h. In 1 yard there are 3 feet. How many feet are there in § of a yard ? If in 1 yard there are 3 feet, in § of a yard there are § Solution. of 3 feet; etc. i. What is the cost of 7 of a ream of paper at $ 6 a ream ? j. What is the cost of of an acre of land at $ 100 an acre? Examples for the Slate. 234. ILLUSTRATIVE EXAMPLE VI. What must I pay for of an acre of land at $165 an acre? 55 183 × 5 12 WRITTEN WORK. (92.) 36 × 5 = ? (93.) 568 × § (94.) 385 × 275 4 85. A man bought a carriage for $ 645, and sold it for § of what it cost him. How much did he receive for it? = ? = 86. If it requires 9 bushels of apples to make a barrel of cider, how many bushels will be required to make 13 barrels of cider? 682. Ans. $683, or $68.75. NOTE. Multiply by the integral and the fractional number, separately; then add the results. = ? 87. In 1 pound there are 16 ounces. How many ounces in 1311⁄2 pounds? 88. 4 quarts equal 1 gallon. How many quarts are there in 653 gallons? 89. At $3 a bundle, what will 263 bundles of shingles cost? 90. If $1 will buy 9 pounds of sugar, how much will $19% buy? 91. What must I pay for 63 yards of flannel at $0.54 a yard? (95.) 3681 × 4g = ? (96.) 5432 × 317 = ? To multiply a Fraction by a Fraction. 235. ILLUSTRATIVE EXAMPLE VII. Multiply by WRITTEN WORK. 15. Explanation. To multiply by is to take 1 third of To find 1 third of, the must be divided into 3 equal parts. Since the entire unit will contain 5 times 3 or 15 such parts, of is. (See 1 of 1 = . one of the parts will be of the unit; hence illustration.) Ans.. In multiplying by, how was the new denominator obtained? 236. Oral Exercises a. Multiply by 1; 1 by 1; 4 by ; † by ; † by 1. b. Multiply by ; by ; by ; by ; by t by 3. 237. ILLUSTRATIVE EXAMPLE VIII. Multiply Explanation. To multiply by is to take of of is(); then of must be (5), and of must be 2 times as much (13), Ans.. org. In multiplying by, how was the new numerator obtained? the new denominator? Then how do you multiply one fraction by another? c. What is of ? 4 of ? % of ? & of ? & of § ? d. Multiply by 7; ‡ hy fr; fo by ; fi by ; 4 by 3. 238. From the previous illustrations may be derived the following Rules. 1. To multiply an integer by a fraction, or a fraction by an integer: Find the product of the integer and the numerator of the fraction for the numerator of the answer, and take the denominator of the fraction for the denominator of the answer. 2. To multiply a fraction by a fraction: Find the product of the numerators for the numerator of the answer, and the product of the denominators for the denominator of the answer. 239. Examples for the Slate. 98. What is % of ? 99. What is 100. What is of? NOTE. Change mixed numbers to fractional numbers before multiplying. 104. Multiply 8 by 244. 105. Multiply 4g by 151. 106. If a man can walk 43 miles in an hour, how far can he walk in of an hour? (101.) × = ? = ? 107. What is the cost of 223 thousand bricks at $113 a thousand? 108. What is the cost of 37 pounds of French carmine at $12 per pound? 109. A woman who inherited of a ship divided 2 of her part equally between her two daughters. What part of the ship did each daughter receive? NOTE. Each daughter received of of of the ship. The operation may be expressed thus: 5×3×1 9×4×2' 3 110. If a dress-pattern cost $113, the trimmings as much, and the making as much as the trimmings, what was the cost of making? 111. In a school of 840 pupils, cipher, as many write as 240. The following, which are usually called "compound fractions," are simply expressions for multiplication of fractions, the finding of a part of a part: 112. What is of 72 of ? 113. What is of 2 of 1? 114. What is of 115. What is 13 of 116. What is off 3 117. What is 18 of of × 4 of 14 × 8 × 15? For other examples in multiplication of fractions, see page 123. ILLUSTRATION. ᄒᄒᄒ of? of 3 of 1 of § To divide a Fraction by an Integral Number. 241. ILLUSTRATIVE EXAMPLE I. If of a melon be divided equally between 2 boys, how many fifths of a melon will each boy have? ILLUSTRATION. of 53? of 63 ? of 31⁄2? × 6% ? DIVISION. 3 용 1 half of = . In dividing the fraction by 2, what was done to the numerator? 242. ILLUSTRATIVE EXAMPLE II. If of a melon be divided equally between 2 boys, what part of a melon will each boy have? As the number of Explanation. parts, 3, cannot be divided by 2 without a remainder, each one of the parts, fifths, may be divided into two equal parts; these parts will be tenths of the whole melon. We shall then have 6 To 10 1 half of fo=1. tenths, of which each boy's share is 3 tenths. (See illustration.) In dividing by 2, what was done to the denominator? Explanation. — If ‡ of a melon be equally divided between two boys, each boy will have 1 half of , or of a melon. (See illustration.) Ans. melon. 243. From the preceding illustrations we learn that to divide a fraction by an integer, we may divide the numerator or multiply the denominator by the integer. NOTE. In dividing by 2, as above, we find one half of a number; this is equivalent to multiplying by . Dividing by 3 is equivalent to multiplying by what? Dividing by 6 is equivalent to multiplying by what? f by 13; § by 25. by 7; § by 8. by 3; § by 2. of a yard of broadcloth, what part of a d. If $3 will buy yard will $1 buy? e. If 1 man can do a piece of work in 3 of a month, in what time can 12 men do it? f. If $2 be paid for 7 pounds of butter, what is the price of 1 pound? NOTE. Change $24 to fifths before dividing. g. At $24 per day, what are the wages of a man for 1 third of a day? h. If by 1 pipe a cistern can be emptied in 113 hours, in what time can the cistern be emptied by 4 like pipes? Examples for the Slate. 245. ILLUSTRATIVE EXAMPLE III. If 5 yards of ribbon are required for 18 knots of trimming, how much is required for 1 knot? 120. If 3913 inches of rain fell in a year, what was the average fall per week? |