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1261 182

1261 +44 =126544 dolls. Hence, to multiply a mixed number : multiply the whole number and the fraction separately; then reduce the fraction to a whole or mixed number, and add it to the product of the whole number.

XVI. We have seen that single things may be divided into parts, and that numbers may be divided into as many parts as they contain units ; that is, 4 may be divided into 4 parts, 7 into 7 parts, &c. It now remains to be shown, how every number may be divided into any number of equal parts.

If 3 yards of cloth cost 12 dollars, what is that a yard ?

It is evident that the price of 3 yards must be divided into 3 equal parts, in order to have the price of 1 yard. That is, į of 12 must be found.

We know by the table of Pythagoras, that 3 times 4 are 12, therefore { of 12, or 4 dollars is the price of 1 yard.

If 5 yards of cloth cost 45 dollars, what is that a yard?

1 yard will cost } of 45 dollars. 5 times 9 are 45, therefore 9 is į of 45, or the price of 1 yard.

The two last examples are similar to the first example Art. 9. If we take 1 dollar for each yard, it will be 5 dollars, then one for each yard again, it will be 5 more, and so on, until the whole is divided. The question, therefore, is to see how many times 5 is contained in 45, and the result will be a number that is contained 5 times in 45. 5 is conLained 9 times, therefore 9 is contained 5 times in 45. This is evident also from Art. III. When a number, therefore, is to be divided into parts, it is done by division. The number to be divided is the dividend, the number of parts the divisor, and the quotient is one of the parts.

A man owned a share in a bank worth 136 dollars, and sold i of it; how many dollars did he sell it for?

136 (2

Ans. 68 dollars. 2 is contained 68 times in 136, therefore 2 times 68 are 136, consequently 68 is of 136.

A ticket drew a prize of 2,845 dollars, of which A owned }; what was his share?

2845 (5

Ans. 569 dollars. Since 5 is contained 569 times in 2,845, 5 times 569 are equal to 2,845, therefore 569 is of 2,845. Division may be explained, as taking a part of a number. In the above example I say, ; of 28(00) is 5000) and 3(00) over. Then supposing 3 at the left of 4, I say, ļ of 34(0) is 6(0) and 4(0) over. Then } of 45 is 9. Writing all together it makes 569, as before. The same explanation will apply when the divisor is a large number.

Bought 43 tons of iron for 4,171 dollars ; how much was it a ton ?

1 ton is as part of 43 tons, therefore the price of 1 ton will be as part of the price of 43 tons.

4171 (43
387

97 dollars.
301
301

Two men A and B traded in company and gained 454 dollars, of which A was to have and B }; what was the share of each ?

The name of the fraction shows how to perform this example. of 456 signifies that 456 must be divided into 8 equal parts, and 5 of the parts taken. of 456 is 57, 5 times 57 are 285, and 3 times 57 are 171. A's share 285, and B's 171 dollars.

456 (8

57 3

57
5

B's share 171 dollars.

A's share 285 dollars.

A man bought 68 barrels of pork for 1224 dollars, and sold 47 barrels, at the same rate that he gave for it. How much did the 47 barrels come to ?

To answer this question it is necessary to find the price of 1 barrel, and then of 47. 1 barrel costs as of 1224 dollars, and 47 barrels cost of it. o of 1224 is 18. 47 times are 18 are 846. Ans. 846 dollars.

To find any fractional part of a number, divide the number by the denominator of the fraction, and multiply the quotient by ihe numerator.

A man bought 5 yards of cloth for 28 dollars ; what was that a yard?

of 25 is 5, and } of 30 is 6. Žof 28 then must be between 5 and 6.

Cases of this kind frequently occur, in which a number cannot be divided into exactly the number of parts proposed, except by taking fractions. But it may easily be done by fractions.

of 25 dollars is 5 dollars. It now remains to find } of 3 dollars. Suppose each dollar divided into 5 equal parts, and take 1 part from each. That will be 3 parts or of a dollar. Ans. 5 dollars. of a dollar is of 100 cents, which is 60 cents. Ans. $5.60.

