h. In 1 yard there are 3 feet. How many feet are there in of a yard? Solution. If in 1 yard there are 3 feet, in § of a yard there are § of 3 feet; etc. 12 i. What is the cost of 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 682. Ans. $682, or $68.75. 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 137 barrels of cider ? 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 $193 buy? 91. What must I pay for 63 yards of flannel at $0.54 a yard? To multiply a Fraction by a Fraction. 235. ILLUSTRATIVE EXAMPLE VII. Multiply by 1. one of the parts will be of the unit; hence of is. (See illustration.) Ans. . In multiplying by 1, how was the new denominator obtained? WRITTEN WORK. 4× 2 5 X 3 of ofis(); then of must be 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 ? d. Multiply by ; of? % of ą? 2 of z? % of § ? hy fr; fo by ; fr by t; 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 NOTE. Change mixed numbers to fractional numbers before multiplying. 104. Multiply 8 by 244. 105. Multiply 4 by 151. 106. If a man can walk 43 miles in an hour, how far can he walk in of an hour? 107. What is the cost of 223 thousand bricks at $113 a thousand? 108. What is the cost of 371⁄2 pounds of French carmine at $12 per pound? 109. A woman who inherited § of a ship divided 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 3 110. If a dress-pattern cost $11g, 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 cipher, as many study grammar as write, as many study geography as study grammar. How many study geography? 240. The following, which are usually called "compound fractions," are simply expressions for multiplication of fractions, the finding of a part of a part: 119. What is of × of 14 × 8 × 15? For other examples in multiplication of fractions, see page 123. DIVISION. 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. 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 ILLUSTRATION. 1 half of fo=8%. each boy have? Explanation. As the number of 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 tenths, of which each boy's share is 3 tenths. (See illustration.) In dividing & by 2, what was done to the denominator? 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? 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? 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 5g yards of ribbon are required for 18 knots of trimming, how much is required for 1 knot? 120. If 391 average fall per 8 × 18 16 • Ans. f yd. inches of rain fell in a year, what was the week? |