for a denominator, (see ART. 33,) and then the general rule will apply. What is the Rule for multiplying fractions? In this example, we cancel the 4 of the numerator against a corresponding factor of the 16 of the denominator; and 5 of the denominator against a corresponding factor of the 10 in the numerator. Thus: 2 10 16 4 Finally, cancelling the 2 in the numerator against the same factor of the 4 in the denominator, we find X 2 6 = 4 Ans. NOTE.-A little practice will enable the student to perform these operations of cancelling with great ease and rapidity. And since, as was remarked under ART. 39, it is immaterial which factors are first cancelled, the simplicity of the work must depend much upon his skill and ingenuity. 8. Multiply together the fractions 3, 4, Expressing the multiplication, after reducing them, we Cancelling the 7 of the numerator against a part of the 14 of the denominator, we have 9. Multiply together the fractions †,†,†,†. Ans. z. 10. Multiply together the fractions, 4, 5, 4. Ans. §. 11. Multiply together the fractions 3, 44, 5. 17. Multiply the sum of,,,, by the sum of 1, 4, t, f. Ans. 14681,383. 18. Multiply the sum of of, of by the sum of, of, of t Ans. 19. Multiply off of of by off of §. Ans. 20. Multiply the sum of 3, 31, 31, 34, by the sum of 24, 31, 4. Ans. 127718. DIVISION OF FRACTIONS. 46. Let us endeavor to divide by . We know that can be divided by 5, by multiplying the denominator by 5, (see PROP. II., ART. 34,) which gives 4 7x5 Now, since is but one eighth of 5, it follows that divided by must give a quotient eight times as great as + divided by 5. Therefore, divided by must give From which we see that has been multiplied by after it was 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. If either the dividend or divisor is a whole number, it may be converted into an improper fraction having 1 for its denominator. Repeat the Rule for the Division of Fractions. Ans. 15. 7. What is the quotient of 44 divided by 171? 8. What is the quotient of divided by 10? Ans. §. 12. Divide the sum of,,,, by the sum of 1,1,1, 13. Divide the sum of 4, 4, 7, 4, t, v, tt, H, H, H, by the sum of 1,t, t, t, t, t, t, to, TI, 12, 13. 7061 Ans. 4088933-4798887. 5693 = 14. Divide of 4 of 4 of 4 by off of of 4. } Ans. 133. 15. Divide the sum of 1, 11, 21, 31, by the sum of 13, 21, 31. Ans. 48=1787. 16. Divide the sum of ¦ of , ¦ of §, by the sum of of +, of t. Ans. 17. Divide of 4 of 4 by of ♣ of §. 19. Ans. 1988. 18. Divide of of by of of 12. Ans. T 19. Divide of 4 of of by of of 8. Ans. 625 RECIPROCALS OF NUMBERS. 47. The reciprocal of a number is the result obtained by dividing 1 by the number. Thus, the reciprocals of 2, 3, 4, and 5, are 1,,, and . From this we discover that the reciprocal of an integer, or whole number, is equal to a vulgar fraction whose numerator is 1, and whose denominator is the given number. The reciprocal of is found by dividing 1 by, which (ART. 46,) is 1÷÷÷=1ק=§. In the same way we find the reciprocal of to be, and in general, the reciprocal of a vulgar fraction is the value of the fraction when inverted. NOTE-From this, we see that dividing by any number is in effect the same as multiplying by the reciprocal of that number. So that operations of division may be included under those of multiplication. A practical application of this principle may be seen under Reduction of Denominate Fractions. (ART. 89.) EXAMPLES. 1. What are the reciprocals of 7, 8, 9, 10, 11? Ans. t,,, To, tr. 2. What are the reciprocals of 18, 23, and 41 ? Ans. 1, 2, . Ans. †,†,†r. 3. What are the reciprocals of 3, 4, 4, 4? |