numerators together for a numerator, and the denominators together for a denominator. Q. What is the product of one-sixth by one-seventh ? Of threefourths by one-half? Of six-ninths by three-fifths ?' Give the general rule for the multiplication of fractions. 2. Multiply 1 of by 8.3. We first reduce the OPERATION. compound fraction to the of fun simple one in, and then the mixed number to the 81=2 equivalent fraction ?; af Hence, x=26=25: ter which, we multiply the Ans. 25. numerators and denominators together. 3. Multiply 51 by 1. Ans. 1=5. 4. Multiply 12 by of 9. Ans. 810 5. Multiply 1 of 3 of 1 by 15. Ans. 6. Multiply á by of . Ans. 7. Required the product of 6 by of 5. Ans. 20. 8. Required the product of of 3 by of 34. Ans. 23. 9. Required the product of 34 by 4.14 Ans. 14131 10. Required the product of 5, 3, of z and 45. ]7 18 Ans. 11. Required the product of 41, of and 18%. Ans. 9140 12. Required the product of 14, 6 of 9 and 63. Ans. $ 115. Note. In multiplying by a mixed number, we may first multiply by the integer, then multiply by the fraction, and then add the two products together. This is the best method when the numerator of the fraction is 1. EXAMPLES. 1. Multiply 26 by 31. OPERATION. 26 3 We first multiply 26 by 3: the product is 78. Afterwards we multiply 26 by 1: the product is 13: hence the entire product is 91. 78 26x}=13 91 Ans. 2. Multiply 48 by 8. We first multiply by 8, and then add a sixth. OPERATION. 48 x8=384 48x}= 8 392 Ans. 3. Multiply 67 by 912 Ans. 608 1. 4. Multiply 842 by 77. Ans.. 59875 5. Multiply 3756 by 31. Ans. 12019. 6. Multiply 2056 by 5. Ans. Q. How may you multiply by a mixed number? When is this the best method ? APPLICATIONS. 1. What will 7 yards of cloth cost at $ per yard ? Ans. $51 2. What will 32 gallons of brandy cost, at $1; per gallon? Ans. #36. 3. If ilb. of tea cost $14, what will 641b. cost? Ans. 4. What will be the cost of 171 yards of cambric at 21 shillings per yard ? Ans. £2 3s 9d. 5. What will 1516 barrels of cider come to at $3 per barrel ? Ans. $4516 6. What will 3ž boxes of raisins cost at $27 per box? Ans. $816 7. What will 152 barrels of sugar cost at 171 dollars Ans. * 8. What will 3. cords of wood cost at $32 per cord ? Ans. $1416 per barrel ? DIVISION OF VULGAR FRACTIONS. § 116. We have seen that division of integer numbers explains the manner of finding how many times a less number is contained in a greater. In division of fractions, the divisor may be larger than the dividend, in which case the quotient will be less than 1. For example, divide 1 apple into 4 equal parts. Here it is plain that each part will be į; or that the dividend will contain the divisor but į times. Again, divide of a pear into 6 equal parts. If a whole pear were divided into 6 equal parts each part would be expressed by . But since the half of the pear was divided, each part will be expressed by 1 of or 12 When we divide a fraction by a whole number we are to divide the fraction into as many equal parts as there are units in the divisor, and this may be done by dividing the numerator as in § 81, or by multiplying the denominator as in § 82. Q. What does division of whole numbers explain? In division of fractions may the divisor exceed the dividend ? How will the quotient then compare with 1? If an apple be divided in 2 equal parts, what will express each part? If half an apple be divided into 4 equal parts what will express one of the parts ? What is one-half of one-half? What is one-sixth of one-half? One-sixth of one-fourth ? One. seventh of three-fourths ? One-eighth of one-half? One-ninth of onethird? One-tenth of two-thirds ? CASE I. § 117. To divide a fraction by a whole number. RULE. Divide the numerator or multiply the denominator by the whole number. EXAMPLES. 1. Divide by 2. OPERATION. divide the fraction by mul 4 4 4 2 tiplying the denominator: in 32= 3X2 63. the second we divide the 4 2)4 2 numerator, giving the same or 3+2=13 3 result in both cases. 2. Divide 39 by 9. 3=1 3. Divide 455 by 15. Ans. 4:5=15 . 4. Divide 3755 by 19. Ans. 5. Divide 379, by 15. Ans. Q. How do you divide a fraction by a whole number? Ans. 18 379 19005: CASE II. § 118. To divide one fraction by another. EXAMPLES. 120 1. Divide 10 by If the divisor were 5 the quo 1st OPERATION. tient would be 10 But, since the divisor is į of 5 the true quo š=5x}. tient must be 8 times 100, for the 3+5=113% eighth of a number will be contained in the dividend 8 times 120*8=180=. more than the number itself. In this operation we have actually multiplied the numerator of the dividend by 8 and the denominator by 5: that is, we have, inverted the terms of the divisor and multiplied the fractions together. Since multiplying the denomina 2nd OPERATION. tor by 5 is the same as dividing the numerator, and multiplying the nu *-=32= merator the same as dividing the denominator, we may, if we please, divide 10 by 5 and 24 by 8. Hence, for the division of vulgar fractions we have the following RULE. Reduce compound fractions to simple ones, and whole numbers to improper fractions; then divide the numerator by the numerator and the denominator by the denominator, if they will exactly divide : but if not, invert the terms of the divisor, and then multiply the divisor and dividend together. Q. How do you divide one vulgar fraction by another ? EXAMPLES. 1st OPERATION. 8)16 1. Divide yo by Here we divide the numerator by the numerator, and the denominator by the denominator. Here, we have inverted the terms of the divisor and multiplied the fractions together. 8388=i=1 2nd OPERATION. 36 = is equal to boxý== 2. Divide by t. Ans. 8. 3. Divide by 16 Ans. In the division of fractions we should note the following principles. 1st. When the dividend is just equal to the divisor, the quotient will be 1. 2nd. When the dividend is greater than the divisor, the quotient will be greater than 1. 3rd. When the dividend is less than the divisor, the quotient will be less than 1. 4th. The quotient will be just so many times greater than 1, as the dividend is greater than the divisor. 5th. The quotient will be just as many times less than 1, as the dividend is less than the divisor. Q. How do you divide one fraction by another ? When will the quotient be 1 ? When greater than 1 ? When less than 1 ? When greater than 1, how many times greater? When less than 1, how many times less ? 4. Divide 1 by . Ans. Ž: 5. Divide 3) by . Ans. 291 6. Divide 16 of by 44. Ans. 7. Divide 413} by 333. Ans. 8. Divide 3713 by ala: Ans. 370215 9. Divide 114 by 21. Ans. 302353. 10. Divide of } by of Ans. 11. Divide 5 by to Ans. 12. Divide 5205} by of 91. Ans. 711. 13. Divide 100 by 43. Ans. 2030 14. Divide of by s. Ans. 15. Divide of 50 by 41. Ans. 16. Divide 141 of by 3 of 6. Ans. 113 1440 APPLICATIONS. 1. If 716. of sugar cost 4 of a dollar, what is the price per pound? 4+7=4 of $1; or 25 of cents=440=8 Ans. 8 & cents. |