1. Multiply by 12. 144÷12=12, denominator. Or thus: H=1, Ans. 13×12=156, numerator. In the first example, it is plain that are multiplied by dividing the denominator by 12, because the number of parts into which the unit is divided, is diminished; and therefore their magnitude is increased. Or if the denominator be considered a divisor, then, if 144 be divided by 12, the value of the fraction is increased in the same proportion. Again; 12 times are 1; therefore, multiplying the numerator by 12, increases the value of the fraction twelve times. 2. Multiply by 7. 3. Multiply by 7. Ans. 56-63. 4. If one ton of hay cost $10, what will 13 tons cost? 5. If a man pays $2 for one week's board, how much must he pay for 11 weeks' board? Ans. 27. 6. If a man builds 3 rods of wall in one day, how many rods of wall can he build in 7 weeks, allowing that he does not work on the Sabbath? Ans. 147. 7. If a man pays of a mill for the use of one dollar one day, how much must he pay for the use of $847 one day? Ans.,141. VIII. TO DIVIDE A FRACTION BY A WHOLE NUMBER. Multiply the denominator by the whole number, or divide the numerator by the whole number, when it can be done without a remainder. 1. Divide by 8. 2. Divide by 7. Ans. Ans. T In each of these examples, the division is performed by dividing the numerator by the whole number. That is 8 times less than 8, is plain; because the denomination of parts in the fraction remains unchanged, while the number of parts is diminished 8 times. 3. Divide by 5. Ans. 3-5=2. In this example, the division is performed by multiplying the denominator by the whole number. That this process divides the fraction is evident, because the parts of the fraction, are 5 times less than the parts of the fraction 3, while the number of parts is the same in both. 4. Divide by 6. Ans. 5. If 13 tons of hay cost $144, what is the price of one ton? 6. Divide 14 by 8. 7. Divide 34 by 11. 8. Divide 8 by 6. Ans. $11. 39 Ans. 13. Ans. . Ans. 18. Ans. 995405429301 12. Divide 889900112233445577667788888 by 1243576. Ans. 71559768943228 85246105608119 124215431904528. 13. Divide 8756253758287 by 125375. Ans. 6984 44661874 879051867 14. Divide 6251887 by 375875. Ans. 4675633 934 $2587 15. Divide §87 by 892756. 16. Divide 1848827582583 by 1155. 281172132 IX. TO MULTIPLY ONE FRACTION BY ANOTHER. Multiply the numerators together for a new numerator, and the denominators for a new denominator. 1. Multiply by . OPERATION. X, Ans. This process may be explained by referring to the definition of multiplication. Multiplying by a whole number consists in taking the multiplicand as many times as there are units in the multiplier. Multiplying by a fraction is taking a certain part of the multiplicand as many times as there are like parts of a unit in the multiplier. Multiplying by, is taking of the multiplicand once. One half of is found by multiplying the denominator by 2. are gare. of . are taken once. Nineteen thirtieths of & are taken once, in example 2; or one thirtieth of nineteen times. One thirtieth of & is found by multiplying the denominator 9 by 30, and writing the product under 8, thus, . Nineteen thirtieths are 19 times as much as one thirtieth; therefore by multiplying to by 19, we have the true answer, 1=7. 76 135 X. TO DIVIDE ONE FRACTION BY ANOTHER. Invert the divisor, and proceed as in multiplication. 1. Divide by . OPERATION. X=1=3, Ans. This process deserves the careful attention of the pupil. To make the illustration of the rule more intelligible, let us suppose the above sum to be given thus: If a man pays of a dollar for of a yard of cloth, what will a yard cost? If of a yard cost of a dollar, one third will cost one half of, or. If of a yard cost, one yard will cos three times, or 13=3. 6. Divide 178 by 6417. 7. Divide 115431187 by 334422jj???? 8. Divide 1374228 by 28356 1 1 8 8 7 7. 9999 9. Divide 33327788 by 1994. 10. Divide 61 by 81. TO REDUCE FRACTIONS TO THE SAME DENOMINATOR. Multiply the numerator and denominator of each fraction by the denominators of all the other fractions. 1. Reduce , 7, and to the same denominator. It will be seen by inspecting the above process, that each fraction retains the same value after it is reduced, as it had before. And this is evident, because both terms of each fraction are multiplied by the same numbers. 2. Reduce 1, 2, to the same denominator. Ans.,, and . 3. Reduce and to the same denominator. §. Ans. 4 and 48. XII. TO REDUCE A FRACTION OF A HIGHER DENOMINATION TO THAT OF A LOWER. Multiply the given fraction by that number of the next lower denomination which makes a unit of the given denomination. Proceed in this manner until the fraction be reduced to the required denomination. 1. Reduce of a shilling to the fraction of a penny. OPERATION. TX12=1=15, Ans. As the value of shillings is 12 times greater than that of pence, therefore to reduce a fraction of a shilling to the fraction of a penny, the number of parts taken in the fraction of a shilling must be increased 12 times, or the value of the parts must be increased 12 times. Multiplying the numerator of a fraction multiplies the number of parts in a given fraction; dividing the denominator of a fraction increases the value of the number of parts in the given fraction. Hence, either of these ways may be adopted, according to the character of the sum. |