6 3 . 120 4. Find the least common denominator (see § 96), and add the fractions to, , , and Ans. 5. Find the least common denominator and add 122 59 , and 30 Ans. 1 9.00 NOTE. § 105. When there are mixed numbers, instead of reducing them to improper fractions we may add the whole numbers and the fractional parts separately, and then add their sums. 6. Add 19, 63, and 4 together. OPERATION. OPERATION. Whole numbers. Fractional parts. 19+6+4=29. 1+3+4=169=1665 Hence, 29+1945=30045, the sum. 7. Add 31, 65, 815, and 65%. Ans. 84578 CASE III. § 106. When the fractions are of different denominations. RULE. Reduce the fractions to the same denomination. Then reduce all the fractions to a common denominator, and then add them as in Case I. 1 EXAMPLES. 1. Add of a £to of a shilling. of a £= of 40=40 of a shilling: Then, 4° +5=246 +15=2558=85s=14s 2d. Or, the 5 of a shilling might have been reduced to the fraction of a £ thus, of b=ião of a £=24 of a £. Then, +3=4+= of a £: which being reduced by $ 100, gives 14s 2d. Ans. 14s. 2d. 2. Add } of a yard to of an inch. Ans. Syds. or 14in. 3. Add } of a week, 1 of a day, and į of an hour together. Ans. da. hr. 4. Add 4 of a cwt., 85lb. and 3102. together. Ans. 2qr. 1716. 13.0%. 5. Add 1 miles, jo furlongs, and 30 rods together. Ans. lm. 3 fur. 18rd. Note. The value of each of the fractions may be found separately, and their several values then added. 6. Add of a year, } of a week, and of a day together. of a year = of 365 days=219 days. 3 hours. Ans. 221da. 11hr. 7. Adds of a yard, of a foot, and of a mile together. Ans. yd. ft. in. 8. Add of a cwt., h of a lb. 13oz. and į of a cwt. 61b. together. Ans. lcwt. lqr. 271b. 130. Q. How do you add fractions of different denominations ? What is the second method ? SUBTRACTION OF VULGAR FRACTIONS. § 107. It has been shown (see § 102), that before fractions can be added together, they must be reduced to. the same unit and to a common denominator. The same reductions must be made before subtraction. SUBTRACTION of Vulgar Fractions teaches how to take a less fraction from a greater. .Q. Can one-third of a shilling be subtracted from one-third of a £ without reduction ? Can one-fourth of a shilling be subtracted from one-fifth of a shilling? What reductions are necessary before subtraction? What is subtraction ? CASE I. § 108. When the fractions are of the same denomination and have a common denominator. RULE. Subtract the less numerator from the greater and place the difference over the common denominator. EXAMPLES. 1. What is the difference between and i? 2. What is the difference between 1 and 38675 Ans. 3. From 185 take 163 Ans. 4. From 975 take 1997 Ans. 5. From 18906 take 909 Ans. 33997 58 365 166 105 327 327 CASE II. § 109. When the fractions are of the same denomination, but have different denominators. RULE. Reduce mixed numbers to improper fractions, compound fractions to simple ones, and all the fractions to a common denominator : then subtract them as in Case I. EXAMPLES. 1. What is the difference between 5 and ? Here, -=-== answer. Q. How do you subtract fractions which have the same unit but dif. 'ferent denominators ? What is the difference between one-half and one-third ? 2. What is the difference between 12 of 1 and 2 ? Ans. iz. 3. What is the difference between 21 of a £, and 1 of a £? Ans. £ 4. From of 6, take 19 of 1. Ans. 5. From of 3 of 7, take of Ans. 6. From 371), take 3 of. Ans. 36 635 19 68• CASE III. $ 110. When the fractions are of different denominations. RULE. Reduce the fractions to the same denomination : then reduce them to a common denominator, after which subtract as in Case I.. EXAMPLES. 1. What is the difference between of a £, and of a shilling? of a shilling=of or of a £. Then, 1-60=20-06=27 of a £=98 8d. Q. How do you subtract fractions which are of different denominations? 2. What is the difference between of a day and of a second ? Ans. 11hr. 59m. 591 sec. 3. What is the difference between of a rod and of an inch ? Ans. 10 ft. 11 in. 4. From 1 of a lb. troy weight, take į of an ounce. Ans. .5. What is the difference between i of a hogshead, and ion of a quart? Ans. 16gal. 2qt. 1 pt. 35gi. 6. From į of a £ take of a shilling? Ans. S. d. 7. From oz. take ž pwt. Ans. 11pwt. 3gr. 8. From 4 cwt. take 4% 16. Ans. 4cwt. Iqr. 151b. 1oz. 93}dr. MULTIPLICATION OF FRACTIONS. § 111. John gave į of a cent for an apple. How much must he give for 2 apples? For 3 apples ? For 4 ? For 5 ? For 6? For 7? For 8? For 9? Charles gave of a cent for a peach ? How much must he give for 2 peaches ? For 3? For 4? For 5 ? For 6 ? EXAMPLES. 1. Multiply the fraction by 4. When it is required to mul OPERATION. tiply a fraction by a whole 3x4=2==2]; number, it is required to in | or by dividing the denocrease the fraction as many times as there are units in the minator by 4, we have multiplier, which may be done by multiplying the numerator (see § 80), or by dividing the denominator (see § 83). CASE I. § 112. To multiply a fraction by a whole number. RULE. Multiply the numerator, or divide the denominator by the whole number. 2. Multiply 37 by 12. Ans. 31. 3. Multiply 43 by 7. Ans. 4. Multiply 175 by 9. Ans. 1575. 5. Multiply 127 by 5. Ans. 6. Multiply by 49. Ans. 124101 Q. How do you multiply a fraction by a whole number ? $ 113. Note. When we multiply by a fraction it is required to repeat the multiplicand as many times as there are units in the fraction. For example, to multiply 8 by is to repeat 8, i times; that is, to take of 8, which is 6. Hence, when the multiplier is less than 1 we do not take the whole of the multiplicand, but only such a part of it as the fraction is of unity. For example, if the multiplier be one-half of unity, the product will be half the multiplicand: if the multiplier be į of unity, the product will be one-third of the multiplicand. Hence, to multiply by a proper fraction does not imply increase, as in the multiplication of whole numbers. Q. What is required when we multiply by a fraction? What is the product of 8 multiplied by one-half? By one-fourth ? By one. eighth? By three-halves? By six-halves ? What is the product of 9 multiplied by one-half ? By one-third ? By one-sixth ? By one-ninth ? When the multiplier is less than 1, how much of the multiplicand is taken? Does the multiplication by a proper fraction imply increase ? CASE II. § 114. To multiply one fraction by another. EXAMPLES. OPERATION. 1. Multiply by In this example à is to be taken ý times. That is, šis product divided by 7, a result which is obtained by multiplying the numerators and denominators together. Hence, we have the following RULE. Reduce all the mixed numbers to improper fractions, and all compound fractions to simple ones: then multiply the |