nator respectively of the fraction which expresses the lowest terms of the given fraction”. Secondly, divide both terms of the given fraction by the greatest 144 24 186 31 common measure 6, thus; 6): = the answer. The greatest common measure by Art. 166 is 12. n Two fractions are equal to each other when the numerator of one has to its denominator the same ratio which the numerator of the other has to its denominator, as is plain from the definition of a fraction; hence it follows that the same fraction may be expressed in a great variety of ways; thus, is the same as or or or or 14 or 41, &c. &c.; this being premised, the above rule teaches to find the least numbers possible that will express any given fraction; if a fraction be not in its lowest terms, both terms must evidently be divided, and they must both be divided by the same number, otherwise the terms would not be proportionals, and therefore the resulting fraction would not equal the given one; (see note on Art. 165.): moreover this divisor must be the greatest possible, (viz. the greatest common measure of both terms,) otherwise the quotients will not be the least possible: whence the rule is 2064 86 31. Reduce to the lowest terms possible. Ans. 3432 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, 4. * 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 dívisible 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 - of subtraction, Here I readily discover by inspection that 6 will divide both terms of the Explanation. 120 given fraction, and the quotient is ; this again will divide by 4, and vide this by 2, and the quotient not being divisible (viz. both terms) by any number greater than 1, 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 X of multiplication between 3+6+9-12 3 them, then only one of the numbers must be divided. Thus, 1+2+3—4= 2, where each number is divided; and 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; and the like in all cases: wherefore the rule is plain. 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× 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 5 39. Reduce 12 to its equivalent improper fraction. Ans. 40. Reduce 12345 to an improper fraction. Ans. 7409 6 35994 41. Reduce 312 to an improper fraction. Ans. 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. 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 29 6 to its equivalent mixed number. Explanation. Here I divide the numerator 29 by the denominator 6, 4 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 uumerator 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. |