18. Divide, seven hundred and ninety by thirty-one and six-tenths.

Ans.

19. Divide fifty-seven ten thousandths by six hundredths.

Ans.

20. Divide fifty-five thousand four hundred and sixty-two by one thousand and eight and four-tenths. Ans.

QUESTIONS FOR EXERCISE.

1. From a given multiplier to find a divisor that gives a quotient equal to the product.

RULE-Divide 1 with ciphers annexed by the given multiplier, and the quotient will be the divisor required.

EXAMPLES.

21. What divisor will give a quotient equal to the product

of 125?

Ans.

.008

22. What divisor will give a quotient equal to the product of 625? Ans. .0016 23. Find a divisor that will give a quotient equal to the product of 3125? Ans. .00032 II. From a given divisor to find a multiplier that gives a product equal to the quotient.

RULE-Divide 1 with ciphers annexed by the given divisor, and the quotient will be the multiplier required.

EXAMPLES.

24. What multiplier will give a product equal to the quo tient arising from the same number divided by .008? Ans. 125 25. What multiplier will give a product equal to the quotient arising from the same number divided by .0016? Ans. 625. 26. Divide 7315 by .008, or multiply by 125. Quere, the result. Ans. The quotient and product equal. 914375. 27. Divide 785 by .00032, or multiply by 3125. Quere the quotient and product? Ans. Both the same. 2453125. 28. The multiplier that will produce equal to the quotient arising from the same number divided by .00032 is required?

Ans.

$125.

29. Divide $115 65 c. among 5 men, so that 4 men may have equal shares; and the 5th only half a share?

Ans.

S$25 60 c. to 4 men each, and $12 85 c. to the 5th.

THE SINGLE RULE OF THREE IN DECIMALS.

63

30. Divide $226 87 c. 5 m. among 3 white men and 1 negro, so as the whites may have equal shares, and the negro only of a share?

Ans. $60 50 c. to each white man and $45 37 c. to the negro.

SINGLE RULE OF THREE IN DECIMALS. RULE-Reduce the fractional parts into decimals, then state the question, and proceed according to the rules given for Direct and Inverse Proportion, or by the following general RULE-In every question in the Rule of Three,

Three given numbers there must always be;
Like what you seek must in the middle stand,
And that which gives it the first place demands.
What asks the question the third term must be,
The first and third in name and kind agree.
If more you want, the less divisor make,
If less the greater for divisor take.

EXAMPLES.

1. Suppose I give 12s. 6d. for 4 yards of linen: what will 48 yards of the same come to at that rate.

Ans.

£6.381578961. 78. 71⁄2 1. 2. A grocer buys 4 chests of tea, each weighing neat 2 cwt. 3qrs. 14lb. for $906 50cts. at what rate did he give per lb. Ans. 8.7038048-70cts. 3.8m.

3. How much persian .75 yards wide, will line 25.5 yards of 5 quarters wide? Ans. 42 yds. 4. If 1 lb. of indigo cost $3.84, what will 24.6 lb. cost at that rate? Ans. $62.976 5. A. owes B. $1851.75 but B. compounds with him for 663 cents on the dollar, what must B. receive for his debt? Ans. $1234

6. A merchant bought 4 tuns, 2014 gallons of Corsica wine, for 2401. 16s. 6d. but by misfortune it chanced to leak out 24 gallons, I demand to know what the remainder stands him in per gallon? Ans. £.2032278 4s. Od. 3qrs.

7. Goliath is said (in 1 Samuel chap. xvii. v. 4.) to have been 6 cubits and a span high; this answers to 10 feet 474 inches. Pray what was the length of the cubit in United States or British measure? Ans. 1 ft. 7.168 in. 8. If a board be 9 inches broad, what length will it require to measure 12 square feet? Ans. 16 feet.

9. The cubic inch of marble, is 1.5688 oz. avoirdupois ? what difference is there, in point of weight, between a figure containing a solid foot and half thereof, and another of equal di mensions in brass 4.63 oz. whereof make a cubic inch?

Ans. 7934.6304oz.-4cwt. 1qr. 19lb. 14oz. 9.88drs. 10. If the cubic inch of olive oil be .52835 decimal parts of

64

REPEATING AND CIRCULATING DECIMALS.

an ounce avoirdupois : what quantity of oil weighing 7 lb. per gallon, will be contained in a cask allowed to hold 15 gallons of water, each 282 solid inches?

