CASE III. § 142. When the fractions are of different denom. inations. RULE. Reduce the fractions to the same denomination. Then reduce all the fractions to a common denominator, and then add them as in Case 1. Ex. 1. Add f of a £. to of a shilling. of a £.=f of Y=y of a shilling: Then, 40+1=4+1}=**s=\'s=14s 2d. Or, the f of a shilling might have been reduced to the fraction of a £. thus, of j;=rty of a £.= of a £. Then, f +=+*+*=*] of a £.: which being reduced by $ 136, gives 14s 2d. Ans. 14s 2d. 2. Add f of a yard to of an inch. Ans. 741yds. or 1474inches. 3. Add } of a week, 1 of a day, and į of an hour together. Ans. 2da. 144hr. 4. Add of a cwt., 8416. and 37602. together. Ans. 2qr. 1726. 1370z. 5. Add 17 miles, fo furlongs, and 30 rods together. Ans. lm. 3fur. 18rd. NOTE. $ 143. The value of each of the fractions may be found separately, and their several values then added. Ex. 6. Add f of a year, of a week, and f of a day together. of a year = of 302 days=219 days į of a week=1 of 7 days 2 days, 8 hours of a day = 3 hours. Ans. 221da. 11hr. 7. Add f of a yard, of a foot, and of a mile together. Ans. 1540yd. 2ft. Iin. 8. Add of a cwt., y lb. 13oz. and į of a cwt. 61b. together. Ans. Icwt. lqr. 2716. 13oz. QUESTIONS. $138. What is addition of integer numbers ? What is ad. dition of fractions ? Can fractions be added while they have different units ? If they have the same unit can they be ad. ded if the parts are unlike ? What then is necessary before fractions can be added ? § 139. How do you add when the fractions have a common denominator ? § 140. How do you add when the fractions have different denominators ? $ 141. When there are mixed numbers, how may the ad. dition be performed ? $ 142. How do you add when the fractions are of different denominations ? § 143. In what other way may they bo added ? SUBTRACTION OF VULGAR FRACTIONS. $ 144. Subtraction of Vulgar Fractions teaches how to take a less fraction from a greater. CASE I. $ 145. When the fractions are of the same denomination, and have a common denominator. RULE. Subtract the less numerator from the greater and pla sho difference over the common denominator. Ex. 1. What is the difference between and ; ? 3-2=1: hence is the answer. 2. What is the difference between it's and 3%} ? Ans. CASE II. $ 146. When the fractions are of the same denomination, but have different denominators. RULE. Reduce mixed numbers to improper fractions, compound fractions to simple onės, and all the fractions to a common denominator : then subtract them as in Case I. Ex. 1. What is the difference between 1 and } ? Here, 441=-==1 answer. 2. What is the difference between 121 of f and 2? Ans. th: 3. What is the difference between 21 of a £., and its of a £. ? Ans. £2 6s. CASE III. $147. 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. Ex. 1. What is the difference between 4 of a £., and of a shilling? of a shilling = of brit of a £. Then, t=28=1# of a £=9s 8d. 2. What is the difference between f of a day or of a second ? Ans. 11hr. 59m. 59 3. What is the difference between $ of a rod and of an inch? Ans. 10ft. 114in. 4. From 17 of a lb. troy weight, take of an ounce. Añs. 126. 8oz. 16pwt. 16gr. 5. What is the difference between is of a hogshead, and of a quart? Ans. 16gal. 2gt. Ipt. 39% gi. QUESTIONS. $ 144. What does Subtraction of Vulgar Fractions teach? g 145. How do you subtract when the fractions are of the same denomination, and have a common denominator ? § 146. How do you subtract when the fractions have dif. ferent denominators ? $ 147. What do you do when the fractions are of different denominations ? MULTIPLICATION OF VULGAR FRACTIONS. Ø 148. When we multiply by a whole number we repeat the multiplicand as many times as there are units in the multiplier, 28. Therefore, if it be required to multiply a fraction, as f, by 4, it is necessary to increase the fraction as many times as there are units in 4; which, as shown in Ø 114 and $ 117, may be done either by multiply. ing the numerator, or dividing the denominator by 4. CASE I. $ 149. To multiply a fraction by a whole number. RULE. Multiply the numerator, or divide the denominator by the whole number. Ex. 1. Multiply by 6. 2. Multiply it by 12. Ans. 315. 3. Multiply 47 by 7. Ans. 64. Ø 150. 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, 1 times; that is, to take of 8, which is 2. 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 of unity, the product will be one third of the multiplicand, &c. Ex. 1. Let it be required to multiply 4 by 4. Here is to be taken times. But #=5X+; hence is to be multiplied by 5, and then of the product taken. But * x 5=Y, and Y-7573=} 116. Hence, for multiplying one fraction by another we have the following RULE. § 151. Reduce all the mixed numbers to improper fractions, and all compound fractions to simple ones : then multiply the numerators together for a numerator, and the denominators together for a denominator. Ex. 1. Multiply 1 of if by 84. of ?=; and 8f=Y: Ans. H= Ans. Na |