ADDITION OF FRACTIONS. 172. Addition of Fractions is the operation of finding the number of fractional units in two or more fractions. 1. What is the sum of 1, 3, and ? ANALYSIS.-The fractional unit is the same in each fraction, viz.: ; but the numerators show how many such units are taken (Art. 148); hence, the sum of the numerators written over the common denominator, expresses the sum of the fractions. 2. What is the sum of and ? ANALYSIS.-In the first, the fractional unit is, in the second it is. These units, not being of the same kind, cannot be expressed in the same collection. But the %, and 3=1, in each of which the unit is : hence, their sum is 7=14. OPERATION. Ans. 41. OPERATION. NOTE. Only units of the same kind, whether fractional or inte gral, 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 the 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.-After the addition is performed, reduce every result to its lowest terms. EXAMPLES. 1. Add 1, 3, §, and §. 3 8. Add ,,, and. 9. Add 9, 3, 1, §, and . 15. What is the sum of 194, 63, and 4? Whole numbers. 19+6+4=29 OPERATION. Fractions. Sum=29+16=30&t. 173. NOTE.-When there are mixed numbers, add the uhole numbers and fractions separately, and then add their sums. Find the sums of the following fractions : 16. Add 31, 7%, 12, 17. 17. Add 16, 92, 257, 11. 20. Add 900, 450, 751%. 21. Addofofto of . 18. Add of, 4 of 9, 14.22. Add 173 to 4 of 27%. 19. Add 2, 63, and 1213. 24. What is the sum cf 25. What is the sum of 174. 1. What is the sum of 23. Add , 7, and 83. of 123 of 74, and § of 25? of 93 and and ? NOTE.-If each of the two fractions has of 328? 1 for a numerator, the sum of the frac+f=z%+zv=}J. tions will be equal to the sum of their denominators divided by their product. }+}= OPERATION. 5+6 5 × 6 4. What is the sum of and ?of and ? of and? SUBTRACTION OF FRACTIONS. 175. SUBTRACTION of Fractions is the operation of finding the difference between two fractions. 172. What is addition of fractions? When the fractional unit is the same, what is the sum of the fractions? What units may be expressed in the same collection? What is the rule for the addition of fractions? 173. When there are mixed numbers, how do you add? 174. When two fractions have 1 for a numerator, what is their sum equal to ? 175. What is subtraction of fractions? 1. What is the difference between § and ? : ANALYSIS.-In this example the fractional unit is there are 5 such units in the minuend and 3 in the subtrahend: their difference is 2 eighths; therefore, 2 is written over the common denominator 8. OPERATION. Ans. 6. What is the difference between § and } ? ANALYSIS.-Reduce both to the same fractional unit: then, there are 10 such units in the minuend and 4 in the subtrahend: hence, the difference is 6 twelfths. OPERATION. 1= £¬A=A=}. Ans.. From the above analysis we have the following RULE.-I. When the fractions have the same denominator, subtract the less numerator from the greater, and place the difference over the common denominator. II. When they have not the same denominator, reduce them lo a common denominator, and then subtract as before. 7. From of of take of 3 of 1. 10. What is the difference between 4 and 24? 176. Therefore: When there are mixed numbers, change both to improper fractions and subtract as in Art. 175; or, subtract the integral and fractional numbers separately, and write the results. NOTE.-Since we cannot take from we OPERATION. borrow 1, or, from the minuend, which added to; then from 7 leaves . We must now carry 1 to the next figure of the subtrahend and proceed as in subtraction of simple numbers. 74=729 43=47 Ans. 2 16. From 363 take 27 17. From 400 take 3275. NOTE.-When the numerators are 1, the difference of the two fractions is equal to the difference of the denominators divided by their product. OPERATION. and ? 19. What is the difference between MULTIPLICATION OF FRACTIONS. 177. MULTIPLICATION of Fractions is the operation of taking one number as many times as there are units in another, when one of the numbers is fractional, or when they are both fractional. 1. If one yard of cloth cost of a dollar, what will 4 yards cost? ANALYSIS. Four yards will cost 4 times as much as 1 yard; if 1 yard costs 5 eighths of a dollar, 4 yards will 5 OPERATION. cost 4 times 5 eighths of a dollar, which are 20 eighths; equal to 2 dollars. 176. When there are mixed numbers, how do you subtract? Explain the case when the fractional part of the subtrahend is the greater? 177. What is multiplication of fractions? To multiply a fraction by a whole number :-Multiply the numerator, or divide the denominator by the multiplier. 7. If 1 dollar will buy of a cord of wood, how much will 15 dollars buy? 8. At § of a dollar a pound, what will 12 pounds of tea cost? 9. If a horse cats of a bushel of oats in a day, how much will 18 horses eat? 10. What will 64 pounds of cheese cost, at of a dollar a pound? 11. If a man travel of a mile an hour, how far will he travel in 16 hours? 12. At of a cent a pound, what will 45 pounds of chalk cost? 13. If a man receive of a dollar for 1 day's labor, how much will he receive for 15 days? 14. If a family consume of a barrel of flour in 1 month, how much will they consume in 9 months? 15. If a person pays of a dollar a month for tobacco, how much does he pay in 18 months? 181. To multiply a whole number by a fraction. 1. At 15 dollars a ton, what will of a ton of hay cost? ANALYSIS.-1st. Four fifths of a ton will cost 4 times as much as 1 fifth of a ton; if 1 ton cost 15 dollars, 1 fifth will cost of 15 dollars, or 3 dollars, and will cost 4 times 3 dollars, which are 12 dollars. OPERATION. = (155) x 4 12. 180. How do you multiply a fraction by a whole number? |