19. The minuend is 16, and the remainder 2. What is the subtrahend? 20. If a boy spends of his money one day, of it the next day, and has $11 left, how much money had he at first? Subtracting mixed numbers. 1. From 7 take 35. 36=1. c. d. 71=65 | 45 35 20 3 Written Work 71 = 6 + 1 + 1 = 65. The integers and fractions are subtracted separately. The least common denominator is 36. Changing the fractions to 36ths we find that, and § = 38. 18 - 38 = 38. 6-3 = 3, which added to 3 = 33%. Hence the difference between 7 and 35 = 33%. In subtracting mixed numbers subtract the integers and frac tions separately. Find differences : 20. If I pay a grocery bill of $221, a water bill of $31, and a gas bill of $54, how much shall I have left from 2 twentydollar bills? 21. The sum of three numbers is 150. The least number is 15, and it is 633 less than the greatest. Find the other number. 22. What fraction added to the sum of, 3, and will make ? 23. If 5 is added to each term of the fraction, is the value of the fraction increased or diminished, and how much? 24. Two boys undertake to save $50 apiece. When one of them lacks $8% of having $50, both together have $843. How much has each? 25. A traveling man's grips, when starting out, weighed as follows: 123 pounds and 19 pounds. Find the weight of both, and the difference in their weight. 16 26. James lives 13 miles east of the schoolhouse, and Harry 1 miles west of the schoolhouse. Find the sum of the distances walked by both each day and the distance James walks farther than Harry. 27. Eight women do different parts in making a finished garment. The cost of the different parts is 1, 2, 417, 38, 41, 411, 14, and 7. Find the cost of making one garment. 10 28. Four automobiles finish a race in the following time: 8 hours, 8 hours, 7 hours, and 93 hours. Find the difference between the time of the winner and each of the others. 29. The average cost per mile of a fleet of freight boats on the Great Lakes was: coal 134, crew 12, repairs 172, supplies 7. The average cost per mile of a freight train hauling the same freight was: coal 743, crew 371, repairs 192, supplies 4. Find the saving per mile by water. 30. A drayman hauls freight by the ton from the depot to a village. Find the amount hauled in 8 loads weighing respectively: 13 tons, 2 tons, 12 tons, 252 tons, 23 tons, 11⁄2 tons, 2 tons, 1 tons. MULTIPLICATION OF FRACTIONS To multiply a fraction by an integer. 1. 1+1+1+1= how many whole units? 3. How many, then, are 4 × 1? 4 × 1 ? 4. Does it make any difference in the process of multiplication in problem 3 whether the multiplicand is a whole number or a fraction? 5. What term of the fraction is multiplied by 4? 23. In multiplying the above fractions by a whole number, did we increase the size or the number of the fractional units? 24. How, then, may any fraction be multiplied by an integer without increasing the size of its fractional units? 25. 2x of a square how many eighths of the square? How many fourths of the same square? Draw figures to illustrate. 26. How does of a square compare in size with 4 of the same square? Draw figure to illustrate. = 27. 3x of a square how many ninths of the square? how many thirds of the same square? Draw figures to illustrate. 28. How does of a square compare with of the same square? 29. How much larger is the fractional unit in fourths than in eighths? in thirds than in ninths? in 30. How can we increase the size of the fractional unit without decreasing the number of fractional units? in 3? 31. How, then, may any fraction be multiplied by an integer without increasing the number of its fractional units? In what two ways, then, may we multiply a fraction by an integer? Multiplying the numerator or dividing the denominator of a fraction by a number multiplies the value of the fraction by that number. Written Work Since multiplying the numerator of a 1. Multiply by 5. fraction multiplies the fraction, 5 times 5×32=2=132 32 = }} = 13%. When possible, use cancellation. Finding fractional parts of an integer. 8 × means that is to be taken as an addend as many times as there are units in the multiplier. 2 + 3 + 2 + 3 + 2 + 3 + 2 + 8. Thus, of 8 means that 8 is to be divided into 4 equal parts and 3 of these parts are to be taken. What is the first step in finding the fractional parts of the whole number? the second step? While the process of finding fractional parts of a whole number is classed as multiplication, the multiplicand at no time is taken as an addend, but is partitioned, that is, divided into equal parts, and a certain number of these parts is taken. The sign is read "of" when the number preceding it is a simple fraction; as, x $6 is read " of $6." |