A man had 853 lb. of butter, which he wished to divide into 7 equal parts ; how many pounds would there be in each part?

} of 847 lb. is 121 lb. Then suppose each of the 6 remaining pounds to be divided into 7 equal parts, and take 1 part from each ; that will be 6 parts, or of a pound.

Ans. 1214.

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A man having travelled 47 days, found that he had travelled 1800 miles; how many miles had he travelled in a day on an average ? How many miles would he travel in 53 days, at that rate ?

In one day he travelled 'y of 1800 miles, and in 53 days he would travel of it. 14 of 1800 is 38, and 14 over. y of 1 is one of 14 is 14 times as much, that is, 4. In one day he travelled 3844 miles. In 53 days he would travel 53 times 3844 miles. 1800 (47

38 53 141

53 14
3844 miles in 1 day.
390

114 212
876

190 53

14

742

2014

1537

un = 1527

Ans. 202937 miles in 53 days. Hence to divide a number into parts ; divide it by the Rember of parts required, and if there be a remainder, make it the numerator of a fraction, of which the divisor is the denominator.

N. B. This rule is substantially the same as the rule in Art. X.

When one part is found, any number of the parts may be found by multiplication.

It was shown in Art. X. that, in a fraction, the denoninator shows into how many parts 1 is supposed to be divided, and that the numerator shows how many of the parts are used. It will appear from the following examples, that the numerator - is a dividend, and the denominator à divisor, and that the fraction expresses a quotient. The denominator shows into how many parts the numerator is to be divided. In this manner division may be expressed without being actually performed. If the fraction be multiplied or divided, the quotient will also be multiplied or divided. Hence division may be first expressed, and the necessary operations performed on the quotient, and the operation of division itself omitted, until the last, which is often more convenient. Also, when the divisor is larger than the dividend, division may be ex pressed, though it cannot be performed.

A gentleman wishes to divide 23 barrels of flour equally among 57 families; how much must he give them apiece ?

In this example, the divisor 57 is greater than the dividend 23. If he had only 1 barrel to divide, he could give them only zų of a barrel apiece ; but since he had 23 barrels, he can give each 23 times as much, that is, 4 of a barrel.

Hence it appears that ; * rightly expresses the quotient of 23 by 57.

If it be asked how many times is 57 contained in 23 ? It is not contained one time, but of one time. If 10 lbs. of copper cost 3 dollars, what is it

per

16.? Here 3 must be divided by 10. 1o of 1 is to, and is of 3 must be is Ans. io of a dollar, that is, 30 cents.

At 43 dollars per hhd., what would be the price of 25 galls. of gin?

25 galls. are it of a hogshead. To find the price of 1 gallon is to find o of 43 dolls., and to find the price of 25 galls. is to find of 43 dolls. o of 1 is os, os of 43 is 43 times as much, that is, 11. lis 25 times as much as ago that is, 25 times 4. 25 times are 1875 = 177 dolls. Ans. If 5 tons of hay cost 138 dolls. what cost 3 tons ?

3 tons will cost of 138 dolls. This may be done as fol lows. of 138 is 27}, and 3 times 27}, are 82 dolls. Ans. Or,

Expressing the division, instead of performing it, { of 138 is 138. Žof 138 are 3 times '38, that is, *}* = 825 dolls. as before.

Note. of 138 by the above rule is 27. But the same result will be obtained, if we say, { of 138 is 38, for 128 are equal to 273.

The process in this Art. is called multiplying a whole num. ber by a fraction. Multiplication strictly speaking is repeating the number a certain number of times, but by exten sion, it is made to apply to this operation. The definition of multiplication, in its most extensive sense, is to take one number, as many times as one is contained in another number. Therefore if the multiplier be greater than 1, the product will be greater than the multiplicand; but if the multi

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