex a cipher; then half that number is the decimal required.

8. Reduce 1, 2, 3, 4, 9, 16, 17, and 19 shillings to decimals. Thus 1,0 2 3,0 4 9,0 16 17,0 19,0

Answers ,05 ,1 ,15 ,2 ,45 ,8 ,85 ,95 9. Reduce £18 2s. 7d. to a decimal expression.

Ans. £18,129166 +. 10. What is the decimal expression of £25 18s. 64d. ?

Ans. £25,926041+. 11. Reduce 9oz. 13pwt. to the decimal of a pound Troy.

Ans. ,80416+. 12. Reduce 3qrs. 25lb. to the decimal of a cwt. Avoirdupois, lbs.

qrs. Thus 28)25,0000(,8928+qrs. ; then 4)3 ,8928

Answer ,9732+: 13. Reduce 3qrs. 2 na. to the decimal of a yard.

Ans. ,875. 14. Reduce 3pks. 2qts. 1 pt. to the decimal of a bushel.

Ans. ,828125. 15. Reduce 45 gals. to the decimal of a hogshead.

Ans. ,7142+. 16. Reduce 2 feet 9 inches to the decimal of a yard.

Ans. ,91666+ 17. Reduce bfur. 26rd. 2yd. to the decimal of a mile. (By Rule I.)

Ans. ,8323+ 18. Reduce 2 roods 24rds. to the decimal of an acre.

Ans. ,65. 19. Reduce 4 months 2 weeks to the decimal of a year,

Ans, ,375.

PROBLEM III.

To find the decimal (to three places of figures,) of any number of shillings, pence and farthings, by Inspection.

RULE.

1. Write half the greatest even number of shillings for the first decimal figure ; and if the shillings be odd, increase the second place, or place of hundredths, by 5.

2. Let the farthings in the given pence and farthings possess the second and third places, increasing the third place by 1 when the farthings exceed 12, and by 2 when they exceed 36.*

EXAMPLES. 1. Reduce 17s. 60. 2qrs. to the decimal of a pound. Thus, ,8 =half the greatest even number of shillings,

5 the s. are odd, therefore we write 5 hund'ths. 26=farthings in 6d. 2qrs.

1 we increase by 1, because the qrs. exceed 12. £ ,877 decimal required. 2. Reduce 8s. 5 d. to the decimal of a £. Ans. ,424.

3. Reduce 19s. 4d. ; 5s. 6 d. ; 6s. 4 d., and 12s. 6d. to decimals of a pound. Ans. ,967; ,277; ,320, and ,625.

4. Express £5 15s. 11 d. decimally. Ans. £5,797. 5. Express £44 18s. 644. decimally, Ans. £44,926.

PROBLEM IV.

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

RULE. 1. Multiply the given decimal by the number of parts in the next less denomination, and point off as many decimal places as there are in the given decimal.

* The reasoning of this Rule is as follows,--shillings are 20ths of a pound. Thus ls. 26=£,05 ; and 25. 2. -£,1; and 4s. £ ,2: thus the number of shillings is so many 10ths of a £ and each odd shilling is ,50 £.

Then each farthing is go of a £. Had it happened that 1000 instead of 960 had made a pound, then the farthings would have been so many thousandths. But 960, increased by 24 part of itself, is=1000. Therefore, any number of farthings, increased by their 24 part, will be an exact decimal. Hence, when the farthings are more than 12, 24 part is more than ] a farthing, and we add 1; and when they are more than 36, z part is greater than 1}, and we add 2:

4 20

2. Multiply this dccimal by the next inferior denomination, and point off the decimals as before : proceed in this manner through all the parts of the integer, and the several denominations standing on the left of the decimal point, constitute the answer.

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EXAMPLES.

1. Reduce ,695 of a pound to its proper value in whole numbers of a smaller denomination.

,695 We multiply the decimal by 20, be20

cause £1=20s., and the product is s. 13,900 13,900s.; and because Is.=12d. we

12 multiply this decimal by 12, which d d. 10,800 gives 10,800d.; lastly we multiply by

4, because ld.=4qrs. 4

Ans. 13s. 10d. 3qrs. qrs. 3,200 2. What is the value of ,95 of a pound ? Ans 196. 3. What is the value of ,625 of a shilling?

Ans. 7 d. 4. Find the value of ,640625 of a £. Ans. 12. 9 d. 5. Find the value of ,0356 of a pound. Ans. 8 d. 6. Reduce ,857 of a shilling to pence and farthings.

Ans. 10d. itr. 7. Reduce ,945 of a lb Troy to oz. pwts. and grs.

Ans. 11oz. 6pwt. 19 Zagrs. 8. Reduce ,6725 of a cwt. to qrs. lbs. oz. &c.

Ans. 2qrs. 19lbs. 5oz. 9. Reduce ,954 of a yard to qrs. and nails.

Ans, 3qrs. 3+na. 10. Reduce ,725 of a hogshead to gals. qts. and pts.

Ans. 45gals. 2t. 1,4pt. 11. Reduce ,4021 of a mile to its proper quantity.

Ans. 3fur. 8rds. 3yds. 2ft. 1+in. 12. What is the value of ,96875 of an acre ?

Ans. 3 roods 35 rods. 13. Reduce ,0546875 of a lb. avoirdupoise to its proper quantity

Ans. 14dr. 14. Change £45,940625 to its proper expression in pounds, shillings, &c.

Ans. £45 18s. 9 d. 15. Reduce ,569 of a year to days, hours, minutes and seconds.

Ans. 207d. 16h. 26m. 24sec.

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