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35. What number must be multiplied by to give a product of 61? (See Proof, page 65.)

Ans. 16.

36. For $735, how many weeks' board can be obtained at $4 per week? Ans. 157. 37. How many cords of wood can be bought for $109, if 1 cord cost $61?

38. If 7 lb. of tea cost 3,

pound?

Ans. 161

dollars, what is it per Ans. 7 of a dollar.

39. If a man make 125 barrels in 60 days, how many barrels does he make per day? 40. 5 men purchased of a ship. equal shares, what part of the whole man purchase?

Ans. 2TT
If they own
ship did each
Ans.

41. Bought 31 yards of cloth for $145, what did I give per yard?

Ans. $44.

Promiscuous Questions in Vulgar Fractions.

1. What is 4 times the difference between 1 and

Ans.

2. What is a 4th part of one third, plus one fourth?

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5. of 30 is equal to 1, of what number?

Explanation. of a number is equal to of the difference between and the whole.

ber be subtracted from the number, therefore be of 3. So if we add

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If of a num

remain, must of a number to the number itself, the will be 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 1, the quotient is the answer, for the whole number 10, being brought into quarters by dividing by, shows what number 10 is of.

6. of 18 equal of what number?

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of 1812. Now if we divide 12 by, the quotient is 48, which shows that 12 is of 48. But 12 can be of a number only as large as the num94 ber of which it is, therefore dividing 48 by 3 gives 16 the answer.

7. A merchant bought a quantity of flour, and sold 150 barrels, which was of the whole quantity. How many barrels did he buy? Ans. 750.

8. What is the difference between

and 1?

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, is the difference between and 1.

Ans.

9. A man sold a quantity of grain to 3 men; to A ; to B; and to Ĉ 20 bushels, which was the remaining part. How many bushels did he sell to the 3 men?

Explanation.-1 equal the sum of and . Since §. represent the parts sold to A and B, 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 i. Since 20 is of the whole quantity, multiplying 20 by 15 must give the whole quantity, for the product will contain the multiplicand 15 times. But if we divide 20 by '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, 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

RULE,

For finding the whole number, or quantity, when a part or parts 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 by subtracting, the quotient will be the whole number or quantity required.

10. A man after spending and of his property, ៖ had 100 dollars left; what was the value of his property? Ans. $444. 11. A butcher killed an ox, and sold of of it to one man; to a second; and the remainder, which was 300 lb., to a third. What was the whole weight

of the ox?

Ans. 738 lb.

12. What costs a job of work performed by 3 men, if the first man do ; the 2d. ; and the third the remainder, for which he receives $56?

Ans. $256

13. A man owning 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?

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$22.

14. A gentleman divided his estate between his 3 sons, as follows: to the 1st he gave 3 of it; to the 2d of the remainder. The difference between the portions of the 1st and 2d, was 500 dollars.

What

was the whole estate, also, what was the 3d son's share?

3

22

Ans.

Whole estate $12000. 3d son's share $2500.

15. of a certain farm is wild land; & pasturing; mowing; tillage, and the remainder, which is 3 acres, is orcharding. How much land in said Ans. 174 A.

farm.

16. A certain school house is owned by 14 proprietors, and rented for $24 annually. A and B own each of the house; C; D and E each; F G and of; Hand of ; I and of '; Kofs; Lof; M; N and of; and O and of. What does each proprietor receive annually for his share of the rent?

Ans. A and B each $1,60; C $3,20; D and E each $1; F and H each $2; G $1,50; I $2,40; K $1,20; L$0,75; M $1,15; N $1,80; and O $2,80.

DECIMAL FRACTIONS.

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) 18 (1,5

1 2

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60
60

It was proved, Contraction 2d, page 45, that annexing a cipher to a number, multiplies that number by 10; 12 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 équal 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. Had we set 12 the divisor under 6 the remainder, the fraction would have been

which equals ;-but equal, 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 á denominator written to express it. The answer to the question is therefore $14, or 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?

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