to each other, although the terms are different, for in each fraction the denominator is just double the numerator. In 1, 2, 3, 4, &c. the denominators are triple the numerators, and consequently the fractions are equal to each other. Now if any of these fractions are to be reduced, you will divide the numerator and denominator by the same number; and as these fractions are equal to each other, they can all be reduced to the same fractional terms. Thus may be reduced to by dividing both terms by 2; by dividing them by 3; may be reduced to by dividing both terms by 3; by dividing them by 4, and the same with the others.

To make this familiar to you I will give you a few ex

amples.

Hence it follows, that in order to simplify a fraction we must divide its two terms by one of the numbers 2, 3, 4, &c.; but by this mode of proceeding it will often be necessary to perform a great number of operations.

by

3. Here it may be proper to mention, that dividing the numerator lessens the fraction, and dividing the denominator increases it. If you divide the numerator in 4, you make it. If you divide the denominator by 4, you increase it, making it 4. Multiplying the numerator increases the value of the fraction; multiplying the denominator decreases its value. If you multiply the denominator in by 4, you make it ; if you multiply the numerator you make it 19.

48

We will now consider the general rule for finding a number by which the two terms of the fraction can be reduced to their lowest terms. This number is called the common divisor or common measure.

If there be two numbers only, divide the greater by the less, and this divisor by the remainder, and so on till nothing remains, always dividing the last divisor by the last remainder; then will the last divisor be the greatest common divisor required.

When there are more than two numbers, find the greatest common divisor of two of them first; then of that common divisor and one of the other numbers, and so on with all the numbers ; when the greatest common divisor last found will be the answer.

If 1 happen to be the common divisor, the numbers are prime to each other, and are incommeasurable, or in their lowest terms.

What is the greatest common divisor of 132 and 356 ?. 132) 356 (2

264

92) 132 (1
92

40) 92 (2
80

12) 40 (3
36

4)12 (3

12

Pup. I do not understand why dividing in this way should give a common divisor; I should like to have you explain it.

4. Tut. It is plain that the greatest common divisor of these two numbers cannot exceed the least, 132; then it is proper to try if 132, which divides itself and gives 1 for a quotient, will divide 356; if it will, and not leave a remainder, 132 is the greatest common divisor of those two numbers. But in this example it is not the case; there is a remainder of 92.

Now it is evident, that every common divisor of the two numbers 132 and 356 should also divide 92, the remainder resulting from their division; because the greater, 356, is just equal to twice 132 with the remainder added. Now the common divisor will divide both numbers without a remainder, and as it will divide 132, 356, and 92, it follows that by dividing 132 by the remainder, after dividing 356 by 132, and continuing to divide in

this manner, you will get a common divisor, which will divide both the terms, 132 and 356, and each of the remainders with exactness.

Thus, in the example given, 4, the greatest common divisor of 132 and 356, will also divide each of the re-. mainders, 92, 40, and 12.

Having thus explained the rule, you will be able, if you have given proper attention to it, to perform the operations of the rule without difficulty.

Pup. The reasons appear very plain, and I think I shall be able to apply the rule in all cases; it is not so difficult as I at first imagined.

Tut. You may find the greatest common divisor for the following numbers.

What is the greatest common divisor of 24 and 36 ?

Ans. 12.

What is the greatest common divisor of 35 and 100 ?

Ans. 5.

What is the greatest common divisor of 1224 and 1080?

Ans. 72.

the

The common divisor of more than two numbers is found by the common divisor of the two numbers, and one of the other numbers, and so on with the whole, last greatest common divisor being the greatest common divisor of all the numbers.

It is not always necessary to find the greatest common divisor, in order to reduce the terms of the fraction, for some numbers are such that they may be known to be divided by certain numbers as their common divisors. Every number terminated by 0, 2, 4, 6, or 8, is divisible by 2, for in dividing any number by 2, only 1 can remain, from the tens; and as no figure prefixed to any of these will make it such a number that, when divided by 2, 1 will remain, it follows that it must be divisible by 2.

Likewise every number terminated on the right by 0 or 5, is divisible by 5, for when the division by tens has been performed, the remainder, if there is any, must be 1, 2, 3, or 4; the remaining part of the operation will be performed on the numbers 0, 5, 10, 15, 20, 25, 30, 35, 40 or 45, all of which are divisible by 5.

That you may make the rule familiar, you may reduce

the following fractions.

You have now been taught how to reduce fractions to their lowest terms; the next step will be to learn how to multiply fractions.

5. In Multiplication, when the multiplier is a whole number, it shows how many times the multiplicand is to be repeated. (2. Con. II.) But the term multiplication, when applied to fractional numbers, does not always imply augmentation, as it does in whole numbers. When it is required to multiply by 4, the multiplicand is 4 times repeated, and consequently the product is 4 times greater than the multiplicand; if it were required to multiply by 2, which is one half of 4, the product would be one half of what it would have been, had the multiplier been 4; and if it be required to multiply by 1, the product will be one half of the product by 2, or merely a repetition of the figures of the multiplicand; thus the smaller the multiplier, the smaller the product will be.bears the same proportion to 1, that 1 does to 2; therefore if you multiply by, the product will be one half what it would have been had you multiplied by 1, yiz. one half the multiplicand.

If, for instance, it be required to multiply any number by, all you want is two thirds of that number. This is obtained by taking one third of it, and multiplying it by

2.

Thus to multiply 24 by 3, you first find what is by division, and then repeat that twice, and it will give the product of 24 by 3. Thus you see that to multiply any number by is to take from it such a part as shall be equal to of that number, or equal to twice.

Therefore, to multiply any whole number by a fraction, Divide the whole number by the denominator of the fraction, and multiply the quotient by the numerator.

Pup. How shall I proceed when there is a remainder after dividing by the denominator?

Tut. When there is a remainder after dividing the whole number by the denominator, this remainder will be a fraction of the quotient, and makes what is called a mixed number. If, for instance, it be required to multiply 22 by, you will have 5 for a quotient figure, and 2 remainder. This remainder is the numerator, and 4 is the denominator of a fraction; and the quotient will be 52, which is a mixed number. This fraction multiplied by 3 will give 15%.

6. Here you have a fraction in which the numerator is larger than the denominator, and may be thus explained. The expression & denotes 6 parts, 4 of which make a unit; and hence it follows that are equivalent to one and two fourths expressed 12 equal to 11, which added to 15 make 161, which is 3 of 22.

From the preceding examples it will appear evident that contains unity, or a whole one, and 2 over, and hence it follows that are equal to 12.

You must now see plainly that denotes 6 parts, 4 of which make a unit, and every fractional expression, in which the numerator exceeds the denominator, contains one or more units or whole ones, and that these whole ones may be extracted by dividing the numerator by the denominator The quotient expresses the number of units there are in the fraction, and the remainder, if there be any, written as a fraction. is what must accompany the whole ones. Hence we have the following rules for reducing mixed numbers to improper fractions, or fractions: whose numerators exceed the denominator; and imoper fractions to mixed or whole numbers.