130. Other Methods of expressing the Value of a Fraction.

(a.) 1. Explain the fraction .

=

of 3, as of a number =

8. Of .4

9. Of .007

10. Of .0346 following method.

2, or of 1, of 3 times the num ber. For if each of 3 equal things should be divided into 4 equal parts, and 1 of the parts of each taken, the result would equal of 1.

NOTE.

- This may be illustrated to the eye, by taking 3 equal lines, and dividing them into 4 equal parts, arranged thus: one of the parts will then contain of 3 lines, which,

as will be seen, is equal to of a line.

11. To what simple fraction is of 3 equal? Answer. of 3 = of 1 = 2.

(c.) To what simple fraction is each of the following

equal?

131. To find a Fractional Part of a Number.

The following forms of writing the work exhibit very clearly the reason for every step which is taken. The letters a, b, &c., are used as explained in the note on the 88th page.

NOTE.

It will be seen that by both methods of solution, we multiply by the numerator, 7, and divide by the denominator, 8. The only difference between them is, that by the first we divide before we multiply, while by the second we multiply before we divide. The second method involves somewhat smaller numbers than the first.

What is the value of each of the following?

8. What is the value of .0014 of 16.427 ?

6. 우궁 of 46935.

7. 1 of 87516.

First Solution. .0014 of 16.427 14 times .0001 of 16.427; .0001 of 16.427, found by removing the point four places towards the left, is 0016427, and 14 times this is .0229978 = Ans.

The work may be written in full after either of the following models

Second Solution. .0014 of 16.427 .0001 of 14 times 16.427; 14 times 16.427229.978, and .0001 of 229.978, found by removing the point four places towards the left, is .0229978 = Ans.

The work may be written in full after either of the following models.

First Model.

a= 16.427

14 times a= b = 229.978

.0001 of b = .0014 of a = .0229978

Second Model.

16.427

14

10000) 229.978

.0229978

NOTE. It will be seen that by both solutions, we multiply by the numerator, 14, and divide by the denominator, 10000; i. e., we multiply by 14, and remove the point four places towards the left. The only dif ference between them is, that in the first we remove the point before we multiply, while in the second we remove it afterwards. Some of the above written work might have been avoided by multiplying by 14, and changing the position of the point at the time of writing the product, thus:

16.427
.0014

.0229978

In like manner we may get .07 of a number by multiplying by 7, and removing the point two places towards the left; we may get 5.2 * of a number by multiplying by 52, and removing the point one place towards the left; &c.

What is the value of each of the following?

132. To multiply by a Fraction.

1. What is the product of 865 multiplied by ?

Suggestion. The product of 865 multiplied by equals times 865, which is the same as of 865, and may be found by methods before explained.

NOTE. This follows from the definition of multiplication - A process by which we ascertain how much any given number will amount to, if taken as many times as there are units in some other given number. In multiplying by more than a unit, then, more than once the multiplicand is taken, while in multiplying by less than a unit, less than once the mul

* It will be more convenient, for purposes of explanation, to regard this às an improper fraction than as a mixed number.

The number of which a fractional part is taken may be regarded either as a mixed number or an improper fraction.

1

tiplicand (i. e., some fractional part of it) is taken; the product being always the same part of the multiplicand that the multiplier is of unity.

When, therefore, the multiplier is greater than unity, the product will be greater than the multiplicand; and when the multiplier is less than unity, the product will be less than the multiplicand. In multiplying by,of once the multiplicand is taken; in multiplying by, of once the multiplicand is taken; &c.

This conclusion is in exact accordance with the principle, that the larger the multiplier the larger the product, and the smaller the multiplier the smaller the product, so long as the multiplicand remains the same; or, to speak more definitely, that the product of a number multiplied by of another, is equal to of its product multiplied by the whole of the other; that the product of a number multiplied by of another, is equal to of its product multiplied by the whole of the other; &c.

Thus, 12 multiplied by 6 = 72.

=

of 72.

of 72.

of 72; &c.

of 12, or 6.

of 12, or 4.

12 multiplied by of 6, or 3, = 36, which is 12 multiplied by 1 of 6, or 2, = =24, which is 12 multiplied by of 6, or 4, 48, which is So, since 12 multiplied by 1 - 12, 12 multiplied by 2 of 1, or 2, must equal 12 multiplied by of 1, or, must equal 12 multiplied by of 1, or, must equal of 12, or 8; &c. NOTE. By the foregoing explanations, it appears that the change of denomination mentioned in the note under 74, (h.) is caused by the fact that to multiply by a fraction requires a division as well as a multiplication.

8. What is the product of 35 times 2948?

Suggestion. 35 times 2948 = 2 times 2948 + of 2948.

1310 of a

*b+c+d=3§ + a = 10481 =Ans.

NOTE. The second of the above forms is preferable to the first, inasmuch as it presents to the eye, in a compact and convenient form, all the numbers which it is necessary to use, and avoids all side work. The references by means of letters may be omitted as soon as the processes and explanations are familiar.

1. How much will of an acre of land cost at $478.36 per acre?

Solution. If 1 acre costs $478.36, of an acre will cost of $478.36, which, found in the usual way, is $418.562.

NOTE. The second form is better than the first, when a full statement and explanation of the successive steps are required; but in other cases the first is preferable.

2. How much will .9 of an acre of land cost at $394.26 per acre?

* Since § = } + §.