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2. Reduce 7s. 6d. to the decimal of a pound. Ans. 375. 3. Reduce 14s. 9d. 3qrs. to the decimal of a pound.

Ans. ,740625.

Ans. ,5

4. Reduce 6d. to the decimal of a shilling.
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.)

7. Reduce 14s. to the decimal of a pound.

Ans. ,875.

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.

16

17,0 19,0

8. Reduce 1, 2, 3, 4, 9, 16, 17, and 19 shillings to decimals. Thus 1,0 2 3,0 4 9,0 Answers ,05,1,15,2,45 9. Reduce £18 2s. 7d. to a decimal expression.

,8,85

,95

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. 2na. to the decimal of a yard.

Ans. ,875.

14. Reduce 3pks. 2qts. 1pt. 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 6fur. 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. 6d. 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. 52d. to the decimal of a £.

Ans. ,424.

3. Reduce 19s. 4d.; 5s. 61d.; 6s. 42d., and 12s. 6d. to Ans. ,967;,277;,320, and,625,

decimals of a pound.

4. Express £5 15s. 114d. decimally. 5. Express £44 18s. 61d. decimally,

PROBLEM IV.

Ans. £5,797.

Ans. £44,926.

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 1s. £,05; and 2s. 20 -£,1; and 4s. 20 £,2: thus the number

4

4

of shillings is so many 10ths of a £ and each odd shilling is,50 £. Then each farthing is 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 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, part is more than a farthing, and we add 1; and when 24 they are more than 36, part is greater than 1, and we add 2,

2. Multiply this decimal 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.

EXAMPLES.

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

,695
20

s. 13,900
12

d. 10,800

4

qrs. 3,200

We multiply the decimal by 20, because £1=20s., and the product is 13,900s.; and because 1s.=12d. we multiply this decimal by 12, which gives 10,800d.; lastly we multiply by 4, because 1d.=4qrs.

Ans. 13s. 10d. 32qrs.

Ans 19s.

Ans. 7d. Ans. 12. 93d. Ans. 81d.

2. What is the value of ,95 of a pound?
3. What is the value of,625 of a shilling?
4. Find the value of ,640625 of a £.
5. Find the value of,0356 of a pound.
6. Reduce ,857 of a shilling to pence and farthings.

Ans. 10d. 1qr.

7. Reduce ,945 of a lb Troy to oz. pwts. and grs.

Ans. 11oz. 6pwt. 19,2 grs.

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. 2qt. 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. 93d.

15. Reduce ,569 of a year to days, hours, minutes and seconds. Ans. 207d. 16h. 26m. 24sec.

PROBLEM V.

To find the Value of any decimal of a pound (£) by Inspection.

RULE.

Double the first figure, or figure of tenths, for shillings; then if the second figure be 5, or more, deduct 5 from it, and reckon another shilling; then call the remaining figures in the second and third places so many farthings, subtracting 1 when they are above 12, and 2 when they are above 36.

EXAMPLES.

1. Find the value of ,785 of a pound.

Double 7, the first figure, or tenths, for s. 14s.
Then the second figure being more than

5, we deduct 5 from it, and add 1 to the
shillings.

Then the remaining figures, 35, we call so many qrs.; abating 1 because they are more than 12 and less than 36, leaves 34qrs. And 34qrs. 8d. 2qrs.

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Ans. 15s. 8d. 2qrs.

2. Reduce,875 of a pound to shillings, pence and farthings.

Ans. 17s. 6d.

3. Reduce,095 of a pound to its proper quantity.

Ans. 1s. 103d.
Ans. 4s. 7 d.

4. Find the value of £,230. Note. When the decimal has but two places of figures, annex a cipher to it, or suppose a cipher to be annexed. 5. Find the value of ,76 of a pound. 6. Find the value of ,34 of a pound. 7. Find the value of ,95 of a pound.

Questions.

1. What is a Decimal Fraction? 2. How is the integer divided? 3. How is the true value of a decimal traction expressed?

4. If the numerator has not so many places as the denominator has ciphers, how do you write it?

5. By what does each figure take its value? What is the first figure on the

Ans. 15s. 2 d.
Ans. 68. 94d.
Ans. 19s.

right hand of units, or the separatrix? -the second?-the third ?-the fourth? &c.

6. What effect do ciphers have when placed on the right of decimals?

7. When placed on the left, what effect do they have?

8. On what does the magnitude of a decimal fraction depend?

9. How are decimals read?

10. When whole numbers and decimals are expressed in the same number, what is it called?

11. What is the Rule for Addition of decimals ?-for Subtraction ?

12. What is the Rule for Multiplication of decimals?

13. What is the Rule for Division of decimals?

tion to its equivalent decimal?
II. How do you reduce quantities of
several denominations to a decimal of
the highest?

III. How do you find the decimal of any number of pounds, shillings, pence and farthings, by Inspection?

IV. How do you find the value of a decimal in whole numbers of a lower denomination?

V. How do you find the value of any

I. How do you reduce a Vulgar frac- decimal of a pound by Inspection?

REDUCTION OF CURRENCIES.

Formerly the pound was of the same value in Great Britain and all the American Colonies, (now States,) and the dollar reckoned at 4s. 6d.

But the Legislatures of the different States issued bills of credit, which depreciated in their value, in some States more, and in others less, which caused the currencies of the several States to differ from each other.

Thus, a dollar is reckoned
In the New England States,
also Virginia, Kentucky,
and Tennessee,
In New York, North Caroli-
na and Ohio,

}

at 6s. called New England currency.

at 8s., called New York

currency.

In N. Jersey, Pennsylvania, at 7s. 6d., called PennsylDelaware and Maryland, S

In South Carolina and Geor

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vania currency.

at 4s. 8d., called Georgia

currency.

at 5s., called Canada cur

PROBLEM I.

rency.

To reduce the currencies of the several States in which a dollar is an even number of shillings, to Federal money,

viz : New England,

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New York,

North Carolina,

and

Ohio.

RULE.

1. When the sum consists of pounds only, annex a cipher to the pounds, and divide by half the number of shillings in a dollar, the quotient will be dollars, &c.

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