quotients again in the same manner; 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 at once, and the quotients will be the terms of the fraction required, of the same value as at first. 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 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 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. s. If a number cannot be divided by some quantity less than the square root of the same, that number is a prime, os cannot be di'vided 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 To Reduce a Mixed Number to its Equivalent Improper Fraction. *MULTIPLY the integer or whole number by the denominator of the fraction, and to the product add the numerathen set that sum above the denominator for the frac tion required. tor; 10. When numbers, with the sign of addition or subtraction be tween them, are to be divided by any number, then each of those 10+8-4 2 numbers must be divided by it. Thus =5+4-2=7. 11. But if the numbers have the sign of multiplication between them, only one of them must be divided. Thus, * 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. 3 CASE CASE III. 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. 1. Reduce Here 2. Reduce Here EXAMPLES. to its equivalent number. or 12÷ 3 = 4, the Answer. to its equivalent number. 3. Reduce 749 to its equivalent number. Thus, 17) 749 (4477 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. EXAMPLES. 1. Reduce 9 to a fraction whose denominator shall be 7. Here 9 × 763: then is the Answer; For 6379, the Proof. 63 2. Reduce 12 to a fraction whose denominator shall be 13. Ans. 156 3. Reduce 27 to a fraction whose denominator shall be 11. To Reduce a Compound Fraction to an Equivalent Simple One. * MULTIPLY all the numerators together for a numerator, and all the denominators together for a denominator, and they will form the simple fraction 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, er cancelled. *The truth of this Rule may be shown as follows: Let the compound fraction be of. Now of is 3, which is r 43 consequently of will be × 2 or 1; 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 faction and a third will be the same as a compound fraction of two parts; and so on to the last of all. 2. Reduce To Reduce Fractions of Different Denominators, to Equivalent Fractions having a Common Denominator. * MULTIPLY each numerator by 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 first be reduced, by their proper Rules, to the form of simple fractions. EXAMPLES. 1. Reduce, and 4, to a common denominator. 1 × 3 × 4 = 12 the new numerator for 2 x 2 x 4 = 16 3 × 2 × 3 = 18 ditto 2 2 × 3 × 4 24 the common denominator. 2 I Therefore the equivalent fractions are 14, 14, and 4. Or the whole operation of multiplying may be best performed mentally, only setting down the results and given fractions thus:,, 2, 16, 14 = 2 T, by abbreviation. 249 249 24 6 2. Reduce and to fractions of a common denominator. *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. 3. Reduce |