4. Divide by 3% Illustration.-To find how many times 3 is contained in 3/4, the 5ths and 4ths are subdivided into parts of the same size by dividing each 5th into four and each 4th into five equal parts, thus making 12/20 and 15/20, whence we see that is contained in 3/4 as many times as 12 is contained in 1515/12 = 11/4. 149. The arithmetical solution is performed on the same principle as the solution by diagram, the folding of paper, etc., thus: Indicating the division by writing the divisor under the dividend (see Art. 72, § 3), we reduce both fractions to 20ths, 20 being the least common multiple of 4 and 5, and divide the numerator of the dividend by the numerator of the divisor. But, since the common denominator does not affect the quotient (15/12), the work, printed in italics, by which it is obtained may be omitted. The written process would then appear as at the right. /12=1% 3/ X5_15 = 3/X5/=15/ /4x/3= /12=1 together for the numerator of the quotient, and the denominators 150. Rule.-Invert the divisor and multiply the numerators for the denominator of the quotient. Note. Since integers and mixed numbers may be expressed in the form of im." proper fractions, this rule applies to all cases of Division in Common Fractions. 151. The following analysis leads to an equivalent rule: 3/5 is contained in 1, or 5/5, five thirds times, and in 3/4 it is contained 3/4 of 5/3 times; hence we have 5/45/15/ = /12-1 152. An inverted fraction shows how many times the fraction is contained in one, and is called the reciprocal of the fraction. Hence, for dividing one fraction by another, we have the Rule.-Multiply the reciprocal of the divisor by the dividend. 153. In dividing a fraction by an integer, a part of the written work required by the rule may be omitted, as follows: 154. Hence, dividing the numerator or multiplying the denominator of a fraction divides the value of the fraction. 155. In dividing a mixed number by an integer, by a fraction, or by a mixed number, the written work may be as follows: Example.-3. Divide 3793 by 6. 6)379 63 24 Explanation. Six is contained in 3793/4 63 times, with a remainder of 13/4 13/47/47/4÷6=7/24, which, being annexed to 63, gives the entire quotient, 637/24 Note. In the two following examples we multiply both divisor and dividend by the denominator of the divisor in order to get rid of the fraction in the divisor. This makes the division more convenient and does not alter the value of the quotient. The process then becomes the same as in the preceding solution. 4. Divide 379/4 by 2/3 2/3) 379/4 3 3 2)11373/4 11 568% Question. In which of the above columns do the quotients increase as you descend? Why do they increase? In which do they decrease? Why? by %; % by 2/3 by 9/32; 5/8 by 7/12; ; 8. How many times may 1/3 of a pound be taken from /, lb., 3 lb., 8 lb., 9/10 lb.? 13. Divide 6 by 4; 5% by 2%; 311 by 113; 21⁄2 by 14. Divide 9/10 by 4%; 2% by 910; 7% by 2% ́; 6% by 33⁄4 19. Divide 5 by 4; 7 by 11; 6 by 5%; 12 by ORAL AND SLATE EXERCISES. Note.-Let the oral exercises be carried as far as time and the ability of the 10. How many times $43% in $26? In $373? In $45/? 11. 33 hours is what part of 19%, 27, 38 hours? 12. What part of 18/16, 2412/16, 3015/16, 498/18 lb. is 63/16 lb.? 14. 10/21÷2/3= 16 13.2+14= 12 9 = ÷57 20.1% 21. 11%÷1/15 (22. 33 ÷ = 23.512÷4% = 33/4-5/12= 2312+2%= 45%÷32/25= 2512÷414= = 21/14/9= 14/57/13 27/15/51= by 223; % by 3; 23 by 1%; % by 2%; 25. Divide 3⁄4 by 7%; % by 33⁄4; % by 4%; % by 5%; 3% by 325° Applications.-1. If 4 yards of ribbon cost $5%, what will 1 yard cost? Analysis.—If 4 yards cost $5/7, one yard will cost 1/4 of 85/7 = $5/28 2. A farmer sold 5 dozen eggs for $11/20 per dozen ? How much was that 3. In six days a man plows 14 of a field. At this rate, how much does he plow in 1 day? 4. If a weaver earns $93/20 per week (6 days), what does she earn per day? 5. If % lb. of coffee cost $38, what will 1 lb. cost? 5 Analysis.—If 1⁄2 lb. cost $3/8, 1/6 will cost 1/4 x $3/8 = $3/32, and 51⁄2 lb. will cost 5×3/32 = $15/32⋅ 6. If a traveler can make % of his journey in 3⁄4 of a month, what time will the entire journey require ? 9 7. If of a bar of gold weighs 5/12 lb., what is the weight of the bar? 8. How many are 2 of 22 dozen? 2% of 2% gross? 9. A garden, containing 148% □ yd., is to be divided up into beds of 12% yd. each. How many such beds will there be? 10. A quantity of grain, weighing 1101⁄4 cwt., is to be put into bags, each holding 134 cwt. How many bags are required? 11. How many yards of cloth at $3% a yard can be bought for $2, $5, $7, $9, $4, $23? 12. How many bushels of potatoes at $24/25 per bu. can be bought for $6, $8, $11, $13 ? 13. How many times may 1% quarts be drawn from a can holding 17 quarts? From one holding 22 quarts? 14. If a laborer can mow a field in 714/19 days, how much of it can he mow in 1 day? 15. Divide $22,500 among the 7 members of a family so that each of the 4 older ones may receive a third more than one of the younger. |