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). 1. From subtract . ANALYSIS. Reducing the 3-3= ' 11310= given fractions to a common denominator, the resulting fractions f; and is express fractional units of the same value, (185, I). Then 12 fifteenths less 10 fifteenths equals 2 fifteenths = 15, the OPERATION. answer. OPERATION. 2. From 2381 take 24 ANALYSIS. We first reduce the frac2381 = 238 tional parts, 1 and 5, to the common denominator, 12. Since we cannot 24 = 24 take ifrom in, we add 1 = ti, to ja, 213. Ans. making i. Then, i, subtracted from 1 leaves ; and carrying 1 to 24, the integral part of the subtrahend, (73, II), and subtracting, we have 213, 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. EXAMPLES FOR PRACTICE. Ans. 15 1 Ans. 11 Ans. is: Ans. Iš 64 1 From 1 take 's 2. From za take 3. From 35 take 3 4. From take. 5 From take zo 6. From 13 take 24. Ans. 3 7 From take a Ans. 15 8. From takes 9. From take is. 10. From ? take 5 11. From take 13 12 12. From take 30 13. From 16á take 76 Ans. 93. 14. From 36.11 take 8:14. Ans. 273 15. From 2510 subtract 1411. 16. From 75 subtract 43. Ans. 704 17. From 187 subtract 55. 48. From 26:14 subtract 251%. 19. From 286 subtract 3 4. Anrs. 2415 20. From 78.145 subtract 323. 21. The sum of two numbers is 261, and the less is 713; what is the greater? Ans. 1972 22. What number is that to which if you will be 978? 23. Whát number must you add to the sum of 1264 and 2401, to make 5605 ? Ans. 1936 24. What number is that which, added to the sum of s, in, and is, will make 36 ? 25. To what fraction must ž be added, that the sum may be 26 From a barrel of vinegar containing 317 gallons, 147 gallons were drawn; how much was then left ? Ans. 16 gallons. 27. Bought a quantity of coal for $140%, and of lumber for $4563. Sold the coal for $775%, and the lumber for $516,6; how much was my whole gain? Ans. $69448 add 184, the sum Ans. 31 ? THEORY OF MULTIPLICATION AND DIVISION OF FRACTIONS. 189. In multiplication and division of fractions, the various operations may be considered in two classes : - Ist. 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. FII. 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. MULTIPLICATION. FIRST OPERATION. In the first opera SECOND OPERATION. 5 THIRD OPERATION. con FIRST OPERATION. 192. 1. Multiply by 4. ANALYSIS. by 4 by multiplying its nume rator by 4; and in the second X4= = 1 operation, we multiply the frac tion by 4 by dividing its denom5 4 inator by 4, (190, I or III). Х = 1; In the third operation, we express the multiplier in the form of a fraction, indicate the multiplication, and obtain the result by cancellation. 2. Multiply 21 by 4. ANALYSIS. To multiply by 4, 21 x 4 = 84 = 12 we must multiply by 4 and di vide by 7, (191, I or III). In the first operation, we first 21 x 4 = 3x4=12 multiply 21 by 4, and then di vide the product, 84, by 7. In the second operation, we 3 first divide 21 by 7, and then RA 4 Х 12 multiply the quotient, 3, by 4. 1 方 In the third operation, we ex press the whole number, 21, in the form of a fraction, indicate the multiplication, and obtain the result by cancellation. 3. Multiply by . Analysis. To multiply by 3, we must multiply by 7 and 24 step, is 8= 36 divide by 8, (191, I or III). In the first operation, we mul. 10 = Ans. tiply s by 7 and obtain it; SECOND OPERATION. THIRD OPERATION. FIRST OPERATION. 1st step, 5 14 35 112 5 16 SECOND OPERATION. 35 5 112 16 THIRD OPERATION. we then divide ii by 8 and obtain XZ ifa, which reduced gives ja, the required product. In the second operation we obtain the same result 5 Х by multiplying the numerators to14 8 gether for the numerator of the pro duct, 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. NOTE8.—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. Ans. 1. Multiply s by 8. Ans. 23 2. Multiply by 27, i by 4, and to by 9. 3. Multiply is by 15. 4. Multiply 8 by : Ans. 6. 5. Multiply 75 by * , 7 by 8, 756 by 5, and 572 by : 6. Multiply by 7. Multiply it by žy, and i5 by 5}. 8. Multiply izby 17, and 3 by zi. 9. Multiply 24 by 38. Ans. 10. 10. Multiply 1by 115. Ans. 21 11. Multiply | by 218 12. Find the value of ý x 6.X i Xž 13. Find the value of į xxx X i X 48 14. Find the value of 31 x 33 x 1's. 15. Find the value of 27 x 24 x î x 168 x 115 x 261. Ans. 2. 16. Find the value of ý xi X 4 x 15 x 17. Find the value of Me Xo X 53430 Ans. 24 Ans. i's: Ans. 188 |