hat will do this, is the greatest common measure: so 6 is the greatest common measure of 18 aud 24; the quotient of the former being 3, and of the latter 4, which will not both divide farther. 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; then 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 so on, through all the numbers; then will the greatest common measure last found be the answer. then the numbers If it happen that the common measure thus found is 1 ; are said to be incommensurable, or to have no common measure. EXAMPLES. 1. To find the greatest common measure of 1998, 918, and 522. 2. What is the greatest common measure of 246 and 372? CASE I. To abbreviate or reduce fractions to their lowest terms. Ans. 0. Ans. 8. RULE.* Divide the terms of the given fraction by any number that will divide them without a remainder; then divide these quotients again in the same man * That dividing both the terms of the fraction by the same number, whatever it be, will give another fraction equal to the former, is evident. And when those 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 erding 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 3. If the right hand place of any number be 0, the whole is divisible by 10; if there be two ciphers, .t is divisible by 100; if 3 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 by 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 will be divisible by 6. 7 A number is divisible by 11, when the sum of the 1st, 3d, 5th, &c., or of 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 of the same, that number is a prime, or cannot be divided by any number whatever. ner; and so on; till it appears that there is no number greater than 1 which will divide them; then the fraction will be in its lowest terms. Or, Divide both the terms of the fraction by their greatest common measure, and the quotients will be the terms of the fraction required; of the same value as at first. EXAMPLES. 1. Reduce to its least terms. #% = 78% = }} = }} ==, the answer. To reduce a mixed number to its equivalent improper fraction. RULE.*-Multiply the whole number by the denominator of the fraction, and add the numerator to the product; then set that sum above the denominator for the fraction required. 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 numbers, with the sign of addition or subtraction between them, are to be divided by any 2 number, then each of those numbers must be divided by it. Thus, 10+8-4 =5+4-2=7 11. But if the numbers have the sign of multiplication between them, only one of them must be 10 X 8 X 310 X 4 X 3. 10 X 4 X 1 divided. Thus, 10 X 2 X 1 6X2 6X1 2 X 1 Txi = 20 = =20. *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 To reduce an improper fraction to its equivalent whole or mixed number. RULE.* Divide the numerato by the denominator, and the quotient will be the whole or mixed number sought. To reduce a whole number to an equivalent fraction, having a giver. denominator. RULE.+-Multiply the whole number by the given denominator, then set the product over the said denominator, and it will form the fraction required. EXAMPLES. 1. Reduce 9 to a fraction whose denominator shall be 7. Here 9 X 763, then is the answer. For 6379, the proof. 2. Reduce 13 to a fraction whose denominator shall be 12. 3. Reduce 27 to a fraction whose denominator shall be 11. Ans. Y • 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. CASE V. To reduce a compound fraction to an equivalent simple one. RULE *— Multiply all the numerators together for a numerator, and all the denominators together for the denominator, and they will form the simple frac tion sought. When part of the compound fraction is a whole or mixed number, it must first be reduced to a fraction by one of the former cases. And, when it can be done, any two terms of the fraction may be divided by the same number, and the quotients used instead of them. Or when there are terms that are common, they may be omitted. EXAMPLES. 1. Reduce of of Here Or, 1 x 2 x 3 2 × 3 × 4 to a simple fraction. 6 = 1 x 2 x 3 I = 2 × 3 × 4 4' by omitting the twos and threes. 2. Reduce of off to a simple fraction. 2 x 3 x 10 60 Here = = 2 X 3 X 10 4 Or, 3 x 5 x 11 = the same as before. To reduce fractions of different denominators to equivalent fractions, having a common denominator. RULE.-Multiply each numerator into all the denominators except its own for the new numerators; and multiply all the denominators together for a common denominator. Note. It is evident, that in this and several other operations, when any of the proposed quantities are integers, or mixed numbers, or compound fractions, they must be reduced, by their proper rules, to the form of simple fractions. • The truth of this rule may be shown as follows: Let the compound fraction be 3 of 4. Now } of is÷3, which is; consequently of will be 2 or 4; that is the numerators are multiplied together, and also the denominators, as in the rule.-When the compound fraction consists of more than two single ones; having first reduced two of them as above, then the resulting fraction and a third will be the same as a compound fraction of two parts; and so on to the last of all. This is evidently no more than multiplying each numerator and its denominator by the same quantity, and consequently the value of the fraction is not altered. EXAMPLES. 1. Reduce, and to a common denominator. 1 x 3 x 4 = 12 the new numerator for §. 3 X 2 X 3 = 18 ditto for . ........ ditto for 2 X 3 X 4 = 24 the common denominator. Therefore the equivalent fractions are, and 1. Or the whole operation of multiplying may be very well performed mentany, and only set down the results and given fractions thus:,,, = 1}, }; } = 12, 17, 12, by abbreviation. 2. Reduce 4 and § to fractions of a common denominator. Ans. 3. Ans. 48, 38, 8. 36 Ans. 3, 38, 38. Note 1. When the denominators of two given fractions have a common measure, let them be divided by it; then multiply the terms of each given fraction by the quotient arising from the other's denominator. 2. When the less denominator of two fractions exactly divides the greater, multiply the terms of that which hath the less denominator by the quotient. 3. When more than two fractions are proposed; it is sometimes convenient, first to reduce two of them to a common denominator; then these and a third; and so on till they be all reduced to their least common denominator. CASE VII. To find the value of a fraction in parts of the integer. RULE.—Multiply the integer by the numerator, and divide the product by the denominator, by Compound Multiplication and Division, if the integer be a compound quantity. Or, if it be a single integer, multiply the numerator by the parts in the next inferior denomination, and divide the product by the denominator. Then, if any thing remains, multiply it by the parts in the next inferior denomination, and divide by the denominator as before; and so on as far as necessary; so shall the quotients, placed in order, be the value of the fraction required.* * The numerator of a fraction being considered as a remainder, in Division, and the denominator as the divisor, this rule is of the same nature as Compound Division, or the valuation of remainders in the Rule of Three, before explained. |