3 8. What is the sum of 1 of 41,31⁄2 of 64, and § ? 9. What is the sum of 1, 1, 1, 1, 2, and 10. What is the sum of 1, 1, 1, ? 5, and 2% ? 11. What is the sum of 11, 21, 31, 41, and 51? SUBTRACTION OF FRACTIONS. 24. To subtract one fraction from another we have this RULE. Reduce the fractions to a common denominator, and subtract the numerator of the subtrahend from that of the minuend; place the common denominator under the difference. These fractions, when reduced to their least common denominator become 23,-4-2-4-341 93 88 893 88 MULTIPLICATION OF FRACTIONS. 25. Multiply by 4. We have seen (under Art. 20) that multiplied by & is the same as of: Therefore we must use the same rule for multiplying fractions, as we would for reducing compound fractions. Hence, to multiply together fractions, we have this RULE. Multiply all the numerators together for a new numerator, and all the denominators together for a new denominator: always observing to reject, or cancel, such factors as are common to both numerators and denominators. Examples. 1. Multiply together the fractions, 2, 3, and §. 3 22 71 Expressing the multiplication, we obtain X. X-X 11 21 9 3 Canceling the 3 and 7 of the numerators, against 21 of the denominators; also the 11 of the denominators against a part 3 2 77 1 2 2 of the 22 of the numerators,we get. X X X 1X 21 9 3 9×3 27 2. Multiply together the fractions 77, 21, 5, and . 7 21 55 4 Indicating the multiplication, we get—X X Can celing the 11 of the denominators, against a part of the 55 of the numerators; also the 7 of the numerators, against a Again, canceling the 5 which is common to both numerators and denominators; also the factor 7, which is common to 21 of the numerators and to 42 of the denominators, we get against a part of the 9 of the denominators; and the factor 2, which is common to the 4 of the numerators, and to 6 of the denominators, we obtain NOTE.-A little practice will enable the student to perform these operations of canceling with great ease and rapidity. And since, as was remarked under Art. 20, it is immaterial which factors are first canceled, the simplicity of the work must depend much upon his skill or ingenuity. 3. Multiply together the fractions,, and . 10 4. Multiply together the fractions 17, 14, 29, 26, and Ans. Ans. 5. Multiply together the fractions, 31, of, and Ans. T 20 DIVISION OF FRACTIONS. 26. Divide by §. We know that can be divided by 5, by multiplying the denominator by 5, (see Prop. II., Art. 15,) which gives x6 4 Now, since is but one-eighth of 5, it follows that divided by § must be eight times as great as 4 divided by 5: ... 4 divided by must be 4X8 7×5. tiplied by & inverted. From this we see that has been mul. Hence, to divide one fraction by another, we have this RULE. Reduce the fractions to their simplest form. Invert the di visor, and then proceed as in multiplication. 1. Divide 42 by 21. Examples. Inverting the divisor, and then multiplying, we obtain 43 COMPLEX FRACTIONS. 27. Sometimes fractions occur in which the numerator or denominator, or both, are already fractional. 2 Thus, fractions. and : such fractions are called complex 13 REDUCTION OF COMPLEX FRACTIONS. 28. Since the value of a fraction is the quotient arising from dividing the numerator by the denominator, it follows that is the same as 2÷4=4=43. Again, 4=1÷7=4• 2 Hence, to reduce a complex fraction to a simple one, we have this RULE. Divide the numerator of the complex fraction by the denominator, according to rule under Art. 26. Dividing 41=2 by 31, we get 17-17. |