A whole or integer pumber may be expressed like a fraction, by writing 1 below it, as a denominator; so 3 is for 4 is *, &c. A fraction denotes division ; and its value is equal to the quotient obtained by dividing the numerator by the deneminator ; so 4 is equal to 3. and is equal to 4. Hence then, if the numerator be less than the denominator, the value of the fraction is less than 1. But if the numerator be the same as the denominator, the fraction is just equal to l. And if the numerator be greater than the denominator, the fraction is greater than 1. REDUCTION OF VULGAR FRACTIONS. REDUCTION of Vulgar Fractions. is the bringing them out of one form or denomination into another ; commonly to prepare them for the operations of Addition, Subtraction, &c. of which there are several cases. PROBLEM To find the Greatest Common Measure of Two or more Numbers. The Common Measure of two or more numbers, is that number which will divide them both without remainder ; so, 3 is a common measure of 18 and 24 ; the quotient of the former being 6, and of the latter 8. And the greatest number that will do this, is the greatest common measure : so 6 is the greatest common measure of 18 and 24; the quotient of the former being 3, and of the latter 4, which will not both divide further. RULE. If there be two numbers only; divide the greater by the less ; then divide the divisor by the remainder ; and so on, dividing always the last divisor by the last remainder, till nothing remains; so shall the last divisor of all be the greatest common measure sought. When there are more than two numbers, find the greatest common measure of two of them, as before ; then do the same for that common measure and another of the numbers ; and and so on, through all the numbers ; so will the greatest common measure last found be the answer. If it happen that the common measure thus found is 1 ; then the numbers are said to be incommensurable, or not having any common measure. EXAMPLES. 1 To find the greatest common measure of 1908, 936, and 630. 936) 1908 (2 So that 36 is the greatest common 1872 measure of 1908 and 936. Hence then 18 is the answer required. 2. What is the greatest common measure of 246 and 372 ? Ans. 3. What is the greatest common measure of 324, 612, and 1032 ? Ans, 12. CASE I. To Abbreviate or Reduce Fractions to their Lowest Terms. Divide the terms of the given fraction by any number that will divide them without a remainder; then divide these quotients That dividing both the terms of the fraction by the same number, whatever it be, will give another fraction equal to the former, is evi. dent. And when these divisions are performed as often as can be done, or when the common divisor is the greatest possible, the terms of the resulting fraction must be the least possible. Note 1. Any number ending with an even number, or a cipher, is divisible, or can be divided, by 2. 2. Any number ending with 5, or 0, is divisible by 5. quotients again in the same manner ; and so on, till it appears that there is no numben greater than I which will divide them ; then the fraction will be in its lowest terms. Or, divide both the terms of the Fraction by thcir greatest common measure at once, and the quotients will be the terms of the fraction required, of the same value as at first. EXAMPLES 1. Reduce or to its least terms. Or thus : 216) 288 (1 Therefore 72 is the greatest commen 216 measure ; and 72) = the An swer, the same as before. 72) 216 (3 216 2 1 6 2. Reduce 3. If the right-hand place of any number be 0, the whole is divisible by 10; if there be two ciphers, it is divisible by 100; if three ciphers by 1000 : and so on; which is only cutting off those ciphers. 4. If the two right-hand figures of any number be divisible hy 4, the whole is divisible by 4. And if the three right-hand figures be divisible by 8, the whole is divisible by 8, And so on. 5. If the sum of the digits in any number be divisible by 3, or by 9, the whole is divisible by 3, or by 9. 6. If the right hand digit be even, and the sum of all the digits be divisible by 6, then the whole is divisible by 6, 7. A number is divisible by 11, when the sum of the 1st, 3d, 5th, &c, or all the odd places, is equal to the sum of the 2d, 4th, 6th, &c. or of all the even places of digits. 8. If a number cannot be divided by some quantity less than the square root of the same, that number is a prime, or cannot be divided by any number whatever. 9 All prime numbers, except 2 and 5, have either 1, 3, 7, or 9, in the place of units ; and all other numbers are composite, or can be divided, 10. When 2. Reduce to its lowest terms. 3. Reduce si to its lowest terms. 4. Reduce to its lowest terms. Ans. Ans. š. Ans. & CASE I To Reduce a Mixed Number to its Equivalent Improper Fraction. MULTIPLY the integer or whole number by the denominaior of the fraction, and to the product add the numerator; then set that sum above the denominator for the fraction required. 10. When numbers, with the sign of addition or subtraction between them, are to be divided by any number, then each of those 10+8-4 numbers must be divided by it. Thus = 5 +4-2=7. 2 11. But if the numbers have the sign of multiplication between them, only one of them must be divided. Thus, 10X8X3 10X4X3 10x4X1 10X2X1 20 6X2 6x1 2X1 1x1 1 * This is no more than first multiplying a quantity by some number, and then dividing the result back again by the same ; which it is evident does not alter the value ; for any fraction represents a division of the numerator by the denominator. CASE 20. CASE UI. To Reduce an Improper Fraction to its Equivalent Whole or Mixed Number. * Divide the numerator by the denominator, and the quotient will be the whole or mixed number sought. EXAMPLES. 1. Reduce to its equivalent number. Here is or 12 • 3 4, the Answer. 2. Reduce * to its equivalent number, Here 4 or 15 = 7 = 24, the Answer. 3. Reduce 44 to its equivalent number. Thus 17) 749 (4411 68 So that 144 = 4447, the Answer. 69 68 1 4. Reduce to its equivalent number. Ans. 8. CASE IV. To Reduce a Whole Number to an Equivalent Fraction, having a Given denominator. + MULTIPLY the whole number by the given denominator : then set the product over the said denominator, and it will form the fraction required. This rule is evidently the reverse of the former; and the reason of it is manifest from the nature of Common Division. † Multiplication and Division being here equally used, the result must be the same as the quantity first proposed. EXAMPLES |