13. If I had 1 acres less land, I should have 1 acres more than you, who have 15% acres. How many acres belong to me? MULTIPLICATION OF FRACTIONS. 139. Under this head, we have the following cases :- 6th, to multiply a mixed quantity by another. 140. 1st, to multiply a fraction by a whole number. Let it be required to multiply by 4. From the definition of multiplication (§ 65), we have to repeat as many times as there are units in 4. Now, we know that 4×2=8; and since are 9 times less than two wholes, 4 x =9 times less than 8, or 3. Therefore, to multiply a fraction by a whole number, multiply the numerator by the whole number, and divide the product by the denominator. 141. 2nd, to multiply an integer by a fraction. Multiply 8 by . If it were necessary to multiply 8 by 6, the product would be 48; but 8 must be multiplied by, or by a quantity 7× less than 6 wholes. Hence, the product must be 7x smaller, or 48=6%. Therefore, to multiply an integer by a fraction, the integer must be multiplied by the numerator, and the product divided by the denominator. 142. These two cases often admit of contractions which must not be neglected. For instance, multiply by 8. Here we have 8×11 ; but (by § 102) to multiply the numerator and the denominator by the same number does not alter the value of the fraction; therefore, 2x4x11 2×11 7. The contraction consists, then, in cancelling a measure common to the terms. Also, x 18= 5 × 18 5 × 3 6 = 15. 143. 3rd, to multiply a mixed quantity by a whole number. For instance, 7 by 12. Every mixed quantity may be transformed into an improper fraction; therefore, the question is, 12x7=933. Thus, we have to reduce the mixed 12 × 39 = quantity to an improper fraction, and proceed as in case 1; or, since 12x7=12x7+12x3=84+18=84+98=933; that is to say, to the product of the integer of the mixed quantity by the multiplier, add the product of the fraction by the multiplier. Example: 15×67. 1st method: 15×67= 15 × 55 =825=1031. 8 2nd method: 15 x 63-15 x 6+15x3=90+13=103. 144. 4th, to multiply a whole number by a mixed quantity. Multiply 10 by 14. Reducing the mixed quantity into an improper fraction, we have: 14359; then, 143 × 10= 59 × 10 4 Another method is this: 14 × 10= 140+=1474. Therefore, reduce the mixed quantity to an improper fraction, and proceed as in case 2; or, add the product of the integer and multiplicand to the product of the fraction and multiplicand. Thus: 2nd method: 98x8=9x8+8×8=72+63=783. 145. 5th, to multiply a fraction by a fraction. Multiply by . If it were required to multiply 2x3 ; but, by the question, by 2, the result would be must be multiplied by a quantity 5 times less than 2 wholes, and (by § 99) it is known that a fraction is made 5 x less by multiplying its denominator by 5; therefore, x=5 times less than 2×3 or "" 2x3 4 X 5 Therefore, multiply the numerators together for the numerator of the product, and the denominators together for its denomi 146. 6th, to multiply a mixed quantity by another. Multiply 31 by 24. Thus, we see that, after having reduced the mixed quantities to improper fractions, we proceed as in case 5. The multiplication might have been effected by multiplying 3 by 2, by 2, by 3, and by, then adding those four products; but this operation would be much longer. Example: Multiply 53 by 743. 1st method: 5×773=43 × 103=42-84. 2nd method: 5x7+8x7+5x+3+1×13=35+28+47% +18=42184. 147. The 7th and 8th cases are easily explained, by § 146. 148. We know that×6=3; also, of 6=3. Therefore,×6= 1⁄2 of 6. Likewise, x8=16=51; also, 16=5. Therefore, x8 of 8. -- of 8=2x of 8=2ק= Since, and of ; therefore, x=1 of 4. Likewise, since x; also, & of 5x of 7=5x+2= ; therefore, 3 of 7. From these different examples, it follows that the word of, connecting two fractions, is equivalent to the sign X. 149. When the word of connects two or more fractions, it is usual to call such an expression a compound fraction. Thus, of of is a compound fraction. 150. It is sometimes necessary to find the result of a quantity expressed thus: 3 of 3×3 of 73x. By what has been said before, it may be written in this manner: ×× 2 × 3 × &= 3x7×2×31 x 5 Cancelling the factors 5x8x5x4x6 and the measures 151. The expression merator and the denominator by a common multiple (the least is preferable) of 3 and 5, or 15; thus, 15 x = 10, and 15 x = 12; is simplified by multiplying the nu 152. It is observed that, in the multiplication of fractions, the product of two quantities is often less than either quantity. Now, this ambiguity, when compared with the multiplication of whole numbers, is easily explained. We know that 1xa quantity that quantity; for instance: 1x=3; and since fractions are less than 1, multiplied by a quantity less than 1=less than. Example: × =}. = 153. RECAPITULATORY EXERCISES. 1. Multiply by 36; 5 by ; by 1; 7 by 65. 2. Required, the product of x×3×13. 5. Find the product of 24 x 12 × 27 × 17. 6. What is of £80 ? 7. How much is % of 34 ? 8. Required, the of 750. 9. Find the sum of 3 of £40+ of the same sum. 10. I paid £ of £36, and my debt was 3 of £74. How much do I owe still ? 11. A does, on the first day, of his work; on the second, of the remainder. How much has he yet to do? 12. A pump gives daily 64 gallons. How much will it give in 6 days? 13. On the first day, A performs of his work; on the second, of the remainder; on the third, of what is left; and on the fourth day, he finishes it. last day? How much did he do on the 14. I spent of + of % of what I had, and I have £10 left. How much had I? 15. With £1, 43 yards are bought. How much can be bought for £3. 16. A traveller had to go 144 miles in 3 days; on the first day he went of the way; on the second, . How many miles did he travel each day? 17. 6 lbs. of copper are melted with 4 lbs. of tin. How much of each metal is there in lb. ? 18. A room is 343 feet in length, and 173 in breadth. Find its area, or the number of square feet in its floor. DIVISION OF FRACTIONS. 154. We are enabled to make as many cases in division as were made in multiplication. 1st, to divide an integer by a fraction. 3rd, to divide an integer by a mixed quantity. 4 155. 1st, to divide an integer by a fraction, as 4÷%. =43. If 4 were divided by 5, the result would be ; but 4 is to be divided by a quantity 6 times less than 5, viz., ; therefore, the quotient must be 6×, or 43. Therefore, to divide an integer by a fraction, multiply the integer by the denominator of the fraction, and divide by the numerator. 9 × 10 14 × 12 = 12; 14÷1}= =15 |