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35. What number must be multiplied by ; to give a product of 64? (See Proof, page 65.) - : - Ans. 161';. 36. For $73}, how many weeks' board can be obtained at $4Fr per week? Ans. 15;}}. 37. How many cords of wood can be bought for $1094, if 1 cord cost $64? Ans. 16}}. 38. If 7 lb. of tea cost 3r', dollars, what is it per pound? Ans. #4 of a dollar. 39. If a man make 125 barrels in 604 days, how many barrels does he make per day ? Ans. 21; r. 40. 5 men purchased of a ship. If they own equal shares, what part of the whole ship did each
Promiscuous Questions in Vulgar Fractions.
1. What is 4 times the difference between #} and
Explanation.—; of a number is equal to , of the difference between , and the whole. If ; of a number be subtracted from the number, 3 remain, ; must, therefore be # of #. So if we add , of a number to the number itself, the 4 will be 4 of the sum... If we take # of 30, and add it to 30, the sum 40 will be the answer. But if we divide the one third 10, by #, the quotient is the answer, for the whole number 10, being brought into quarters by dividing by #, shows what number 10 is ; of. 2. - r 6. # of 18 equal # of *: number 2
# of 18= 12. Now if we divide 12 by 4, the quotient is 48, which shows that 12 is # of 48. But 12 can be # of a number 5hly £ as large as the number of which it is 4, therefore dividing 48 by 3 gives 16 the answer.
7. A merchant bought a quantity of flour, and sold 150 barrels, which was 4 of the whole quantity. How many barrels did he buy 7 Ans. 750.
8. What is the difference between 2, and 12
Explanation.—As the denominator represents an unit divided into 15 equal parts, if we take 9 the numerator from the denominator, 6 the remainder, exhibits the difference between 9 and 15 ; or 1%, is the difference between or and 1. Ans. ".
9. A man sold a quantity of grain to 3 men ; to A # ; to B ; ; and to § 20 bushels, which was the remaining part. How many bushels did he sell to the 3 men 2 Explanation.—#4 equal the sum of 4 and #. Since # represent the parts sold to A and B, or the remaining part must represent the 20 bushels sold to C. 15 the denominator of the fraction represents the whole quantity, and we may suppose each bushel divided into 15 equal parts, and that A has 5 of those parts, B 9, and C 1. Since 20 is or of the whole quantity, multiplying 20 by 15 must give the whole quantity, for the product will contain the mul-, tiplicand 15 times. But if we divide 20 by T's, we multiply by 15 and divide by 1, which gives the same number for a quotient, that we had for a product before. Had the numerator of the fraction Ps, expressing C's part, been more than 1, C's part must have been a greater part of the whole quantity, and the whole quantity a less quantity:—but in that case, we should have divided by the numerator, which would have made the quotient as much less than it now is, as the numerator was greater than 1. To
prove the work of this question, take # of 300, which will denote A and B's part, and subtract it from 300, the remainder will be C's part. From the three last explanations, we derive the accuracy of the following
For finding the whole number, or quantity, when a part orparts of the number or quantity, are represented by fractions, and a part by a simple number.
Add the given fractions; subtract their sum from 1, (or the numerator from the denominator,) and divide the simple number by the fraction found b subtracting, the quotient will be the whole number or quantity required.
10. A man after spending ; and 3 of his property, had 100 dollars left; what was the value of his proerty 2 - * * Ans. $444;. 11. A butcher killed an ox, and sold # of # of it to one man; so to a second; and the remainder, which was 300 lb., to a third. What was the whole weight of the ox 2 * Ans. 738;4 lb. 12...What costs a job of work performed by 3 men, if the first man do #; the 3d. }; ; and the third the remainder, for which he receives $56 Ans. $256,or. 13. A man owning or of a house, sold } of his share to D, and # to F. Now, if all the owners except D and F, receive yearly $208 for their part of the rent, what is the whole rent, and what is D's share of it? Who! hole rent $24743. Ans. D’s share $22.É. 14. A gentleman divided his estate between his 3 sons, as follows: to the 1st he gave ; of it; to the 2d # of the remainder. The difference between the
portions of the 1st and 2d, was 500 dollars. What
The only difference between a vulgar and a decimal fraction, is this, the denominator to a decimal fraction is always 1, with a cipher or ciphers at the right hand of it, while the denominator to a vulgar fraction may be any other number.
Derivation and explanation of Decimal Fractions.
We have already explained how vulgar fractions are derived from division ; we shall now illustrate the manner in which decimal fractions are obtained. To do this, we shall make use of several examples.
If 18 dollars be divided equally between 12 men, how many dollars will each man receive *
Operation. - * 2) 1 8 (1, 5 It was proved, Contraction 1 2 2d, page 45, that annexing a - cipher to a number, multi6 0 plies that number by 10; 12 6 0 can be taken from 18 once,
- and 6 remains. By annexing a cipher to 6, the remainder, each of the 6 dollars is divided into 10 equal parts, since 60 contains 6 tens. Annexing the cipher does not increase the number of dollars, but denotes the remaining number of dollars divided into parts. By subtracting 12, the number of men, from 60 once, we take 12 tenths of a dollar, which is enough to give each person 1 tenth, and as 12 can be taken 5 times from 60, the whole number of tenths, each man will have 5 tenths besides the 1 dollar. A comma is set between the 1 and the 5 to denote that the 5 is a fraction. The denominator 10 is understood to stand under the 5, and if written, the fraction would stand thus F. Had we set 12 the divisor under 6 the remainder, the fraction would have been # which equals #;—but to equal 4, for 5 is half of the denominator. The denominator is not written, since any number obtained by adding one cipher to the remainder, will be tenths, and will always be known as such without a denominator written to express it. The answer to the question is therefore $1}, of which is the same thing $1,5. When any figure stands at the right hand of units’ place, and is separated from it by a comma, (which is called a separatrix) it denotes as many tenths of an unit as it contains 1's, whether it be actually obtained from division or not, for it always represents the same thing as though it had been.
If 15 dollars be equally divided between 12 men, how many dollars will each man receive *