SUBTRACTION. 187. The process of subtracting one fraction from another is based upon the following principles: I. One number can be subtracted from another only when the two numbers have the same unit value. Hence, II. In subtraction of fractions, the minuend and subtrahend must have a common denominator, (185, I). fractions and 19 express fractional units of the same value, (185, I). Then 12 fifteenths less 10 fifteenths equals 2 fifteenths, the answer. 2. From 2381 take 248. OPERATION. 2381 2383 213 Ans. ANALYSIS. We first reduce the fractional parts, and, to the common denominator, 12. Since we cannot take 12 from, we add 1=1, to fa, making 15. Then, subtracted from 13 leaves; and carrying 1 to 24, the integral part of the subtrahend, (73, II), and subtracting, we have 2135 for the entire remainder. 188. From these principles and illustrations we derive the following general RULE. I. To subtract fractions.- When necessary, reduce the fractions to their least common denominator. Subtract the numerator of the subtrahend from the numerator of the minuend, and place the difference of the new numerators over the common denominator. II. To subtract mixed numbers. Reduce the fractional parts to a common denominator, and then subtract the fractional and integral parts separately. NOTE. We may reduce mixed numbers to improper fractions, and subtract by the rule for fractions. But this method generally imposes the useless labor of reducing integral numbers to fractions, and fractions to integers again. 21. The sum of two numbers is 261, and the less is 7; what is the greater? Ans. 19. 22. What number is that to which if you add 184, the sum will be 978? 23. What number must you add to the sum of 1261 and 240%, to make 560§? Ans. 1934 24. What number is that which, added to the sum of 1,2, and, will make 35? 36 Ans. 1. 25. To what fraction must be added, that the sum may 26 From a barrel of vinegar containing 31 were drawn; how much was then left? 27. Bought a quantity of coal for $140%, be &? gallons, 14 gallons Ans. 16 gallons. and of lumber for $4563. Sold the coal for $7753, and the lumber for $516,3; how much was my whole gain? Ans. $694. /THEORY OF MULTIPLICATION AND DIVISION OF FRACTIONS. 189. In multiplication and division of fractions, the various operations may be considered in two classes: 1st. Multiplying or dividing a fraction. 2d. Multiplying or dividing by a fraction. 190. The methods of multiplying and dividing fractions may be derived directly from the General Principles of Fractions, (174); as follows: I. To multiply a fraction.—Multiply its numerator or divide its denominator, (174, I. and II). II. To divide a fraction.-Divide its numerator or multiply its denominator, (174, I. and II). GENERAL LAW. HII. Perform the required operation upon the numerator, or the opposite upon the denominator, (174, III). 191. The methods of multiplying and dividing by a fraction may be deduced as follows: 1st. The value of a fraction is the quotient of the numerator divided by the denominator (168, I). Hence, 2d. The numerator alone is as many times the value of the fraction, as there are units in the denominator, 3d. If, therefore, in multiplying by a fraction, we multiply by the numerator, this result will be too great, and must be divided by the denominator. 4th. But if in dividing by a fraction, we divide by the numerator, the resulting quotient will be too small, and must be multiplied by the denominator. Hence, the methods of multiplying and dividing by a fraction may be stated as follows: I. To multiply by a fraction. Multiply by the numerator and divide by the denominator, (3d). II. To divide by a fraction.-Divide by the numerator and multiply by the denominator, (4th). GENERAL LAW. III. Perform the required operation by the numerator and the opposite by the denominator. ANALYSIS. To multiply by 4, we must multiply by 4 and divide by 7, (191, I or III). In the first operation, we first multiply 21 by 4, and then divide the product, 84, by 7.. In the second operation, we first divide 21 by 7, and then multiply the quotient, 3, by 4. In the third operation, we express the whole number, 21, in the form of a fraction, indicate the multiplication, and obtain the SECOND OPERATION. 35 1312 = 18 THIRD OPERATION. 5 t 5 8 = 16 we then divide by 8 and obtain 35 , which reduced gives f, the required product. In the second operation we obtain the same result by multiplying the numerators together for the numerator of the product, and the denominators together for the denominator of the product. In the third operation, we indicate the multiplication, and obtain the result by cancellation. 193. From these principles and illustrations we derive the following general RULE. I. Reduce all integers and mixed numbers to improper fractions. II. Multiply together the numerators for a new numerator, and the denominators for a new denominator. NOTES.-1. Cancel all factors common to numerators and denominators. 2. If a fraction be multiplied by its denominator, the product will be the numerator. EXAMPLES FOR PRACTICE. 1. Multiply by 8. Ans. 21 2. Multiply by 27, by 4, and by 9. 3. Multiply by 15. Ans. . 4. Multiply 8 by 3. Ans. 6. 5. Multiply 75 by, 7 by 1, 756 by %, and 572 by 5 |