number that will divide both 204 and 60, will also divide 468, which is equal to 2 × 204 +60. By the next division, the problem is reduced to finding the greatest common divisor of 24 and 60; and by the last division to finding the greatest common divisor of 12 and 24; and since 12 will divide both itself and 24 without remainder, it is the greatest common divisor of these two numbers, and also of the given numbers, 204 and 468. For, since 12 divides 24, it will also divide 2 × 24 +-12, or 48+12=60; and as it divides without remainder both 24 and 60, it will also divide 3 × 60+24, or 180+24=204; and dividing both 60 and 204, it will also divide, without remainder, 2 x204 + 60, or 408 +60=468; and therefore, 12 is the greatest common divisor of 204 and 468. EXAMPLES. 1. Find the greatest common divisor of 1908 and 936. Ans. 36. 2. Find the greatest common divisor of 246 and 372. Ans. 6. 3. Find the greatest common divisor of 324 and 612. Ans. 36. 4. Find the greatest common divisor of 327 and 932. 5. Find the greatest common divisor of 462 and 743. EXAMPLES IN REDUCTION OF FRACTIONS. 1. Reduce to its lowest terms. 39 2. Reduce 453, to its lowest terms. 1057 Ans.. To reduce a whole number to a fraction having a given denominator. RULE. (28.) Multiply the whole number by the given denominator, and write the product with the given denominator beneath it. Demonstration.-A whole number may be considered a fraction having unity for its denominator; as the quotient of any number divided by unity is equal to the number itself. Hence, if we would reduce a whole number to a fraction having a given denominator, we have only to write it in a fractional form with unity for its denominator, and multiply both terms by the given denominator. For example, let 4 be reduced to a fraction whose denominator shall be 3: 4 is the same as 4, and multiplying both terms by 3, we have 12 or 4. The fraction 4 having been both multiplied and divided by 3, the value remains the same. This method is evidently the same as multiplying 4 by 3, and writing 3 beneath the product. 3 EXAMPLES. 1. Reduce 7 to a fraction whose denominator shall be 9. 7x9= 63. 3 the required fraction. 2. Reduce 12 to a fraction whose denominator shall be 7. To reduce a mixed number to an improper fraction. RULE. (29.) Multiply the whole number by the denominator of the fractional part, and to the product add the numerator, and write the sum over the denominator. This rule depends upon the same principle as the preceding; the whole number being multiplied and divided by the denominator of the fractional part, its value will not be altered. 3. Reduce 1313 to an improper fraction. 4. Reduce 67 to an improper fraction. Ans. 15. Ans. 395. Ans. . To reduce an improper fraction to a whole or mixed number. RULE. (30.) Divide the numerator by the denominator, and the quotient will be the whole or mixed number sought. EXAMPLES. 1. Reduce 2 to a whole or mixed number. Ans. 27+462. 2. Reduce 1 to a whole or mixed number. 3. Reduce 73 to a whole or mixed number. 8 4. Reduce 7321 to a whole or mixed number. Ans. 9. Ans. 41. Ans. 430 5. Reduce to a whole or mixed number. 1 Ans. 14. To reduce a compound fraction to a simple one. RULE. (31.) Multiply all the numerators together for a numerator, and all the denominators for a denominator; the result will be the simple fraction required. 5 Demonstration.--To show the correctness of this rule, let it be required to reduce the compound fraction, of 5, to a simple one; of is the same as twice of; to take of , we should divide by 3; +3, and multiplying this result by 2, we have 10, or § × 3 = 10. It is evident that the same principle would apply to any compound fraction whatever. When the compound fraction consists of more than two single ones, we may at first reduce two of them, and afterwards reduce the resulting fraction with the next, and so on. This would be the same as multiplying all the nume rators and all the denominators together. 4* Ans.. 3. Reduce of of 1 to a simple fraction. 4. Reduce of 13 of 8 to a simple fraction. To reduce a complex fraction to a simple one. RULE. (32.) If the denominator be a fraction, multiply the numerator of the given fraction by the denominator of the denominator; if the numerator be a fraction, multiply its denominator by the denominator of the given fraction; if both numerator and denominator are fractions, multiply the numerator of the numerator by the denominator of the denominator, and the denominator of the numerator by the numerator of the denominator. 3 329 3 For an application of the rule, take the complex fractions, and The first has a fractional denominator, and reduced, by the rule, gives; the second has a fractional numerator, and reduced is; and the third, in which both terms are 27 fractional, becomes 3 × 9 35 Demonstration of the Kule.--Considering fractions as unexecuted divisions, we shall have in a complex fraction the denominator of which is fractional, the numerator to be divided by a fraction, which is done by inverting the divisor and multiplying (23). This is the same as multiplying the numerator by the denominator of the fractional part. Thus, 3 might be written 3+, or (23) 3 × §. When the numerator is fractional, it denotes that a fraction is to be divided 9 by a whole number, which is done by multiplying its denominator (21); may be written ÷ ÷ 9, which becomes, by performing the division, indicated. When both terms of the complex fraction are fractional, we have a fraction to be divided by a fraction; might be written +7, or (23) §×4 = 33. This operation, though written in a different form, is evidently the same as that given in the rule. |