LESSON LXVI. CASE IV.-TO DIVIDE A MIXED BY A WHOLE NUMBER. 8. How is a mixed number divided by a whole number? RULE. ·Divide the integer of the mixed number by the whole one, until you get a remainder; then reduce this remainder and the fraction to an improper fraction, and complete the division as in Case I. EXAMPLE SUPPLEMENTAL TO FEDERAL MONEY. 9. Bought 167 barrels of flour for $2,393.663; what was the price per barrel? In this operation we divide first the integer 2,393.66 by 167, and get in the quotient 14.33 with a remainder 55; this we multiply by 3 to change it into thirds; and, by adding in the numerator 2 of the fraction, we get 17, which divided by 167, gives the quotient; making the complete answer $14.331. This is an example of division of Federal money, with fractions of cents instead of mills, which could not be introduced before vulgar fractions. OPERATION. 2393.66 167 ⚫723 •556 •556 501 55 3 167 2494 by 12. CASE V.-TO DIVIDE EITHER A WHOLE OR MIXED NUMBER BY A MIXED NUMBER. 10. How do you divide either a whole number, or a mixed number, by a mixed number? RULE. Reduce the mixed number or numbers to improper fractions; invert the divisor fraction, and then multiply the two fractions together: if the result is an improper fraction, reduce it to a mixed number. The dividend transformed into a fractional form is 97 and the divisor 47; the quotient, by Case III., is 155. 582 517 PROOF OF MULTIPLICATION AND DIVISION OF FRACTIONS. 12. How do you prove multiplication and division of fractions? Ans. Multiplication is proved by division, and division by multiplication, as in other cases. 7,8 by 310. 7 39 by 9 15,6 by 17, 22.* 21,201 by 21% = 999. 16 12 by 1710 = 14. 31. By what number should 3 be multiplied to make ? 32. By what number should make 11? 55 be multiplied to Ans.. 33. By what number should 7g be multiplied to make 323? 34. At of a dollar per bushel, how much can be bought for of a dollar? 35. At $4 a yard, how many yards can you buy for $125? 36. A locomotive engine has gone 3789 miles in 13 hours; how many miles an hour did it run? Ans. 29 miles. This and following are reduced agreeably to Proposition III., Lesson LXIII. 37. What part of an hour did it take to run one mile? In the first case you divide the distance by the time; in the last the time by the distance. Ans. 8 233. 38. I paid 7 dollars 183 cents for a barrel of flour which contains 196 pounds; what was it a pound? 39. If of a gallon of wine cost 23 dollars, how much is it a gallon? LESSON LXVII. ADDITION OF FRACTIONS. 1. How do you add fractions together which have the same denominator? Ans. When fractions have the same denominator, add the numerators and place the sum over the common deno minator. The sum of+12+2 would be 1 2. Why do you add the numerators only, and not the denominators also? Because the denominator is only the name of the things added, and The result being, in both cases, the sum of 8 things of the same kind. 8 Suppose, for instance, that it was inches we added; the sum would be 8 inches, that is, (8 twelfths) of one foot, whether we added them as 1+ 2+ 5 inches, or as 1 1⁄2 + 1⁄2 + 1⁄2 of a foot. Only, in one case we use the common name of the thing; in the other, we wish to express its numerical relation to the unit of comparison, which here is the foot. 2. How would you add fractions which have different denominators? Ans. Only things of the same kind can be added together numerically. Therefore: before adding fractions with different denominators, they must be reduced to the same denomination.* 3. How are fractions reduced to the same denominator? RULE. To reduce fractions to a common denominator, multiply both terms of each fraction by the denominator of all the other fractions. EXAMPLE. Reduce,, to the same denominator. OPERATION. The common denominator will be 3 × 4 × 6 = 72. 1st fraction, both terms multiplied by 46 4757 4. Are the new fractions equal in value to the original ones? Ans. The new fractions are exactly equal in value to the original ones; because: If both terms of a fraction are either multiplied or divided by the same number, the value of the fraction remains the same; its form alone is changed. (LXIII.) * I have purposely used here the word denomination instead of denominator. Suppose, for instance, that you count 2 cows and 3 oxen; you cannot add them as either cows or oxen, but you can class them under the common denomination of cattle, and call them together 5 head of cattle. Nor could you add one foot and 2 yards as either feet or yards, without changing one of the denominations into the other, and making the sum either 7 feet or 21 yards. |