(ART. 41.) To investigate and form a rule to multiply a fraction by a whole number, we can take some simple fraction, as 4, and if it be multiplied by 2, the result must evidently be 2; and if it be multiplied by 3, the result must be ; if by 5, the result must be &c. Also, take, and multiply it by 4: the result will be 4, or 4. Also, multiplied by 2 will give; multiplied by 3 is, or ; and from these observations, to multiply a fraction by a whole number, we deduce the following

RULE. Multiply the numerator by the whole number; or, when you can, divide the denominator by the whole number.

To multiply a whole number by a fraction is the same as to multiply a fraction by a whole number; for, when two numbers are multiplied together, it is indifferent which we call the multiplier, or which the multiplicand. (See Art. 11.) That is, in the last example, 7 multiplied by 11, is the same as 11 multiplied by; and so of the other examples.

(ART. 42.) If we multiply a fraction by its denominator, it will produce the numerator for a product.

EXAMPLES.

1. Multiply by 7. By the above rule, the product must be 3,

N. B. Let the pupil remember this article, when he comes to clearing equations of fractions in Algebra.

(ART. 43.) We have already defined Complex Fractions to be such as have fractions in the numerator or denominator, or in both, as

2}

7'

Now, if we multiply both numerator and denominator of this fraction by 2, we shall have 5

11, a simple fraction.

But, why did we multiply by 2, in preference to any other number? Ans. Because it was the denominator in the fractional part of the

and we can clear its numerator of the fraction by multiplying it by 3, and we can clear its denominator by multiplying it by 8. Hence, we can banish fractions from both numerators by multiplying by 3 and then by 8, or multiplying both numerator and denominator by 24

at once.

Thus: 93X24=232; 83X24=201. equivalent simple fraction.

Therefore, 232 is the

It is now apparent that we can change complex fractions to simple fractions, by the following

RULE. Multiply both numerator and denominator by the denomi nators of the partial fractions, or by their product, or by their least common multiple.

If we divide by 2, that is, one-half cut into 2 equal parts, each part will be. If we divide by 3, or in other words, cut into 3 equal parts, each part must be 1. But, we can obtain these results mechanically, by multiplying the denominator of the fraction by the divisor.

5

If we were required to divide into 3 equal parts, each part will be In this case, then, we can divide the fraction by dividing the numerator, because it is susceptible of being divided into the parts required.

Therefore, to divide a fraction by a whole number, multiply the denominator of the fraction by the whole number, and place the numerator over the product. Or, when you can, divide the numerator by the whole number, and let the denominator remain unchanged.

7. Divide 17 by 34.

8. Divide 15 by 15.

Ans. 3

Ans. 5

32°

This is the reverse operation of Art. 41.

[N. B. Let pupils omit the following Article the first time going through.]

CONTINUED FRACTIONS.

(ART. 44.) We have a clearer conception of the value of a fraction when its numerator is unity, than when it is in any other form. For instance, we have a clearer understanding of the fraction than of although they differ in value but little. very

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1478'

As it does not change the value of a fraction by dividing both numerator and denominator by the same number, (Art. 36,) we can always change the numerator to unity, by dividing both terms of the fraction by the numerator; but this will make the denominator a mixed number,* and the fraction a complex fraction.

If we take the fraction 27, and divide both numerator and deno

minator by 287, we have

1

31379

287

which shows that the value of the

fraction is between and 4, and is its first approximate value. If we take 131, and divide its numerator and denominator by 131,

1319

until we find the

last fraction with unity for a numerator, the original fraction, 287, will take the following form, which is a continued fraction.

1
3+1

0927

241

5+1

4+1

*That is: In case the numerator and denominator are prime to each other, as it is supposed they are. If they are not, the fraction must be reduced by Article 36,

If we neglect the fractional part of the last denominator, the second approximate fraction will be

1

or 2. 3+1

The first approximate value of the original fraction, 387, is, as

992

we have just observed, . The second approximate value is

1

3+1

And the fifth and last approximate value is the fraction itself.

Let it be observed, that the principle here involved is all encompassed in Article 36, and that we reduce fractions to this shape by division; and, of course, to reduce them back again, all we would have to do is to inspect the work, and take the reverse operation. For example, what simple fraction expresses the value of the third approximate value of the fraction under consideration?

We must commence with the last denominator, 1, and multiply

1

the numerator and denominator of the fraction by 5, which gives

2+1

In this manner we shall find the approximate values of the fraction

By inspection, we perceive that a continued fraction may be defined thus:

A continued fraction is one that has a whole number and a fraction for its denominator, which fraction has also a whole number and a fraction for its denominator, and so on, as far as the fraction may extend; and each fraction must have unity for its

numerator.

H

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