As multiplying by 10, 100, 1000, &c. is only removing the separating point of the multiplicand so many places to the right hand, as there are ciphers in the multiplier, so to divide by the same is only removing the separatrix, in the same manner, to the left.

REDUCTION OF DECIMALS.

CASE I.

To reduce a vulgar fraction to its equivalent decimal. RULE. Divide the numerator by the denominator, and the quotient will be the decimal required.

To reduce numbers of different denominations to their equivalent decimal values.

RULE. 1. Write the given numbers perpendicularly under one another for dividends, proceeding orderly from the least to the greatest.

2. Opposite to each dividend, on the left hand, place such a number for a divisor as will bring it to the next superior name, and draw a line between them.

3. Begin with the highest, and write the quotient of each division, as decimal parts, on the right hand of the dividend next below it, and the last quotient will be the decimal sought.

EXAMPLES.

1. Reduce 14s. 54d. to the decimal of a pound.

2. Reduce 15s. to the decimal of a pound.
3. Reduce 3 qrs. 18 lbs. to the decimal of a cwt.

Ans.,75,

Ans. ,910714+

4. Reduce 2 qrs. 2 nails to the decimal of a yard.

Ans. ,625.

5. Reduce 14 gals. 3 quarts to the decimal of a hogshead.

CASE III.

Ans.,2341+

To find the decimal of any number of shillings, pence, and farthings, by inspection.

RULE. Write half the greatest even number of shillings for the first decimal figure, and let the farthings, in the given pence and farthings, possess the second and third places; observing to increase the second place by 5, if the shillings be odd, and the third place by 1, when the farthings exceed 12, and by 2 when they exceed 37.

EXAMPLES.

1. Find the decimal of 13s. 9ąd. by inspection.

2. Find by inspection the decimal of 15s. 84d. 9s. 3 d. 19s. 62d. 3s. 6d. and 2s. 114d. Ans.,784,465,978,175,148.

CASE IV

To find the value of any given decimal in the terms of the integer.

RULE. 1. Multiply the decimal by the number of parts in the next lowest denomination, and cut off as many places for the remainder to the right hand as there are places in the given decimal.

2. Multiply the remainder by the parts in the next inferior denomination, and cut off a remainder as before.

3. Proceed in this manner through all the parts of the integer, and the several denominations, standing on the left hand, make the answer. EXAMPLES.

1. Find the value of ,691 of a pound.

,691

20

13,820

12

9,840
4

3,360 Ans. 13s. 93d.

2. What is the value of ,9 of a shilling? 3. What is the value of ,592 of a cwt.?

Ans. 10 d.

Ans. 2 qrs. 10 lbs. 4 oz. 13+drs. 4. What is the value of ,258 of a tun of wine?

Ans. 1 hhd. 2+gals.

5. What is the value of ,12785 of a year?

Ans. 46 days 15 hours 57 minutes 57+seconds.

SINGLE RULE OF THREE DIRECT.

THE SINGLE RULE OF THREE DIRECT teaches, from three numbers given, to find a fourth, that shall be in the same proportion to the third, as the second is to the first.

If more require more, or less require less, the proportion is direct.

RULE. Write down the number which is of the same name or kind with the answer for the second term. Consider whether the answer ought to be greater or less than this number; then, if greater, write down the greater of the remaining numbers on the right of it, for the third term; but if less, make the less of those numbers the third term; and on the left of the second term, write down the remaining number for the first term.

Reduce the first and third numbers into the same, and the second into the lowest denomination mentioned in it.

2. Multiply the second and third numbers together, and

divide the product by the first, and the quotient (if there be no remainder) is the answer, or fourth number required.

NOTE. In multiplying the second and third numbers together, if either be 1, it is unnecessary to multiply by it: and the same remark applies to it, when used as a divisor.

If, after division, there be a remainder, reduce it to the next denomination below that to which the second number was reduced, and divide by the same divisor as before, and the quotient will be of this last denomination. Proceed thus with all the remainders till you have reduced them to the lowest denomination which the second number admits of, and the several quotients taken together will be the answer required.

The method of proof is by reversing the question.

EXAMPLES.

1. If 2 yards of cloth cost 4s. what will 125 yds. come to?

2. When 891 gallons of oil cost £176 6 101, how many gallons may be bought for 13s. 104d.?

Ans. 34 gallons

3. 871 yards
4. 12
5. 794
6.61d.
1 lb.

:

8. £23 14 3

: £8 561 :: 5

11 yard: £10 6 8

: 17 cts. : 2 cwt. 1 qr. 21 lb.
19 yards: £142 5 6.

9. 6 lb. 8 oz. 4 dt.: £21 65: 40 lb. 1 oz. 4 dt.
10. 7 cwt. 3 qrs. 15 lb. : £33 2 3: 1 qr. 21 lb.
11. £24 4 8
12. 33 lb.
13. £9 0 10

14. $163,35

:

117 yards: £141 13 10.
$3,51 cts. :: 3 qrs. 17 lb. 8 oz.
1 cwt. £89 5 83.

Ans. $2,61. 21,00. 18.0 d.

496 yards.

$48,452. 114 yards. £127 18 6. £1 16 9. 687 yards.

$105,56.

9 cwt. 3 qrs. 14 lb.

:

12 cwt. 3 qrs. 24 lbs. : : $3,15.

28 lbs.

15. B. owes £2119 17s. 6d. and he is worth but £1324 18s. 51d.; if he delivers this to his creditors, how much do they receive on the pound? Ans. 12s. 6d.

16. How many crowns of 110 cents each will pay a debt of £82 168. 74d. N. E. ? - Ans. 251 crowns.

17. If 203 tons 9 cwt. 3 qrs. 3 lbs. of tallow cost £4558 3s. Od. what does 1 ton cost? Ans. £22 8 0.

18. How many cwt. of rice may be bought for 487 dols. 50 cts. when 7 lb. cost 25 cents? Ans. 121 cwt. 3 qrs. 14 lb. 19. A captain of a ship is provided with 18000 lb. of bread for 150 seamen, of which each man eats 4 lb. per week, how long will it last them? Ans. 30 weeks.

20. How long would 2295 lb. of beef last for 45 seamen, if they get 1 lb. each, and that 3 times a week?

Ans. 17 weeks. 21. Suppose 120 seamen are provided with 7200 gallons of water for a cruise of 4 months, each month 30 days; how much is each man's share per day? Ans. 2 quarts.

22. A ship's company of 16 men is on allowance of 6 ounces of bread per day, when meeting with a vessel from which they are supplied with 2 cwt. of bread, what addition will this make to their daily allowance, if they suppose their voyage to last 28 days? Ans. 8 ounces.

23. A person failing in trade owes, viz. to A. $818,73; B. $3673,46; C. $1800,40; D. 117,41, and to E. 814,50; and his property, worth $4009,593 cts. he gives up to his creditors; how much does he pay on the dollar, and what is E.'s loss on receiving his dividend ?

Ans. He pays 55 per dollar-E.'s loss is $362,451.