PRACTICAL PROBLEMS. IN ADDITION AND SUBTRACTION OF FRACTIONS. 1. A has $51, B has $41, and C has ; how much money nave they all? Ans. $16. 2. A miller ground 7 bushels of corn for A, 9 for B, 10 for C; how much did he grind in all? Ans. 2717 bu. 3. A lady bought material for a wrapper costing $11†, and buttons costing $; what change should she receive from a $5 bill? Ans. $31. 4. A lady went shopping with $100 and paid $12 for e bonnet, $32 for a dress, and $52§ for a cloak; how much money did she bring home? Ans. $2.124. 5. A boy gave 12 cents for a slate, 183 cents for a knife, 37 cents for a grammar, and 62 cents for an arithmetic; what did they all cost? Ans. $1.314. 6. A bought two pieces of muslin, each containing 41 yd.; after selling 57 yd., how much remained? Ans. 25 yd. 7. Mr. Weeks finds that his family burned last winter 1 tons of coal in December, 2 tons of coal in January, 2,3 tons of coal in February, and in March 13 tons; how much was burned during the four months? Ans. 71 tons. 8. Four loads of hay weighed upon the scales 49,7 hundredweight, 43 hundredweight, 39, hundredweight, and 458 hundredweight; what was the weight of the hay, the wagon weighing 15 hundred weight? 100 Ans. 115 MULTIPLICATION OF FRACTIONS. cwt. 173. Multiplication of Fractions is the process of finding a product when one or both factors are fractions. 174. There are Three Cases: 1st. A fraction by an integer; 2d. An integer by a fraction; 3d. A fraction by a fraction. CASE I. 175. To multiply a fraction by an integer. 1. Multiply by 4. SOLUTION.-Multiplying the numerator (Prin. 1), we have 4 times equals 1; or, dividing the denominator (Prin. 4), we have 4 times, or 3. OPERATION. H×4=11 Rule. To multiply a fraction by an integer, multiply - SOLUTION.-7 multiplied by one equals 7, hence 7 multiplied by equals of 7, which is 3, and 7 multiplied by equals 4 times, which equals 28. Therefore, etc. 2. Multiply 28 by 78. SOLUTION.-Multiplying 28 by 3 and dividing by 5, we have 28x equals 163; and then multiplying 28 by 7, we have 196; adding 196 to 163, we have 2123. Therefore, etc. NOTE. This method of multiplying by a mixed number is more conve nient than the one usually presented. Rule.-Multiply the integer by the numerator of the mul tiplier, and divide the product by the denominator; or divide CASE III. 177. To multiply a fraction by a fraction. 1. What is the product of by ? SOLUTION.- multiplied by one equals, hence multiplied by equals of, which is, and multiplied by equals 7 times fore, etc. 48, SOLUTION 2D.- times (Prin. 1); and of 35 equals 35 There which is 9. OPERATION. equals of 5 times; 5 times equals (Prin.3). Rule.-Multiply the numerators together and the denomi nators together, cancelling common factors. NOTE.-Practically this case is the same as finding a fractional part of a fraction, but theoretically the two cases are entirely distinct. Required the cost of 1. 96 lb. of paper, at 25 cts. per pound? 8. 6 yards of cloth, at $54 a yard? Ans. $2.72. Ans. $32.06. Ans. $24.374. Ans. $33284. Ans. $1912. Ans. $43.78. Ans. $56. Ans. $3315. Ans. $2 Ans. $21 Ans. $461} 12. 8 cords of wood, at $43 a cord? Ans. $38. 13. 15 barrels of vinegar, at $4 a barrel? Ans. $71. 14. 164 quarts of beans, at $0.183 a quart? Ans. $3.09. 15. What is the value of 3×××29? Ans. 335. 16. What is the value of 2×2×14×48? Ans. 224 X 17. Value of 2×10035×12824×1024? 992 729 X 870 256? 18. Value of 2 × 3 × 700 × 83 8×1998? X 441 340 841 935 19. Value of (+18) × 13 + ( + + §) ×3? 3 15 3 1 63 20. Value of (+19) × (3+75) × 7 323 × 3 8 3 ? 3 32 243 Ans. 12584• Ans. 9. 1870 Ans. 3 $276. Ans. 2348. 21. Of 7X43⁄41⁄2+(41—34)—(5§+6,3⁄4)? 22. Find the value of 73211-(33+374) diminished by (61+67)×4915. Ans. 20,415. 23. Mr. Fish rented a house at $423 a month, taking a lease for 5 years; but disposed of the lease at the end of 31 years; how much rent did he pay? Ans. $16674. 24. A bill of books at retail amounts to $376%, but I get a reduction of for wholesale and from wholesale for cash; what was the exact amount of the bill? Ans. $23517. 25. Mr. Alden bought 52 bushels of wheat at 623 cts. per bushel, and sold of it at 69 cts., and the remainder at 713 cts. per bushel; what did he clear by the transaction? Ans. $4.434. NOTE. In examples like 15, 16, 17, 18, etc., resolve into factors and then cancel. DIVISION OF FRACTIONS. 178. Division of Fractions is the process of dividing when one or both of the terms are fractional. 179. There are Three Cases: 1st. A fraction by an integer; 2d. An integer by a fraction; 3d. A fraction by a fraction. CASE I. 180. To divide a fraction by an integer. 1. Divide by 5, and also by 6. we have 15 equals (Prin. 2). SOLUTION 2D.-Multiplying the denominator of f by 6, we have f÷6 equals 18, or, (Prin. 3). OPERATION. 1÷÷÷÷5= OPERATION. 19÷618. 2. Divide 822 by 4. SOLUTION.-Dividing 822 by 4, we have 205 and a remainder of 2; 2 equals 42, which, added to , equals 17; 2 divided by 4 equals 14; hence the quotient is 20517. OPERATION. 4)8228. 20511 Rule.-Divide the numerator or multiply the denominator of the dividend by the divisor. NOTE. Reduce a mixed number to a fraction; or divide the integer unite the remainder with the fraction, and divide the result. 1. Divide 10 by the fraction 5. SOLUTION.-10 divided by one equals 10, hence 10 divided by equals 6 times 10, and 10 divided by 5 sixths equals of 6 times 10, which is times 10, which equals 12. Hence the OPERATION. 10÷÷ 10x=12 Ans. Rule I.-Multiply the dividend by the denominator of the fraction, and divide the product by the numerator. Rule II.-Invert the divisor and proceed as in multipl cation of fractions. |