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5. Divide 48 by 3715| 12. Divide .594868 by 7.18 6. Divide 35.7485 by 21€ 13. Divide .274869 by .756 7. Divide 2718.5685 by 37.005 14. Divide .81748 by 075 8. Divide 675.008 by .075, 15. Divide .9 by .000125 9. Divide
.007586 by 215 16. Divide 7854 by 10 10. Divide
.41312 by 17:17. Divide 7854 by 1000 11. Divide .087945 by 37.5
18. Divide, seven hundred and ninety by thirty-one and six-tenths.
ns. 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 quotient arising from the same number divided by @08 ? 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. 5125. 29. Divide $115 65 c. among 5 men, so that 4 men may have equal shares; and the 5th only half a share ?
Ş$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 negio, so as the whites may have equal shares, and the negro only # of a share ? Ans. 865 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 ROLE-In every question in the Rule of Three,
Three given numbers there must always be ;
EXAMPI.ES. 1. Suppose I give 12s. 6d. for 4} yards of linen: what will 485 yards of the same come to at that rate.
Ans. £6.3815789=67. 78.74 1. 2. A grocer buys 4 chests of tea, each weighing neat 2cwt. 3qrs. 14 lb. for $906 50cts. at what rate did he give per lb.
Ans. $.7038048=70cts. 3.8m. 3. How much persian .75 yards wide, will line 25.5 yards of 5 quarters wide ?
42żyds. 4. If 13 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 66j cents on the dollar, what must B. receive for his debt?
Ans. $12341 6. A merchant bought 4 tuns, 201) gallons of Corsica wine, for 2401. 16s. 6d. but by misfortune it chanced to leak out 245 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 4125 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. 7934.630402.4cwt. lqr. 1916. 14oz. 9.88dr3. 10. If the cubic inch of olive oil be .52835 decimal parts of
2 EPEATING AND CIRCULATING DECIMALS,
an ounce avoirdupois : what quantity of oil weighing 7} lb. per gallon, will be contained in a cask allowed to hold 13} galions of water, each 282 solid inches?
Ans. 16.555gall.=16gall. 2.22cts. 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. 34.196402. of gold, and 28.8036oz. of silver.
12. Suppose A. lend to B. $1675 for 6 months, what sum m!!st B. lend A. for 3 years to requite him ? Ans. $223 335.
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?
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 ?
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 furs, 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 logether.
Ans. 2 ho. 13 m. 20 s.
REPEATING AND CIRCULATING DECIMALS. As the arithmetic of infinite Decimals is generally considera ed a subject as much of curiosity as of real use, we shall at present only give the rules by which they are reduced to vul. gar fractions; and by means of which any question where they occur may be wrought.
1. If the decimal be å 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?
II. If the decimal be a pure circulate, place the circulating figures for the numerator, and as many O's as there are places in the circle for the denominator.
EXAMPLE 2. When the pure circulating decimal is .259259259, &c. or.259 what is its vulgar fraction?
Ans. 111. If there be ciphers prefixed to the repeating or circulating figures, annex à like number to the 9's in the denominator.
EXAMPLE 3. Let .0555, &c. or.05 be the bepeating, and .0148148, &c. or .0148 the circulating figures, required their equivalent vulgar fractions ?
Ans. 1 and 196=15
==" 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.
From the whole decimal .583
7 denominator 900 12
QUESTIONS FOR EXERCISE. 5. Reduce .1666, &c. or .1ô to a vulgar fraction.
Ans. 6. Reduce .08333, &c. or .083 to a vulgar fraction.
900=i. 7. Reduce .04545, &c. or.045 to a vulgar fraction.
ਠ=ਸੁ 8. Reduce ,04166, &c. or.0416 to a vulgar fraction.
Ans. 9. Reduce .1153846 to a vulgar fraction.
Ans. 20. Reduce 8.3 to a vulgar fraction. Ans. 21. Reduce 65.296 to a vulgar fraction.
115 3845 999 99.90
12. Reduce 4.16 to a vulgar fraction.
5 13. Reduce .53571428 to a vulgar fraction.
Ans. 14. Reduce 8.32142857 to vulgar fraction.
COMPOUND PROPORTION. Compound Proportion teaches how to resolve such ques. tions as require two or more statings by Simple Proportion. * Rule--Set down in the middle place that term of suppos sition 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 ternr; and in like manner, multiply all those on the right-hand side of it. Then multiply the middle term by the Intter product, and divide the result by the former prodict, so shail 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 ?
men If 8 days 7
24 x 28 x 40
=480 acres Ans.
2. If 14 horses eat 56 bushels of corn in 16 days, how many bushels will 20 horses eat in 24 days ? Ans. 120 bush.
3. If 48 bishels pf corn yield 576 in one year, how much will 240 Bushels yield in 6 years at that rate ? Ans. 17280 b.
4. If TË roods of ditching be done by 3 men in 6 days, how many roods will be done by 8 men in 4 days ? Ans. 32 roods.