3 and the fraction, we should say, 3 times 7 are 21, which added to the numerator 5 make 26, which is therefore the numerator of the equivalent improper fraction, the denominator being the same as that in the given fraction: therefore 326. In like manner, 48=35; 163=5; 13= 128 8; and so on. And this merely amounts to writing the given whole number in a fractional form with the given denominator for its denominator; thus, taking the last instance above, namely 13, the 13 is the same as 9 times 13 divided by 9; that is, 13=117; therefore 13 and 11, or, as it is written, 13=128. The reason of the rule is thus evident.

9

9

(50.) To reduce an Improper Fraction to a Mixed Number.

RULE. Actually perform the division denoted by the fraction, and to the quotient annex the remainder with the divisor underneath; that is, complete the quotient by adding the fractional correction. Thus, performing the division implied in 2, the integral part of the quotient is 3, with 5 for remainder; so that the fractional part of the quotient is therefore the complete quotient is the mixed number 35.

These two rules are so easy and obvious, that but a very few exercises in them need be given.

4. Reduce 2075 to a mixed number.

5. Reduce 23819 to an improper fraction. 6. Reduce 31872 to a mixed number.

123

7. Reduce 201627 to an improper fraction. 8. Reduce 12731 to a mixed number.

317

(51.) To reduce Fractions with Different Denominators to others with Equal Denominators.

This reduction is called the reduction of fractions to a common denominator: it is a change necessary to be made in fractions of different denominators, before they can be

You

either added together, or subtracted one from another. will be prepared to expect this; for you know that things of one denomination cannot be added to or subtracted from things of a different denomination, till they are prepared in this way: shillings and pence, or cwts. and lbs., cannot be united together in one result, without distinction of denomination, unless the differing denominations be first changed to common denominations, that is, to denominations the same in, or common to, both. In like manner, cannot be either added to or subtracted from, till the fractions are changed into others, equal to them in value, and of a common denominator; since thirds and fifths, being different denominations, cannot be united together in a single denomination. The desired change may always be effected by aid of the following principle; namely, If both terms of a fraction be multiplied by any number, the two products may be put for the terms themselves; and this is only saying, that we may multiply a dividend and its divisor by any number we please, without altering the quotient: and it is plain, that the quotient of a dividend by its divisor, is the same as the quotient of twice the dividend by twice the divisor; three times the dividend by three times the divisor; and so on, for any number of times: if, for instance, a sum of money is to be divided among a certain number of people, the share of each must be the same as if twice the sum were to be divided among twice the number of people, or as if 8 times the sum were to be divided among 8 times the number; 10 times the sum among 10 times the number; and so on. The liberty thus given to us, to multiply the terms of a fraction by any number we please, enables us to change those fractions having different denominators into others, equal to them in value, but with the same denominators, by the following rule:

(52.) RULE 1. Multiply the numerator of each fraction by the product of the denominators of all the other fractions: the several results will be the several numerators of the changed fractions.

2. Multiply all the denominators together: the product will be the denominator common to all.

Thus, to change the fractions,, into others of the same value, and with a denominator common to all, we multiply the numerator 1, of the first, by 45, the product of the denominators 5 and 9 of the other fractions; we then multiply the numerator 3, of the next fraction, by 18, the product

54

9

of the denominators 2 and 9 of the other fractions; and, lastly, we multiply the numerator 7, of the next fraction, by 10, the product of the denominators 2 and 5 of the other fractions: we thus get for the numerators of the new fractions, 45, 54, and 70; and for the common denominator, 2 × 5 × 990. Hence the proposed fractions,,,, are, respectively, equal to 45, 53, 38: for these are no other than the former fractions, after numerator and denominator of each are both multiplied by the same number: both terms of the first fraction,, are multiplied by 45; both terms of the second,, by 18; and both terms of the third, 7, by 10. And similarly, in all cases, by following the directions of the rule, we multiply the terms of each fraction by the product of the denominators of all the other fractions; so that though the fractions are changed in appearance, they remain unchanged in value.

Exercises.

Reduce the following fractions to others of equal values, having a common denominator.