Ans. 16.555gall. 16gall. 2.22qts. 11. Hiero, King of Sicily, ordered his jeweller to make him a crown, containing 63 ounces of gold; the workman thought substituting part silver therein, to have a proper perquisite, which taking air, Archimedes was appointed to examine it, who on putting it into a vessel of water found it raised the fluid, or that itself contained 8.2245 cubic inches of metal, and having discovered that the cubic inch of gold more critically weighed 10.36 ounces, and that of silver but 5.85 ounces, he, by calculation, found what part of his Majesty's gold had been changed, and you are desired to repeat the process. Ans. 34.1964oz. of gold, and 28.8036oz. of silver. 12. Suppose A. lend to B. $1675 for 6 months, what sum must B. lend A. for 3 years to requite him? Ans. $223 33c.

13. If 60 gallons of water in 1 hour and 15 minutes fall into a cistern containing 225 gallons, and by a pipe in the cistern there runs out 45 gallons in the same time, in what time will it be filled? Ans. 18 hours. 14. What length of a board 9 inches broad will make a square foot, or as much as 1 foot in length & 1 foot in breadth contains? Ans. 1 ft. 4 in. 15. There is a cistern having a pipe which will empty it in 2 hours 45 minutes. How many pipes of the same capacity will empty it in 15 minutes? Ans. 11 pipes. 16. How many yards of matting that is 2 feet 6 inches broad, will cover a floor that is 27 feet long and 20 broad ? Ans. 72 yds.

17. A cistern having a pipe running into it which will fill it in 12 minutes; this same cistern has four emptying pipes, the first of which will empty it in 2 hours, the second will empty it in 1 hour 15 minutes; the third will empty it in 48 minutes, and the fourth will empty it in 30 minutes: required in what time the cistern will be filled if all five run together.

Ans. 2 ho. 13 m. 20 s.

REPEATING AND CIRCULATING DECIMALS. As the arithmetic of infinite Decimals is generally considered a subject as much of curiosity as of real use, we shall at present only give the rules by which they are reduced to vulgar fractions; and by means of which any question where they occur may be wrought.

I. If the decimal be a pure repeater, place the repeating figure for the numerator, and 9 for the denominator.

REPEATING AND CIRCULATING DECIMALS. 65

EXAMPLE 1.

Suppose .2222, &c. or 2 to be a pure repeating decimal, what is its equivalent vulgar fraction? Ans. 20 II. If the decimal be a pure circulate, place the circulating figures for the numerator, and as many 9's as there are places in the circle for the denominator.

EXAMPLE 2.

999

When the pure circulating decimal is .259259259, &c. or .259 what is its vulgar fraction? Ans. 259 I1I. If there be ciphers prefixed to the repeating or circulating figures, annex à like number to the 9's in the denomina

tor.

EXAMPLE 3.

Let .0555, &c. or .05 be the Fepeating, and .0148148, &c. or .0148 the circulating figures, required their equivalent vulgar fractions? Ans. and 148-133 1 9990

་

IV. If the decimal be mixed, subtract the finite part from the whole decimal; the remainder is the numerator, and the denominator consists of as many 9's as there are places in the circle, with as many ciphers annexed as there are finite places in the decimal before the circle.

EXAMPLE 4.

What is the vulgar fraction of .58333, &c. or .583.

From the whole decimal .583

Subtract the finite part.

58

QUESTIONS FOR EXERCISE.

5. Reduce .1666, &c. or .16 to a vulgar fraction.

6

6. Reduce .08333, &c. or .083 to a vulgar fraction.

7. Reduce .04545, &c. or.045 to a vulgar fraction.

8. Reduce.04166, &c. or .0416 to a vulgar fraction.

Bedur

COMPOUND PROPORTION.'

Compound Proportion teaches how to resolve such questions as require two or more statings by Simple Proportion. RULE--Set down in the middle place that term of supposition which is of the same kind with the answer sought.Take one of the other terms of supposition, and one of the demanding terms of the same kind with it; then place one of them for a first term, and the other for a third, according to the directions given in the Rule of Three. Do the same with another term of supposition and its corresponding demanding term; and so on if there be more terms of each kind, setting the numbers under each other which fall all on the left-hand side of the middle term, and the same for the others on the right-hand side. Multiply together all the terms standing under each other on the left-hand side of the middle term; and in like manner, multiply all those on the right-hand side of it. Then multiply the middle term by the latter product, and divide the result by the former product, so shall the quotient be the answer.

Note--When you find any of the proportions to be inverse, place the demanding term on the left, and its corresponding term of supposition on the right-hand side.

EXAMPLES.

1. If 40 acres of grass, be cut down by 8 men in 7 days, how many acres will 24 men cut down in 28 days?

2. If 14 horses cat 56 bushels of corn in 16 days, how ma my bushels will 20 horses eat in 24 days? Ans. 120 bush.

3. If 48 shels of corn yield 576 in one year, how much will 240 bushels yield in 6 years at that rate? Ans. 17280 b. 4. If Troods of ditching be done by 3 men in 6 days, how many roeds will be done by 8 men in 4 days? Ans. 32roods.