7. Reduce 2, 3 and · 8. Reduce 3, 6, 8 9. Reduce & and 99 27 to their least common denominator. to their least common denominator. to their least common denominator. 10. Reduce 4,73 and 5 to their least common denominator. 36 68 15. Reduce 53, 65, 37 and 2 to their least common denominator. ADDITION OF COMMON FRACTIONS. 126. THE SUM of two or more fractions is a number which contains the unit 1 as many times as it is contained in the fractions separately. ADDITION OF FRACTIONS is the operation of finding the sum of two or more fractions. There are two cases: 1st. When the fractions have the same unit. 2d. When they have different units. CASE I. 127. When the fractions have the same unit. 3 and ? 1. What is the sum of 1, 2, ANALYSIS.—In this example, the unit of the fraction is 1, unit. There is one and the fractional OPERATION. 1+ 3 + 6 +3 = 13 in the first, 3 one hence, 186 sum. halves in the second, 6 in the third, and 3 in the fourth: hence, there are 13 halves in all, equal to 6. 2. What is the sum of £ and £? ANALYSIS.-The unit of both fractions is 1£. In the first, the fractional unit is £, and in the second, £. These fractional units, being different, cannot be expressed. OPERATION. }£= f£ £+ f£ = £7 = £17. What is addition of 126. What is the sum of two or more fractions? fractions? How many cases are there? What are they? 127. How do you add fractions which have the same unit? = £, in each of which = in one collection. But £ £ and 3£ = the fractional unit is £: hence, their sum is NOTE.-Only units of the same value, whether fractional or integral, can be expressed in the same collection. From the above analysis, we have the following RULE.-I. When the fractions have the same denominator, add their numerators, and place the sum over the common denominator: II. When they have not the same denominator, reduce them to a common denominator, and then add as before. NOTE.-1. After the addition is performed, reduce every result to its simplest form; that is, improper fractions to mixed numbers, and the fractional parts to their lowest terms. 2. It often abridges the operations in fractions to reduce them to their least common denominators, before adding (Art. 125*.) NOTE.-Reduce each fraction to its least common denominator be 128. The sum of two fractions whose numerators are each 1, is equal to the sum of their denominators divided by their product. 128. What is the sum of two fractions equal to when each numerator is equal to 1 ? 18. What is the sum of and ? of and ? of and of and? 19. What is the sum of and? of and? of and ? of and? 20. What is the sum of 123, 113 and 155? OPERATION. Whole Numbers. 12+11+15=38 then, Fractions. 68 8+3+4=18085+10% +708-788=1188: 38+1188=39188. Ans. NOTE. When there are mixed numbers, add the whole numbers and the fractions separately, and then add their sums. Find the sums of the following fractions: 21. Add 19, 34 and 1 of 7. of of 8 and 21. 27. Add, 30. Add 3, 4 of 1 of 15. and of 16. 31. Add 63, 133, 181 and 1321. 32. Add 125, 26, and 4013. 33. Bought a cord of wood for 25 dollars; a barrel of flour for $95; and some pork for $53: what was the entire cost? 34. A person travelled in one day 35 miles; the next, 284 miles; and the next, 2571 miles: how many miles did he travel in the three days? 35. A grocer bought 4 firkins of butter, weighing respectively 543, 553, 517 and 50 pounds: what was their entire weight? 36. I paid for groceries at one time of a dollar; at another, 3 dollars; at another, 73 dollars; and at another, 5 dollars what was the whole amount paid? 37. A merchant had three pieces of Irish linen; the first piece contained 225 yards; the second 20 yards; and the third 21 yards: how many yards in the three pieces? 38. A man sold 5 loads of hay; the first weighed 18,7cwt.; the second 19cwt.; the third 195cwt.; the fourth 211cwt.; and the fifth 201cwt.: what was the weight of the whole? 39. A farmer has three fields; the first contains 17 acres; the second 25 acres; and the third 46 acres: how many acres in the three fields? 40. A man sold 112 bushels of wheat for 250 dollars; 91⁄2 bushels of corn for 623 dollars; 225 bushels of oats for 1047 dollars: how many bushels of grain did he sell, and how much did he receive for the whole ? CASE II. 129. When the fractions have different units. 1. What is the sum of 7b. and 3oz.? ANALYSIS.-In lb. there are oz. (Art. 41). Then, the units of the fractions being the same, viz., 1oz., we reduce to a common denominator and add, and obtain 1311oz. SECOND METHOD.- Threefourths of an ounce is equal to lb. (Art. 41). Then, by adding, we find the sum to be OPERATIONS. lb. = × 16oz. 6402. oz. = 302.116.34lb. b=13110.=13oz. 84dr. lb.=160z.oz. 12oz. 12 dr. oz.=&×16dr.=&dr.= 12 THIRD METHOD.-Find the value of each fractional part in terms of integers of the lower denominations, and then add. RULE.-Reduce the given fractions to the same unit, and then add as in Case 1. Or: Reduce the fractions separately to integers of lower denominations, and then add the denominate numbers. EXAMPLES. 1. Add of a yard to of an inch. 2. Add together of a week, of a day, and 1⁄2 of an hour. cwt. and 676. together. of an ounce. 129. How do you add fractions when they have different units? 5. Add of a ton to 5 of a hundred weight. 6. Add 1⁄2 of a chaldron to 3 of a bushel. 7. What is the sum of 3 of a tun, and of a hogshead of wine? 8. Add of of a year, of of a day, and 3 of 3 of 2 of 19 hours, together. 9. Add of an acre, 3 of 19 square feet, and of a square inch, together. 10. What is the sum of of a yard, of a foot, and of an inch? 11. What is the sum of of a £, and § of a shilling? 12. What is the sum of 1 of a week, of a day, of an hour, and of a minute? 13. Add together of a mile, of a yard, and of a foot. 14. What is the sum of 3 of a year, of a week, and of a day? 15. Add of a ton to 5 of a hundred weight. 16. Add 3b. troy, 17. Add together of 5 minutes. oz. and pwt. 3 19 of a circle, 35 signs, of a degree, and 18. What is the sum of yd., 3 of §qr. and 3}na.? 19. Add of a cord, feet. cubic feet, and of of 243 cubic 6 20. What is the sum of 2 of of 4 cords, of of 15 cord feet, and of 31 cubic feet? 21. Add of 3 ell English to of a yard. 22. Add together of 3A. 1R. 20P., & of an acre, and 3 of 3R. 15P. 23. What is the sum of of a ton, of a cwt., and of an ounce ? 24. What is the sum of of of a mile, of a furlong, of a rod and of a foot? 25. What is the sum of of a year, of a week, of a day and of an hour? |