denominators of the other fractions; if, therefore, we multiply 2, the numerator, into the same numbers, 4 and 5, the value of the fraction will remain the same; since 48. Again, we multiply 1, the second numerator, into 3 and 5, by which its denominator has been multiplied, and obtain 15, a new numerator. Lastly, we multiply 2, the third numerator, into 3 and 4, by which its denominator has been multiplied, and obtain 24, a new numerator. The value of the fractions remains unaltered, since 183, 18= 1, and 24-2. 1. Reduce,, denominator. 5 EXAMPLES. and to fractions having a common 3396 336 6930 384 504 8316 3024 2. Reduce, 2, 14 and 2 to fractions having a common denominator. Ans., 16632, 16632, 16632, 16632 3. Add together,, and . Ans., 960 2941-36 NOTE. Reduce the fractions to a common denominator, and add the numerators. Nore. Mixed Numbers may be reduced to Improper Fractions, or the fractional parts may be reduced to a common denominator, and added as in the foregoing examples. If their sum amount to an integer add it to the whole numbers. 9. Add together 14 and 163. Operation. 1412 3,3=12+12=}}=12. 16, 8 We find the common denominator to be 12, and the new numerators to be 9 and 8, which when added are 13=11⁄21⁄2· under the fractions, and carry 1 to the whole 31 Answer. Write the numbers. 10. Add together 173, 18, 193. Ans. 557. 11. A grocer sold the following parcels of sugar, viz: 16 lbs., 191, 132, 20, 25, 30%, and 11 lbs. How many pounds did he sell in all? Ans., 136. Subtraction of Fractions. RULE. Prepare the fractions as in Addition, and subtract the less numerator from the greater, and under the difference write the denominator. Ans., . 418. 7. Add together ‡ and §, and from their sum subtract 43 of T 8. A owns of of a vessel, B owns 3 of 4-how much greater is A's share than B's? 9. Subtract 132 from 153. 152 13,9/2 8 9 8 -12, 12. 9 Ans., . Having reduced the fractions to a common denominator, and found new numerators as in Addition, we 11 Answer. have to be taken from; we therefore borrow the integer, and say from 12 and add 3, the remainder, to 8, the numerator of the subtrahend. 3+8=11, which we write under the fractions and carry 1 to 13, the whole number, which makes 14-and 14 from 15 and 1 remains-the answer, then, is 112. 10. A man bought a horse for of of $150, and sold him for of of of $60; did he gain or lose, and how much? Ans., gain $40. To find the least Common Multiple. The common denominator found by the preceding rule, is a common multiple of the denominators of the given fraction; for every product must be divisible by all its factors; but it was not the least common multiple. 1. What is the least common multiple of 4, 6, 8, 10? 4X6X8X10=1920. 1920 is evidently a common multiple of 4, 6, 8 and 10, because they are its factors; but it is not the least common multiple. We find, also, that each of these numbers is a multiple of 2, because 2 QUESTIONS. 1. What is the rule for the subtraction of fractions? is a factor in each. Dividing by 2, we find the other factors, which are 2, 3, 4 and 5. Again: As the quotient, 4, is a multiple of 2, we may substitute for it, 2, one of its factors; and as we employ the other factor for a divisor, we erase the other quotient, 2. We now have 3, 2 and 5, undivided numbers, which are found to be factors of the dividends 6, 8 and 10; the other factors are the divisors. If we multiply 3, 2 and 5 separately into the other factor or factors, the product will be each respective dividend. Thus the divisors 2 and 2 are the factors of 4, the first dividend 2x2=4. If 4 be divisible by 4, then 2, 3, and 5 times 4 must be divisible by 4; also 3, the first undivided number, is a factor of 6, the second dividend; and 2, the first divisor, is the other factor, 3×2=6; if 6 be divisible by 6, then 2 and 5 times 6 must be divisible by 6. The same may be said of 2 and 5, the other undivided numbers. Hence it appears, that the product of the continued multiplication of the remainders and divisors is divisible by the several dividends; and by examining the operation, it will be found to be the least number which can be divisible by them. Therefore, to find the least common multiple of two or more numbers, we have the following RULE. Divide by any number which will divide two or more of the given numbers without a remainder. Bring down the quotients with the undivided numbers on a line under the given numbers. Continue to divide until no two numbers are left capable of being divided by any number greater than 1. The product of the continued multiplication of the divisors and undivided numbers, will be the least common multiple required. EXAMPLES. 2. What is the least common multiple of 3, 4, 9 and 12? Answer, 36. 3. What is the least number which can be divided by Answer, 840. 7, 8, 10 and 12, without a remainder ? 4. What is the least common multiple of 7, 14, 28, 35! Answer, 140. 5. What is the least number which can be divided by the nine digits without a remainder ? Ans., 2520. QUESTIONS. 1. How is the least common multiple of two or more numbers found? 1. Reduce,, to equivalent fractions having the least common denominator. 2)4, 5, 6 60 being a new denominator for each of the given fractions, we must find new 2×5×3×2=60 numerators which shall bear the same ratio to 60 that each of the given numerators does to its own denominator. If we take from 60, the common denominator, the value of each fraction, for new numerators to the common denominator, we shall then have fractions of the same value of the given fractions. This is done by dividing 60 by the denominator of the fraction, and multiplying the quotient by the numerator, (see p. 96) =15×3=45, and 45=3. The same is true of the other fractions, and §. We have, then, this RULE for reducing fractions of different denominations to equivalent fractions having the least common denominator. Find the least common multiple of all the denominators, for a common denominator. Divide the common denominator by the denominators of each of the given fractions, and multiply the quotients by the numerators of the given fractions, and the products will be the new numerators required. 2. What is the least common denominator of, 4, 3, and ?? Then 28, 18, 24, 48, Answer.] 40 Answer, 33, 34, 336. 396' 56 462 4. Reduce,,and, to fractions having the least common denominator. Answer, 14, 104, 504, 504 5. A merchant buys 5 pieces of cloth. The first contains 40 yards; the second, 27; the third, 343; the fourth 43; and the fifth 39 yards. How many were there in the whole? Answer, 1851%. QUESTIONS. 2. How are fractions of different denominators reduced to equivalent fractions, having the same denominator? 6. Which is the greater fraction, or ? Answer, is greater by T REDUCTION OF FRACTIONS. 1. Reduce of a pound to the fraction of a penny. We have seen that integers of a higher denomination are brought into integers of lower, by multiplication; (see p. 65,) and also that fractions are multiplied either by multiplying the numerator, or dividing the denominator. (See p. 92.) Pounds are reduced to shillings by multiplying by 20, and shillings to pence by multiplying by 12. Therefore, to reduce of a pound to the fraction of a penny, multiply the fraction by 20 and 12, thus: 28X20×12=248= Answer. Or thus: As the numerator of the fraction is to be multiplied, place it with the multipliers on the right of the line, cross both numbers, and write 24 in the place of 288; 4 is contained in 20, 5 times, and in 24, 6 times. The answer, then, is, in the lowest terms of the fraction. To change fractions of a higher denomination into fractions of a lower denomination, we have the following RULE. Multiply the numerator of the fraction, or divide the denominator by all the denominations between it and that denomination into which it is to be reduced, including the lower denomination. EXAMPLES. 1. Reduce ʊ of a pound to the fraction of a penny. Answer, 2. 2. Reduce 3. What part of a pound is 84 cwt.? Answer, 4. Reduce of a yard to the fraction of a nail. Answer, . £ to the fraction of a farthing. Ans., 4. QUESTIONS. 1. How are fractions of a higher, reduced to fractions of a lower denomination? |