EXAMPLES. 1. What is the least common multiple of 6, 10, 16, and 20? *5)6 10 16 20 *2)3 1 8 *3 1 *4 1 2 We survey the given numbers, and find that 5 will divide two of them, viz. 10 and 20, which we divide by 5, bringing into a line with the quotients, the numbers, which 5 will not measure: again, we view the numbers in the second line, and find 2 will measure them all, and we get 3, 1, 8, 2, in the third line, and find that 2 will measure 8 and 2, and in the fourth line get 3, 1, 4, 1 all 5×2×2×3×4=240 Ans. prime; we then multiply the prime numbers and the divisors continually into each other, for the number sought, and find it to be 240. * * * * * 2. What is the least number, which can be divided by the 9 digits separately, without a remainder? Ans. 2520.. REDUCTION OF VULGAR FRACTIONS, Is the bringing them out of one form into another, in order to prepare them for the operations of Addition, Subtraction, &c. CASE I. To abbreviate or reduce fractions to their lowest terms.* * A fraction is in its lowest terms, when represented by the least numbers possible thus 5 when reduced to its lowest terms is }. 152. What is reduction of vulgur fractions ?-153. When is a fraction in its lowest terms? K RULE. Divide the terms of the given fraction by any number, which will divide them without a remainder, and the quotients again in the same manner; and so on, till it appears there is no number greater than 1, which will divide them, and 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. NOTE. That dividing both the terms, that is, both numerator and denominator of the fractions, equally by any number, whatever, will give another fraction, equal to the former, is evident; and if those divisions be performed as often as can be done, or the common divisor be the greatest possible, the terms of the resulting fraction must be the least possible. 1. Any number ending with an even number or cipher is divisible by 2. 2. Any number ending with 5, or 0, is divisible by 5. 3. If the right hand figure of any number be 0, the whole is divisible by 10. 4. If the two right hand figures of any number be divisible by 4, the whole is divisible by 4. 5. If the sum of the digits constituting any number, be divisible by 3, or 9, the whole is divisible by 3 or 9. 6. All prime numbers, except 2 and 5, have 1, 3, 7, or 9, in the place of units, and consequently all other numbers are composite and capable of being divided. 154. What are the rules for reducing fractions to their lowest terms? 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 of the fraction to the product; under which subjoin the denominator, and it will form the fraction required. * All fractions represent a division of a numerator by the denominator, and are taken altogether as proper and adequate expressions of the quotient. Thus the quotient of 3, divided by 4, is 4. 155. How do you reduce a mixed number to its equivalent improper fraction? 3. Reduce 653, to its equivalent improper fraction. Ans. 12410 CASE III. To reduce an improper fraction to its equivalent whole or mixed number. RULE.-Divide the numerator by the denominator, the quotient will be the whole number, and the remainder, if any, written over the given denominator, will form the fractional part. NOTE. This rule is, evidently, the reverse of the rule in Case II, and has its reason in the nature of common division. 156. What is the rule for reducing an improper fraction to its equivalent whole or mixed number? EXAMPLES. 1. Reduce 223 to its equivalent whole or mixed number. 8)293(36§ Ans. 24 53 48 Or, 283 293-8-36 as before. 5 Reduce 2143 to its equivalent whole or mixed number. Ans. 127 To reduce a compound fraction to an equivalent simple one. RULE.-Multiply all the numerators continually together for a new numerator, and all the denominators, for a new denominator, and they will form the simple fraction required. NOTE. If any part of the compound fraction be a whole or mixed number, it must first be reduced to an inproper fraction. That a compound fraction may be represented by a single one, is evident, since a part of a part must be equal to some part of the whole. The truth of the rale for this reduction may be shewn as follows. Let the compound fraction to be reduced be of 4. Then of # 43, and consequently of 4X2 the same as by the rule, and the like will be found true in all cases. 21 If the compound fraction consist of more numbers than two, the two first may be reduced to one, and that one and the third will be the same as a fraction of two numbers, and so on. 157. How do you reduce a compound to a simple fraction? |