2064 3432 31. Reduce to the lowest terms possible. Ans. 86 143 171. To reduce a fraction to its lowest terms, without first. finding the greatest common measure. RULE I. Divide both terms of the fraction by any number which will divide both without remainder, and the quotients will form another fraction equal to the former, but in lower terms. II. Divide both terms of this latter fraction by any number that will exactly divide them, and the quotients will form another fraction in still lower terms than the last, and equal to it. III. Proceed in this manner as long as division can be made, and the last fraction will be the lowest terms required. • Any number that will divide both terms of a fraction will evidently produce an equal fraction in lower terms; now if this last fraction be in like manner reduced, and likewise the resulting one, and so on continually as long as division can be made, it is plain the last fraction arising from this continued operation will be the given fraction in its least terms. To assist the operations in this rule it may be remarked, that, 1. Any number ending with an even number or a cipher is divisible by 2. 2. If the right hand place be a cipher, the number is divisible by 10. 3. Any number ending with 5 or 0 is divisible by 5. 4. If the two right hand figures be divisible by 4, the number is divisible by 5. If the three right hand figures be divisible by 8, the number is divisible by 8. 6. If the sum of the digits constituting any number be divisible by 3 or 9, the whole is divisible by 3 or 9. 7. If the right hand digit be even, and the sum of all the digits divisible by 6, the whole will be divisible by 6.,^,`, 8. If the sum of the first, third, fifth, digits be equal to the sum of the second, fourth, sixth, &c. the number is divisible by 11. 9. A number which is not a square, nor divisible by some number less than its square root, is a prime. 10. All prime numbers, except 2 and 4, have either 1, 3, 7, or 9, in the units place all other numbers are composite. 11. When numbers having the sign of addition, or Here I readily discover by inspection that 6 will divide both terms of the the lowest terms required. 120 given fraction, and the quotient is ; this again will divide by 4, and 144 10 12 30 the quotient is ; this I find will divide by 3, and the quotient is ; I di36 5 6 vide this by 2, and the quotient not being divisible (viz. both terms) by any number greater than I, is the answer. If the numbers you divided by, viz. 6, 4, 3, and 2, be multiplied together, the product 144 is the greatest common measure of the given fraction, Here dividing successively by 100, and 12, the Ans. is 247 1070 between them, and it is required to divide them by any number, each of the numbers must be divided. But if they have the sign them, then only one of the numbers must be divided. of multiplication between Thus, 1+2+3—4 = 2, where each number is divided; and 3+6+9-12 3 6 × 9 × 12, or 3 × 2 × 9 × 12, or 3 × 6 × 3 × 12, or 3 × 6 × 9 × 4 (=648), where one factor only in each case is divided. P This rule is nothing more than reducing an integral quantity into such parts as the denominator expresses: thus in ex. 37 we are required to reduce 4; now here we bring the 4 into sixths, viz. 24 sixths, to this we add the 5, 29 which is likewise sixths, making in the whole 29 sixths, or all cases: wherefore the rule is plain. 6 ; and the like in 172. To reduce a mixed number to its equivalent improper fraction. RULE I. Multiply the whole number by the denominator of the fraction, and to the product add the numerator. II. Place this number over the denominator of the fraction, and it will be the answer required a. 37. Reduce 4 to its equivalent improper fraction. OPERATION. 4 X 6 + 5 = 29 numerator. Then the answer. 29 Explanation. I multiply the whole number 4 by the denominator 6, and to the product 24 I add the numerator 5, which makes 29; this I place over the denominator 6 for the answer. 38. Reduce 5 and 8 to improper fractions. 63 39. Reduce 123 to its equivalent improper fraction. Ans. 5 6 7409 40. Reduce 1234 to an improper fraction. Ans. 41. Reduce 31211 to an improper fraction. Ans. 35994 115 173. To reduce an improper fraction to its equivalent whole or mixed number. RULE I. Divide the numerator by the denominator, and the quotient will be the whole number. II. Place the remainder (if any) over the denominator, and it will be the fraction, which must be subjoined to the whole number for the answer 9. ¶ This mode of operating is the converse of the former. Here we have a` number of parts given, and are required to find how many wholes can be made out of them, supposing that one whole contains as many parts as are expressed by the denominator; in order to this, we must evidently divide the numerator OPERATION. 29 6 to its equivalent mixed number. Explanation. Here I divide the numerator 29 by the denominator 6, 45 Ans. and the quotient is the whole number; also the remainder 5 placed over the denominator 6 gives for the fraction; these connected give 45 for the answer. 174. To reduce a whole number to an equivalent fraction, RULE. Multiply the whole number by the given denominator, and under the product place the said denominator, and it will be the fraction required'. 47. Reduce 3 to an equivalent fraction, having 4 for a denominator. by the denominator, the quotient will then express the number of wholes contained in the given fraction, and the remainder will be the parts over; this is plain from ex. 42. This rule and the preceding prove each other. Here the given number is reduced into such parts as are denoted by the denominator; under these the denominator is placed, to designate those parts. The truth of the rule may be shewn by dividing the numerator of this fraction by its denominator, by which the given number will be produced. Hence any whole number is reduced to the form of a fraction, by placing 1 under it for a denominator. 48. Reduce 8 and 9 to equal fractions, having 10 for the denominator of each. 49. Reduce 2, 3, and 4 to equal fractions, having 25 for a 50. Reduce 9, 8, 7, 6 to fractions, which will have 5, 4, 3, 175. To reduce a compound fraction to its equivalent simple one, or to a simpler form. RULE. Multiply all the numerators continually together for a numerator, and the denominators together for a denominator. Reduce this new fraction, if it be a proper fraction and requires it, to its lowest terms, (by Art. 170. 171.); but if the new fraction be an improper one, reduce it to a whole or mixed number, by Art. 173'. 2 5 13 Thus of = the answer. Here we multiply 2 and 5 3 7 20 together for the numerator, and 3 and 7 together for the denominator. • Having a part of a part given, we are here taught how to reduce it to its proper part of the whole. Let of be a compound fraction proposed to 2 3 3 4 ofis evidently, wherefore |