4. Having the product of two numbers, and one of them given, to find the other.

RULE. Divide the product by the given number, and the quotient will be the number required.

If the product of two numbers be 196, and one of them 4, required the other. Thus: 196÷4-49. Ans.

5. Having the dividend and quotient, to find the divisor.

RULE. Divide the dividend by the quotient. This will prove division.

Let the dividend be 144, and the quotient 16, required the divisor? 144÷16=9. Ans.

6. Having the divisor and quotient given, to find the dividend. RULE. Multiply them together.

Let the divisor be 6, and the quotient 72, required the dividend. 72X6=432. Ans.

EXAMPLES.

1. Suppose a man born in the year 1743, when will he be 77 years of age? Ans. 1820. 2. What number is that, which if it be added to 19418 will make 21802 ?

Ans. 2384.

3. What number must you multiply by 9, that the product may be 675 ?

Ans. 75.

4. What is the difference between thrice five and thirty, and thrice thirty-five?

5. If a man spend 192D. in a year, how much is that

endar month?

Ans. 60.

per

cal

6. A. borrowed at different times the following sums, namely: of B. D625; of C. D721.50; of D. D842; and he is indebted to others as much as he has borrowed, abating D125.50; he is now prepared to retire from business; required the amount of his debts. Ans. D4251.50

7. General Burgoyne and his army were captured at Saratoga, N. Y., by General Gates, October 17, 1777, and Earl Cornwallis and his army surrendered to General Washington at York, Va., October 19, 1781; required the space of time between. Ans. 4 years, 2 days

8. Bought 5 loads of wood: the first containing 1 cord, 32 feet; the second, 1 cord, 64 feet; the third, 112 feet; the fourth, 1 cord, 28 feet; the fifth, 1 cord, 20 feet; how much in all? Ans. 6 cords

9. How many bottles holding 1 pint, 2 gills each, are required for bottling 4 barrels of cider?

10. 9 A. 7 R. 50 po. +17 A. 11 R. 70 po.; then-12 A. 5 R. 45 po.; then by 62; then by 49.

70

REDUCTION OF DECIMALS.

625

70

THE two following cases of Reduction of Decimals are the very reverse of each other, and by reversing the rule, one will prove the other. The use of the rule is to change a denomination, or several denominations, from their given expression into a decimal quantity having the same value, for the purposes of multiplication, division, &c., which will then become the same as whole numbers. Thus, if you wish to find the decimal expression of 2 quarters, 14 pounds, avoirdupois, first reduce it to pounds, 2×28=56+14=70 pounds; now annex ciphers to the 70 pounds, and divide by 112 lbs. 1 cwt. of which you wish to make it a decimal, and you have .625; 200=112-3 for 1471 Again, you wish to reverse the rule, and give the .625 the expression in the terms of the integer, or to reduce it to its proper value. Thus .625 cwt. multiply by the next less denomination, which is 4 qrs., and count off the three places for decimals, according to the rule in multiplication of decimals, and the quantity at the left of the decimal point is of the same name with the multiplier, that is, 2 qrs.; then multiply the remaining decimal, .500, by the next less denomination, namely, 28 lbs., and count off as before, and you have 2 quarters, 14 pounds=12112= cwt., &c. The correctness of the operation of the rule is evident from the nature of decimals, and the explanations already given in the preceding pages.

625

70

In the third case: to reduce a vulgar fraction to its equivalent decimal, by annexing one, two, three, or more ciphers to the numerator, the value of the fraction is increased ten, a hundred, or more times. After dividing, the quotient will, of course, be ten, a hundred, or more times, too much; the quotient must, therefore, be divided by ten, a hundred, &c., to give the true quotient or fraction. In the first example, is 1000-125, which divided by 1000 is 1.125, and this is the rule.

To reduce numbers of different denominations, as of money, weight, measure, &c., to their equivalent decimal value.

RULE.

1. Begin at the left hand and multiply by as many as it takes of the next lower denomination to make one of the higher, adding in the denominations respectively, as you multiply, until they are reduced to the lowest denomination in the question, and this is the dividend.

2. Then take one of that denomination of which you wish to make it a decimal, and reduce it to the same denomination with the one above-mentioned, and this last number is the divisor. Divide as in whole numbers, and the quotient is the answer. Put the point before it, for it is always a decimal.

EXAMPLES.

1. Reduce 3 roods, 20 poles, to the decimal of an acre.

Explanation.You will first multiply the 3 roods by 40, because 40 poles make 1 rood,

po. 160)140.000(875 Ans. 40 po.=1R. and add in the 20

4X

40

1280

po. 20.000 proof.

