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. 3. Multiply 51 by 1. 4 4. Multiply by of 9. 10 5. Multiply of 3 of by 151. 6. Multiply by of 7. Required the product of 6 by 3 of 5. Ans. 8. Required the product of 3 of 3 by 3 of 32. 9. Required the product of 32 by 413. 10. Required the product of 5, 3, 20. Ans. 23. Ans. 14124. 231 of 3 and 4. Ans. 11. Required the product of 41, of and 18. Ans. 91 140 12. Required the product of 14, 5, 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. 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 $3 per yard? Ans. $51. 2. What will 32 gallons of brandy cost, at $11 per gallon? Ans. $36. 3. If 1lb. of tea cost $11, what will 6176. cost? Ans. $ 4. What will be the cost of 17 yards of cambric at 21 shillings per yard? Ans. £2 3s 9d. 5. What will 151 barrels of cider come to at $3 per barrel? per 6. What will 33 boxes of raisins cost at $2 Ans. $45. per box? Ans. $816. 7. What will 15 barrels of sugar cost at 17 dollars barrel ? Ans. $ 8. What will 32 cords of wood cost at $33 per cord? Ans. $14 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; or that the dividend will contain the divisor but 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 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? Oneseventh 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. Q. How do you divide a fraction by a whole number? 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. 2nd OPERATION. 24 = 5)10 Since multiplying the denominator by 5 is the same as dividing the numerator, and multiplying the numerator 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? 1. Divide 16 by . 90 EXAMPLES. 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. 2. Divide by 1 3. Divide by 16 Ans. 8. Ans. In the division of fractions we should note the follow ng 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? 1. If 77b. of sugar cost of a dollar, what is the price per pound? 147=44 of $1; or 44 of 100 cents=4400-81 525 525 Ans. 8 cents. |