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 s of 3 feet; etc. i. What is the cost of io of a ream of paper at $6 a ream ? j. What is the cost of f of an acre of land at $ 100 an acre ? Examples for the Slate. pay for 234. ILLUSTRATIVE EXAMPLE VI. What must I of an acre of land at $ 165 an acre ? WRITTEN WORK. 55 = 275 = 684. Ans. $ 684, or $68.75. 12 4 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 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 1375 pounds ? 88. 4 quarts equal 1 gallon. How many quarts are there in 65} gallons ? 89. At $3 a bundle, what will 267 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 ? (92.) 36 x 53 = ? (95.) 3681 x 43 = ? (93.) 568 x 8 = ? (96.) 5432 x 347 = ? (94.) 385 x 70 = ? (97.) 87036 x = ? To multiply a Fraction by a Fraction. 235. ILLUSTRATIVE EXAMPLE VII. Multiply } by 1. ILLUSTRATION. ho Explanation. -- To multiply } by } is to take 1 third of 1. 품 } 금 중 To find i third of }, the } must be divided into 3 equal parts. 't of = 1 Since the entire unit will con tain 5 times 3 or 15 such parts, one of the parts will be to of the unit; hence } of } is 1. (See illustration.) Ans. t. In multiplying by }, how was the new denominator obtained ? 236. Oral Exercises a. Multiply } by t; } by }; ! by ly; by t; } by }. b. Multiply } by $; by }; To by }; } by }; il by #. 4 X 2 5 X 3 237. ILLUSTRATIVE EXAMPLE VIII. Multiply by s. Explanation. — To multiply by f is to take WRITTEN WORK. off of is its (5x5); then 5 of 6 must be t ਤ (63), and f of must be 2 times as much (4*3), or it. Ans. . 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 1 ? ļof ; ? of ? of 3? 2 of 8 ? d. Multiply by ; 4 hy i lo by &; A by }; by %. 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 yo of ? 99. What is As of %? 100. What is P of $? (101.) fg x 5 = ? NOTE. Change mixed numbers to fractional numbers before multiplying. 104. Multiply 814 by 244. 105. Multiply 48 by 1518. 106. If a man can walk 43 miles in an hour, how far can he walk in its of an hour ? 107. What is the cost of 224 thousand bricks at $118 a thousand ? 108. What is the cost of 373 pounds of French carmine at $ 124 per pound ? 109. A woman who inherited of a ship divided if of her part equally between her two daughters. What part of the ship did each daughter receive? Nore. Each daughter received } of 4 of 5 of the ship. The operation 5x8x1 may be expressed thus : gX4 X 2' 110. If a dress-pattern cost $ 11%, the trimmings 14 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: 112. What is f of Ya of f? 113. What is & of is of It? 114. What is it of of 2 ? 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 each boy have ? ILLUSTRATION. 1 1 } Explanation. — As the number of parts, 3, cannot be divided by 2 with out a remainder, each one of the parts, I' io fifths, may be divided into two equal 1 half of for B. 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 š. Dividing by 3 is equivalent to multiplying by what? Dividing by 6 is equivalent to multiplying by what? 244. Oral Exercises. a. Divide } by 5; by 11; 38 by 7; ff by 13; }} by 25. b. Divide i by 5; } by 3; 11 by 9; }} by 7; 8 by 8. c. Divide f by 3; 8 by 4; 1 by 7; A by 3; 8 by 2. d. If $3 will buy 1 of a yard of broadcloth, what part of a yard will $ 1 buy? e. If 1 man can do a piece of work in ft of a month, in what time can 12 men do it ? f. If $ 27 be paid for 7 pounds of butter, what is the price of 1 pound? NOTE. Change $24 to fifths before dividing. g. At $ 21 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 177 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 ? |