1200

1120

800
800

po., this will make 140 poles; then annex ciphers, and the dividend will be formed; now you wish to make it the decimal of an an acre, then take

1 acre and reduce it to the same denomination as the dividend, that is, poles, 160=1 acre, and this is the divisor; then to prove it, multiply the quotient by the next lower denomination, observing to count off for the decimal point, and the denominations at the left of the point make the answer, &c.

Ans. .125.

Ans. .525.

2. Reduce 20 poles to the decimal of an acre.
3. Reduce 2 R. 4 po. to the decimal of an acre.
4. Reduce 3 quarters, 2 nails, to the decimal of a yard.

4 qrs.
4 na. X

nails, 16 divisor.

3 qrs. 2 na. 4

16)14.000(.875 Ans.

Ans. .015873

5. Reduce 1 gall. to the decimal of a hhd.

6. Reduce 7 oz. 19 dwt. to the decimal of a pound troy. 240)159.0000(.6625. Ans 7. Reduce 3 quarters, 21 pounds, avoirdupois, to the decimal

of a cwt.

Ans. .9375.

8. Reduce 2 feet, 6 inches, to the decimal of a yard?

Ans. .83333+

9. Reduce 5 furlongs, 16 poles to the decimal of a mile.

Ans. .675

10. Reduce 4.5 calendar months to the decimal of a year!

Ans. .375. .666+

.333+

.1785.

11. Reduce 40 minutes to the decimal of an hour. 12. Reduce 20 minutes to the decimal of an hour. 13. Reduce 1 day, 6 hours, to the decimal of a week. 14. Required the cost of 1 yard, 2 quarters, 2 nails of cloth, at D5.00 per yard.

2 qrs. 2 na.

4

yd. 1.625

5

D8.12.5 Ans.

16)10.000(.625.

15. What is the value of 3 yards, 1 quarter, 3 nails of cloth, at D4.50 per yard?

1 qr. 3 na.

4

16)7.0000(.4375.

yd. 3.4375

4.50D.

D15.46.87.5 Ans.

16. Required the value of 1 acre, 2 roods, 18 poles, at D20 per acre?

17. What cost 2 acres, 1 rood, 30 poles, at D25 per acre?

Ans. D60.93.75 18. What cost 2 pecks, 7 quarts of wheat, at D1.50 per bush Ans. D1.07.8.125. 19. What cost 7 bushels, 2 pecks, 7 quarts of corn at 52 cts. per bushel? D4.01.375. 20. What is the cost of 3 bushels, 3 pecks, 4 quarts of rye, at 75 cents per bushel? Ans. D2.90.6.25. 21. What is the cost of 2 quarters, 15 pounds of coffee, at D11 per cwt.? Ans. D6.97.312.

22. Reduce 375678 feet to miles and decimals. 71.151+ Note. This last question is an exception to the general rule, or at least questions of this kind are not of frequent occurrence

although the rule will apply in this as well as in other cases; the number of feet given are equal to a number of miles; therefore, take the feet for the dividend, reduce 1 mile to feet=5280 for a divisor, then divide according to the rules of Division of Decimals, and you have 71.151+ miles. This example will apply and be sufficient in cases of this kind.

To reduce a Decimal to its proper value.

RULE II.

1. Multiply the decimal by the number of parts in the next less denomination, and cut off so many places for a remainder, counting from the right, as there are decimal places in the given decimal, and there make the decimal point.

2. Multiply the remainder, that is, the decimal, by the next less denomination, and cut off a remainder as before; continue in this way through all the parts of the integer, and the several denominations standing on the left hand of the decimal point make the answer.

23. Required the value of .9075 of an acre.

.9075
X4 R.

Roods, 3.6300

40 po.

Poles, 25.2000 30.25 yd.

Yards, 6.050000

9 feet.

Feet, 0.450000

Explanation.-Begin and multiply by the next lower denomination thar the one mentioned in the question which is roods, 4 of which make ar acre. After multiplying, count off as many places or figures as there are in the given question for decimals, and there place the decimal point, and the figure or figures at the left of the decimal point, will be of the same name as the multiplier, namely, roods. Then take the next less denomination, which is poles, 40 of which make 1 rood; multiply all the figures on the right of the decimal point, then count off from right to left as many figures as there were decimals in the number multiplied; there place the decimal point, and the figures at the left of the point will be poles. Thus far we have 3 roods, 25 poles, and .2000; now 30.25 yards make one pole in square measure; therefore, multiply by that number those figures at the right of the decimal point, which are .2000, and count off in the product 4 figures for those in the multiplicand, and 2 for the multiplier, 6 figures, and there place the decimal

144 in.

1800000 1800000

450000

In. 64.800000