Reduce to thirty-sixths. In the above examples it is seen that when several fractions are to be reduced to a common denominator, a unit is changed first to a fractional form with the required deno. minator. Then it is divided by the denominator of each fraction, to obtain one part, and multiplied by the numera. tor, to obtain the required number of parts. Thus changing and each to twelfths, we first change a unit to a fraction with the required denominator 12; thus, it. We then divide it by the denominator of j, to obtain one fourth, and multiply the answer by 3, to obtain three fourths. In like manner with them. We divide in by the denominator 6, to obtain one sixth, and multiply by the numerator to obtain two sixths. In changing fractions to common denominators then, the unit must be changed to that fractional form which will enable us to divide it by all the denominators of the frac. tions (which are to be reduced) without remainder. Thus if we wish to reduce and to a common deno. minator, we cannot reduce them to twelfths, because is cannot be divided by either the denominator 5, or 7, with. out remainder. We must therefore seek a number that can be thus divided, both by 7 and 5. 35 is such a num. ber. We now take of 3 and 4 of it and the two frac. tions are then reduced to a common denominator. 15 . ONE MODE OF REDUCING FRACTIONS TO A COMMON DENO. MINATOR. Change a unit to a fraction whose denominator can be divided by all the denominators of the fractions to be reduced, without remainder. Divide this fraction by the denominator of each fraction to obtain one part, and multiply by the numerator to obtain the required number of parts. FURTHER EXAMPLES FOR MENTAL EXERCISE. Reduce $ } { to a common denominator. Let the unit be reduced to 14. Reduce and A to a common denominator. But there is another method of reducing fractions to a common denominator, which is more convenient for ope. rations on the slate. When a fraction has both its terms (that is, its numerator and denominator) multiplied by the same number, its value remains the same. For example ; multiply both the numerator and deno. minator of by 4, and it becomes it. But į and are the same value, with different names. The effect, then, of multiplying both terms of a fraction by the same number, is to change their name, but not their value. If therefore we have two fractions, and wish to change them so as to have both their denominators alike, we can do it by multiplication. For example; Let and be changed, so as to have the same deno. minator. This can be done by multiplying both ternis of the by 9, and of by 3. The answers are in and ji, and the value of both fractions is unaltered. In this case both terms of each fraction were multiplied by the denominator of the other fraction. What is a rule for reducing fractions to a common denominator? What is the effect of multiplying both terms of a fraction by the same number? Let the following fractions be reduced to a common denominator in the same way. 1. Reduce ; and to a common denominator. Multiply the } by the denominator 7, and the by the denomi. nator 5. 2. Reduce and A to a common denominator. 3. Reduce and to a common denominator. 4. Reduce io and to a common denominator, The same course can be pursued, where there are several fractions, to be reduced to a common denominator. Thus if į and i are to be reduced to a common deno. minator, we can multiply both terms of the į first by the denominator 3, and then multiply both terms of the an. swer by the denominator 4, and it becomes ją, and its va. lue remains unaltered. For ļ and i have the same value with a different name. Then we can multiply both terms of the , first by the denominator 2, and then by the denominator 4, and it be. comes y, and its value remains unaltered. Then may be multiplied, first by the denominator 2, and then by the denominator 3, and it becomes , and its value is unaltered. The three fractions į { and are thus changed to it to and , which have a common denominator, and yet their value is unaltered. But instead of multiplying each fraction, by each separate denominator, it is a shorter way to multiply by the product of these denominators. Thus in the above example, instead of multiplying the , first by 3, and then the answer by 4, it is shorter to mul. tiply by 12 (the product of 3 and 4), and the answer will be the same. In like manner, if we were to reduce i i and to a common denominator, we should multiply both terms of each fraction by the denominators of all the other frac. tions. But instead of each denominator separately, as multiplier, we can take the product of them for the multiplier. Reduce and { to a common denominator. Here both terms of the are first multiplied by the pro. duct of the other two denominators (which is 12). Then both terms of are multiplied in the same way by the product of the other two denominators (15). Then both terms of are multiplied by the product of the other two denominators (20). RULE FOR REDUCING FRACTIONS TO A COMMON DENOMI. NATOR, Multiply both terms of each fraction by the product of all the denominators except its own. Reduce a to a common denominator. Ans. 448 44 and . Reduce in and to a common denominator. Ans. 14 i 31 and 31. Reduce and to a common denominator. Reduce and 12} to a common denominator. Ans. 4 41 42 Reduce j and of il to a common denominator. Ans. 268 259 2 1982, % . 888 REDUCTION OF FRACTIONS TO THEIR LOWEST TERMS. Which fraction has the smallest numbers employed to express its value ? In the two fractions and is there any difference in the value ? Which fraction has its value expressed by the smallest numbers ? A fraction is reduced to its lowest terms, when its value is expressed by the smallest numbers which can be used, to express that value. For example, is reduced to its lowest terms, because no smaller numbers than 3 and 4 can express this value. The value of a fraction is not altered if both terms of it are divided by the same number. What is the rule for reducing fractions to a common denominator? When is a fraction reduced to its lowest terms ? Thus if has both its terms divided by 2, it becomes and the value remains the same. If it is divided by 4, it becomes j and its value remains unaltered. When it was divided by 2, it was not reduced to its lowest terms, because smaller numbers can express the same value, as 4. But when it was divided by 4, it was reduced to its lowest terms, because no smaller numbers than 1 and 2 can express its value. The shortest way to reduce a fraction to its lowest terms is, to divide it by the largest number which will di. vide both terms, without a remainder. Any number which will divide two or more numbers without a remainder is called a common measure, and the largest number which will do this, is called the greatest common measure. In many operations it saves much time to have a frac. tion reduced to its lowest terms. Thus for exa xample, if we are to multiply 3429 by 7 it would be much easier to re. duce the fraction to (which are its lowest terms) and then multiply. There are many fractions which can be reduced to their lowest terms without much trouble. For example let the pupil reduce these fractions. Reduce it to their lowest terms. But there are many fractions, which it is much more difficult to reduce. Thus if we wish to reduce 34 to its lowest terms, we could not so readily do it. In such a case as this there are two ways of doing it; the first is as follows. RULE FOR REDUCING A FRACTION TO ITS LOWEST TERMS. Divide the terms of the fraction by any number that will divide both, without a remainder. Divide the answer ob. tained in the same way. Continue thus, till no number can be found that will divide both terms without a remainder. Thus, Reduce to its lowest terms. What is the shortest way to reduce a fraction to its lowest terms ? What is meant by a common measure? What is the rule for reducing a fraction to its lowest terms ? |