EXAMPLES FOR PRACTICE. 1. What is the sum of 4 and 11? Ans. 21 4 2. What is the sum of 13, 15, 15 3. What is the sum of 5 8 16 2 and 18? Ans. 14. 4. What is the sum of 711, 833, 217, 518 and 43%? Ans. 28. 5. What is the sum of 37%, 127, 1337 and §3? 6. Add, and . 4 7. Add 3, 4, 13 and 7. Ans. 233 8. Add, and and 5%. Ans. 141. Ans. 37 Ans. 188. Ans. 116138. 13. Add 16,1⁄2 and 24 ̧‰· 14. Add 14, 23, 33, 4 15. Add 4,7, 85, and 28. 1 21 16. Add 1,, and . 4' 12 17. Add,, and . 13 21. Add 41, 21, 116, 224, 576, 73, 4 and 68. 22. Four cheeses weighed respectively 365, 423, 39,7% and 511 pounds; what was their entire weight? 23. What number is that from which mainder will be 33?? Ans. 16947 pounds. if 44 be taken, the re Ans. 83 from 24. What fraction is that which exceeds by 54 ? 25. A beggar obtained of a dollar from one person, another, from another, and from another; how much did he get from all? 26. A merchant sold 464 yards of cloth for $127,7, 644 yards for $2265, and 763 yards for $3123; how many yards of cloth did he sell, and how much did he receive for the whole? Ans. 1877 yards, for $66618. 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 24%. OPERATION. 2381 = 238,3 248-2419 213 Ans. We first reduce the frac ANALYSIS. tional parts, and, to the common denominator, 12. Since we cannot take from, we add 1 = 12, to making. Then, 12 subtracted from 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 denom inator. II. To subtract mixed numbers. Reduce the fractional parts to a common denominator, and then subtract the fractional ana 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. what 17. From 18 subtract 55. 19. From 2819 subtract 394. 63 20. From 78, subtract 323. 21. The sum of two numbers is 261, and the less is 713 ; 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 1264 and 240, to make 560ğ ? Ans. 1934. 24. What number is that which, added to the sum of 1,2, and, will make ? 25 Ans. 3 25. To what fraction must be added, that the sum may be §? 26 From a barrel of vinegar containing 31 were drawn; how much was then left? 27. Bought a quantity of coal for $140%, gallons, 143 gallons Ans. 16 gallons. and of lumber for $456. Sold the coal for $7753, and the lumber for $516,3⁄4; how much was my whole gain? Ans. $6943. 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. III. 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). |