CANCELLATION. 123. Cancellation consists in rejecting equal factors from dividend and divisor. 124. All arithmetical operations in division may be expressed in the form of a fraction, the dividend being the numerator, and the divisor the denominator; since dividing both terms of a fraction by the same number does not alter its value, it follows, that we may strike out, or cancel, any factors common to the dividend and divisor without changing their relative value. N. B. All operations upon arithmetical quantities should first be expressed, as far as possible, by signs, that the processes may be clearly indicated to the teacher, and that the work to be done may be reduced, if possible, by cancellation. ILL. Ex. Divide 3 times 4 times 6 times 5 times 7, by 2 times 8 times 6 times 9 times 10. Express, cancel, and perform the following: 1. Divide 8 X 6X3 X 9 X 7 X 4, by 2 X 5 X 7 X 10 X 8. 2. Divide 81 X 42, by 99 × 7. Ans. 61. 3. Multiply 75 × 10, by 3 × 6, and divide that product by 15 X 25 X 12. 4. Divide 7 X 8 X 48, by 63 X 4 X 5 X 17, and multiply the quotient by 51. 5. Divide 99 X 28 × 6, by 5 × 8 × 18, multiply the quotient by 4 X 4, and divide by 22 × 27. 6. Spent of $75, which I received for work, for flour at $5 a barrel; how many barrels did I buy? 7. If 25 pounds of lead costs $4.60, what do 8 pounds cost? 8. What will be received for 27 pieces of broadcloth, if 6 pieces bring $864? 9. If it requires 13 bushels of wheat to make 3 barrels of flour, how many bushels will be required to make 78 barrels of flour? Ans. 338 bushels 10. If a tree 69 feet high casts a shadow of 90 feet, what length of shadow will be cast by a tree 92 feet high? Ans. 120 feet. 11. A merchant exchanged 561 pounds of sugar, at 9 cents per pound, for eggs at 11 cents per dozen; how many dozen were received? 12. If 12 pieces of cloth, each piece containing 62 yards, cost $372, what cost 24 yards? 13. If a bar of iron 8 feet long weighs 36 pounds, what will a bar of the same size 100 feet long weigh? 14. How many boxes of oranges can be bought for $420, if $28 be paid for 7 boxes? 15. If the work of 7 men is equal to the work of 9 boys, how many men's work will equal the work of 63 boys? 16. If 15 men consume a barrel of flour in 6 weeks, how long would it last 9 men? Ans. 10 weeks. 17* If the interest of $650 for 12 months is $52, what is the interest of three times that sum for eight months? Ans. $104. 18. If 12 men can build a wall in 42 days, how many days will be required for 21 men to build it? 19. If $15 purchase 12 yards of cloth, how many yards will $48 purchase? Ans. 38 yards. 20. A ship has provision for 15 men 12 months; how long will it last 45 men? 21. How many overcoats, each containing 4 yards, can be made from 10 bales of cloth, 12 pieces each, 42 yards in each piece? 22. If 375 barrels of pork, each 200 pounds, cost $6000, what is the cost of 5 barrels, each 195 pounds? 23* Sold 20 barrels of apples at $2.50 per barrel, and spent the money thus obtained for cloth at $.50 a yard, which I sold at $.60 a yard, and bought a horse with the proceeds. What did I pay for the horse? 126. REDUCTION OF WHOLE OR MIXED NUMBERS TO IMPROPER FRACTIONS. ILL. EX. Change 27 to an improper fraction. OPERATION. 2} = Hence the 23. Ans. .. In 1 unit there are 8, in 2 units there are 2 or 12. 16+7=23, Ang RULE. To reduce a mixed number to an improper fraction ;Multiply the whole number by the denominator of the fraction; to that product add the numerator; and write the result over the denominator. 10. Reduce 5848, 368, 875, to ninths. 11. Add 784 to 916, and express the answer in sevenths. 12 Reduce 7 × 98 to eighths. 13. Reduce (158) × 16 to fifths. 14. Reduce 8692 to a fraction whose denominator is 25. 15. Reduce 367 to an improper fraction. 16. Change 45678 to ninths, 43862 to elevenths. 17. Change 36864 to an improper fraction. For Dictation Exercises, see Key. 127. REDUCTION OF IMPROPER FRACTIONS TO WHOLE OR MIXED NUMBERS. ILL. EX. Change 29 to a mixed number. OPERATION. 29=48, Ans. times. Hence the There are in 1 unit, in 22 there are as many units as & is contained times in 22, which is 48 RULE. To reduce an improper fraction to a mixed number; Divide the numerator by the denominator. For Dictation Exercises, see Key. 128. MULTIPLICATION OF FRACTIONS BY WHOLE NUMBERS. As multiplying the numerator of a fraction multiplies the number of parts, their size remaining the same, and dividing the denominator multiplies the size of the parts, their number remaining the same (Art. 119), it follows that, To multiply a fraction by a whole number, we may either multiply the numerator by the whole number, or divide the denominator. The latter method is preferable when the denominator can be divided without a remainder, as it gives the answer in lower We might have cancelled in the first operation, and thus have obtained the same result as in the second; thus, 7 X 4 7 31 17. If one yard of cloth costs of a dollar, what will 17 yards cost? 18. If a ton of coal costs of an Eagle, how much will 15 tons cost? 19. Required the cost of 28 pounds of candles, at § of a dollar a pound. 20. Multiply 2567 by 18. 21. Multiply 3768 by 21. For Dictation Exercises, see Key. Ans. 46231. 129. MULTIPLICATION OF WHOLE NUMBERS BY FRACTIONS. ILL. EX. Multiply 8 by §. OPERATION. 8.X 5 4 X 5 6 3 tiplier, or 8 multiplied by 5 is 5 X 8; if it is =63, Ans. multiplied by §, a number one sixth as large as 5, the product must be one sixth as large as if 5 had been the mul of 5 X 8. The expression then becomes 8 times 5, divided by 6; after cancelling, 4 X 5 20 3 3 =6, Ans. Hence the RULE. To multiply a whole number by a fraction; -Multiply the whole number by the numerator of the fraction, and divide hat product by the denominator. 7. What cost of 1 ton of hay, at $12 a ton? OPERATIΟΝ. 1.50 $12.00 × 5 —$7.50, Ans. If 1 ton of hay costs $12, of a ton will cost of $12, and § of a ton will cost 5 times of $12. Cancelling, we have $1.50 x 5 = $7.50, Ans. 8. What cost of an acre of land, at $100 an acre? of an acre of land, at $150 an acre? of a gross of pens, at $.96 a gross ? 11. What cost 5 cords of wood, at $7.56 a cord? 12. What cost 3 hogsheads of molasses, at $18.80 a hogshead? |