(53.) You see, from these examples, that the rule just given will always enable you to convert a set of fractions, with different denominators, into another set equal to them in value, with the same denominators. Any rule would do that would always supply us with a set of multipliers, for the terms of the several fractions, such that the products derived from the denominators should be all alike: the smaller such suitable multipliers are, the neater and simpler will be the changed forms; and such smaller multipliers often suggest themselves, by merely passing the eye along the row of denominators; for instance, if the original fractions were,,, you would see in a moment, that the denominators would give equal products if the first were multiplied by 4, the second by 2, and the third left as it is, without any multiplication at all: therefore, multiplying both terms of the first fraction by 4,

48

649

5

both terms of the second by 2, and leaving the third untouched, we have the changed forms,,, :-three fractions equivalent to the original ones, and all of the same denomination-eighths. By the rule, the given fractions would have been changed into 32, 3, 4, which are less simple in appearance than the other set, though the same in value: the former would be converted into these by multiplying both terms of each by 8. You thus perceive, that before applying the general rule to a set of fractions, it will always be prudent to look a little at the row of denominators, and try to find out, whether smaller multipliers than those which the rule would give you cannot be discovered: the smallest possible always can be discovered by a mode of proceeding which I will show you presently: but the fewer the rules you depend upon the better; a little thought and attention will often supply their place. I shall therefore give you a few fractions to be reduced to a common denominator, without appealing to the rule; first, however, noticing that, as the terms of a fraction may be multiplied by any number, so they may be divided by any number, whenever such division is possible: thus, is reducible to; to; and so on, as is plain, because by multiplying both terms, we know that = 4,

=14&c. It would be considered as an arithmetical fault in a person, who pretended to a knowledge of fractions, to leave a fraction at the close of his work, of which the numerator and denominator have an obvious common divisor. Be careful to avoid this fault: never allow your work to end with such a form as 1, or 2, or 15, &c., where the simplifying divisors are obvious: the final forms in which these fractions should be put, are 4, 3, and, which are incapable of further simplification. In the fractions which form part of the complete quotients, in the exercises on Division (p. 35), instances occur, where what would now seem an obvious simplification, is neglected; but you could not be supposed to know then, that the terms of a fraction might be divided by a number without changing its value.

It may be of use to you to know and to remember, that a number is divisible by 2, if its last figure be either an even number or 0; that it is divisible by 3, if the sum of its digits (or figures) be divisible by 3; by 4, if the number expressed by its last two figures be divisible by 4; by 8, if the number denoted by its last three figures be divisible by 8; by 5, if its last figure be either 5 or 0; and by 9, if the sum of its digits be divisible by 9.

E

In the exercises below you will have to attend to two things: first, to see whether any individual fraction can be readily changed for another of lower terms; if it can, to make the change at once: second, to seek the smallest multipliers for reducing each set of fractions to others with a common denominator.*

(54.) To give you some guidance in this search, I will show you how to proceed with Example 21, below. You observe here, that the denominators 4 and 6, of the first and third fractions, have a common factor, 2; the one denominator being 2 x 2, and the other 3 x 2. Now these denominators will become equal if the factor 3 be introduced into the first, and the factor 2 into the second; for then each will be composed of the same factors, 2, 2, and 3; therefore, multiplying the terms of the first fraction by 3, and the terms of the second by 2, the fractions become, the other two fractions are,. The only differing denominators are now 12 and 10; these have a common factor, 2; for they are 6 x 2, and 5 × 2; they are therefore made equal by multiplying the first by 5, and the second by 6. Hence, as before, using these multipliers for the fractions last deduced, they become 15, 18, 18, 18.

The reduction of fractions to others with a common denominator, besides being a needful preparation for adding and subtracting, is often also necessary to enable us to see which of two fractions is the greater: thus, and differ so little, that till you find, by multiplying each numerator by the denominator of the other fraction, that 2 x 11 exceeds 3×7, you would not know that the first is the greater.

(55.) ADDITION OF FRACTIONS.

RULE. Reduce the fractions to others having a common denominator. Add the numerators of the latter together,

* I need scarcely say, that when fractions are changed to others with equal denominators, it is not possible to preserve this equality, and at the same time to express each fraction in its lowest terms: as the terms of one or more of the given fractions have to be multiplied by some number to bring about the equality of the denominators, both objects cannot, of course, be accomplished.