MULTIPLICATION OF FRACTIONS. 25. LET it be required to multiply by . 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 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 denomi nators. EXAMPLES. 1. Multiply together the fractions,,, and . Expressing the multiplication, we obtain × 2 Canceling the 3 and 7 of the numerators, against 21 of the denominators, also the 11 of the denominators against a part of the 22 of the numerators, we get 2 22 7 1 X X X XX 2X 9 3 9 × 3 27 2. Multiply together the fractions,,, and . Indicating the multiplication, we get ××× Canceling the 11 of the denominators, against a part of the 55 of the numerators, also the 7 of the numerators, against a part of the 35 of the denominators, we obtain 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 3 Finally, canceling the 3 of the numerators, against a part of the 9 of the denominators, and the factor 2, which is common to the 4 of the numerators, and to the 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 1. 14 Ans. 4. Multiply together the fractions 11, 11, 11, 7. 289 Ans. 5. Multiply together the fractions, 31, of, and Ans. T 6. Multiply together the fractions §, †, †, and 71. 3 7. Multiply together, 4, 1, and §. Ans. 32 Ans. 33. Ans. . Ans. or. 10. Multiply together, 3, 4, 1, 3, and §. Ans. T DIVISION OF FRACTIONS. by §. 26. LET it be required to divide We know that can be divided by 5, by multiplying the denominator by 5, (see Prop. II., Art. 16,) which Now, since is but one-eighth of 5, it follows that 4, divided by, must be eight times as great as divided 4 x 8 by 5. .. ‡, divided by §, must be From this, we see that has been multiplied by, when inverted. Hence, to divide one fraction by another, we have this RULE. Reduce the fractions to their simplest form. Invert the divisor, and then proceed as in multiplication. Inverting the divisor, and then multiplying, we obtain x; which, by canceling, becomes. 43 2X 1 X - 4 2 Ans. 13505-200 4802 Ans. 3. Ans. 27. SOMETIMES fractions occur, in which the numerator, or denominator, or both, are already fractional. 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 the complex fraction is the same as 2 2÷4=4=43. Again,=‡÷1=+ 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 4 by 3=0, we get 27=1. 42 2. Reduce to a simple fraction. Ans. 3. Reduce to a simple fraction. Ans. =3}} 29. SUPPOSE we wish to change the fraction to an equivalent one, having 6 for its denominator. It is obvious that if we first multiply by 6, and then divide the product by 6, its value will not be altered. 4×624_4 By this means, we find that |