A whole number may be considered as a fraction whose denominator is 1 (thus 16 is 4) (89); and the same rule will apply when one or more of the quantities are whole numbers.

EXERCISES.

Multiply together 1, 3, 3, 4 and 7%,

3 and 4.

From 100 acres of ground, two-thirds of them are taken away; 50 acres are then added to the result, and of the whole is taken. What number 97 of acres does this produce?

A can reap a field of corn by himself in 6 days, B can do it in of that time, and C in of that time. How much can all do together in half a-day?

(103.) In dividing one whole number by another, for example 108 by 9, this question is asked:— Can we by the addition of any number of nines produce 108?-and if so, how many nines will be sufficient for that purpose?

Suppose we take two fractions, for example and, and ask, Can we by dividing into some number of equal parts, and adding a number of these parts together, produce?-if so, into how many parts must we divide, and how many of them must we add together? The solution of this question is still called the division of by ; and the fraction whose denominator is the number of parts into which is divided, and whose numerator is the number of them which is taken, is called the quotient. The solution of this question is as follows:

45

Reduce both these fractions to a common denominator (93), which does not alter their value (91.): they then become 19 and . The question now is, to divide into a number of parts, and to produce by taking a number of these parts. Since is made by dividing 1 into 15 parts and taking 12 of them, if we divide into 12 equal parts, each of these parts is these parts, the result is. to produce 19 or (91), we into 12 parts, and take 10 of them; that is, the quotient is 1. If we call the dividend and the divisor, as before, the quotient in this case is derived from the following rule, which the same reasoning will show to apply to other cases:

if we take 10 of Therefore, in order must divide or

The numerator of the quotient is the numerator of the dividend multiplied by the denominator of the divisor. The denominator of the quotient is the denominator of the dividend multiplied by the numerator of the divisor.

(104.) By taking the following instance, we shall see that this rule can be sometimes simplified.

Di

vide by Observe that 16 is 4 x 4, and 28 is 1 2. 4 x 7: 33 is 3x 11, and 15 is 3 x 5; therefore the

which 4x 3 is found both in the numerator and de

nominator. The fraction is therefore (91) the same

as

4 X 5

11 × 7'

or 29. The rule of the last article therefore admits of this addition :-If the two numerators or the two denominators have a common measure, divide by that common measure, and use the quotients instead of the dividends.

(105.) In dividing a fraction by a whole number, for example by 15, consider 15 as the fraction

. The rule gives as the quotient. Therefore, to divide a fraction by a whole number, multiply the denominator by that whole number.

EXERCISES.

Divide by 108, 163 by 34, and 25 by 160. 1 × 1 × 1 - 1 × 14 × 17

What are

91

What part is 10 of 23, 462 of 1803, and 1688 of 1830?

A can reap a field in 12 days, B in 6, and C in 4 days. In what time can they all do it together?

SECTION VI.

DECIMAL FRACTIONS.

(106.) We have seen (94) (103) the necessity of reducing fractions to a common denominator, in order to compare their magnitudes. We have seen

also how much more readily operations are performed upon fractions which have the same, than upon those which have different, denominators. On this account it has long been customary, in all those parts of mathematics where fractions are often required, to use none but such as either have, or can be very easily reduced to others having, the same denominator. Now, of all numbers, those which can be most easily managed are such as 10, 100, 1000, &c., where 1 is followed by ciphers. These are called decimal numbers, and a fraction, whose denominator is any one of them, is called a DECIMAL FRACTION, or, more commonly, a DECIMAL.

(107.) A whole number may be reduced to a decimal fraction, or one decimal fraction to another, with the greatest ease. For example, 94 is 240

or

3000 (91). The addition of a cipher to the righthand of any number is the same thing as multiplying. that number by 10 (47), and this may be done as often as we please in the numerator of a fraction, provided it is done as often in the denominator (91).

(108.) The next question is, is it possible to reduce a fraction which is not a decimal to another which is, without altering its value? Take, for example, the fraction, multiply both the numerator and denominator successively by 10, 100, 1000, &c., which will give a series of fractions, each of which is equal to (91), viz. 76%, 100%, 760005 70000 &c. The denominator of each of these

160000"

fractions can be divided without remainder by 16,

E

the quotients of which divisions form the series, of decimal numbers 10, 100, 1000, 10000, &c. If therefore one of the numerators is divisible by 16, the fraction to which that numerator belongs has a numerator and denominator both divisible by 16. When that division has been made which (91) does not alter the value of the fraction, we shall have a fraction whose denominator is one of the series 10, 100, 1000, &c., and which is equal in value to

The question is then reduced to finding the first of the numbers 70, 700, 7000, 70000, &c., which can be divided by 16 without remainder.

Divide these numbers, one after the other, by 16, as follows:

16)70(4 16)700(43 16)7000(437 16)70000(4375

It appears, then, that 70000 is the first of the numerators which is divisible by 16. But it is not necessary to write down each of these divisions, since it is plain that the last contains all which came