130. Halves and halves, thirds and thirds, &c., can be added, just as we add pears and pears, dollars and dollars. EXAMPLE.-What is the sum of 5 halves and 4 halves? Answer, 9 halves. Here the denominators are the same, and we simply add the numerators, and place the sum over the common denominator. 131. Halves and thirds, halves and fourths, &c., can not be thus directly added, any more than we can add pears and dollars. They are things of different kinds. 130. Can halves and halves, thirds and thirds, &c., be added? In what way? Give an example.-131. Can halves and thirds, halves and fourths, &c., be added directly? Why not? ADDITION OF FRACTIONS. EXAMPLE. What is the sum of 5 thirds and 3 halves? 79 The parts being of different value, we can not put them together, and say they make 8 halves or 8 thirds. But, if we reduce them to parts of the same kind or value, we can then add them. By reducing to a common denominator, we find that 5 thirds are equal to 10 sixths, and 3 halves to 9 sixths. 10 sixths and 9 sixths make 94 1 9 6 fraction, we reduce it to 3%. Answer, 3. 10 6 ee 19 v 6 31 Ans. 132. RULE.—1. To add fractions, when they have a common denominator, add their numerators, and place the sum over the common denominator. When they have not a common denominator, reduce them to fractions that have, and then proceed as above. If the resulting fraction is not in its lowest terms, reduce it. If it is an improper fraction, reduce it to a whole or mixed number. 2. To add mixed numbers, or fractions and whole numbers together, find the sum of the fractions separately, and add it to the sum of the whole numbers. EXAMPLE.-Add together 3, 4, 3, and 5. Show how we add and .-132. Give the rule for adding fractions. Give the rule for adding mixed numbers, or fractions and whole numbers. Arply this latter rule in the given example. 11. A hackman earns $2 one day, $31 the next, $4 the next, and $5 the next. How much does he earn in all four days? Ans. $15. 12. How much land is there in 3 fields, containing 141, 7%, and 233 acres? Ans. 45 acres. 1 13. If I buy $23 worth of paper, and $61 worth of books, and give the storekeeper a ten-dollar bill, how much change will I receive? Ans. $1. 14. Three men, buying a meadow, put in respectively $30, $25, and $19,7%. What does the meadow cost? 15. A five-pound jar contains 33 pounds of bread and 24 pounds of cake; what does the whole weigh? 16. A peddler walks 8 miles one day, 51 the next, 101 the next, and 12 the next; how far does he walk in all? 17. How many pecks of peaches in four baskets, containing respectively 21, 31, 21, and 3} pecks? 10 18. If 5 gallons of brandy are mixed with 1 gallons of water, how many gallons are there of the mixture? 5 19. A lady hires a gardener for 15 cents an hour. How much must she pay him, if he works 6 hours the first day, 7 the second, and 5g the third? 1 2 Ans. 300 cents. SUBTRACTION OF FRACTIONS. 81 Subtraction of Fractions. 133. The same principle applies in subtracting, as in adding, fractions. Before subtracting, the parts must be made of the same kind or value, if they are not already so; that is, the denominators must be made the same. EXAMPLES.-1. From 5 halves take 4 halves. 2. From 5 thirds take 3 halves. 2 9 C Thirds and halves being parts of different value, we must reduce them to parts of the same value. Reducing to a common denominator, we find that 5 thirds are equal to 10 sixths, and 3 halves to 9 sixths. 9 sixths from 10 sixths leave 1 sixth. Answer, . Ans. 1 134. RULE.-1. To subtract one fraction from another, when they have a common denominator, take the numerator of the subtrahend from that of the minuend, and place the remainder over the common denominator. When they have not a common denominator, reduce them to fractions that have, and then proceed as above. Reduce the resulting fraction to its lowest terms, or to a whole or mixed number, as may be necessary. 2. Whole and mixed numbers may be reduced to improper fractions, before subtracting. 133. What principle applies in subtracting fractions? Illustrate this, with the examples given.-134. Recite the rule. What may be done with whole and mixed numbers, before subtracting? Ans.. 17. From 1 subtract 3. [Reduce 1 to thirds; then proceed as before.] From 1 take 3 20 18. From 1 take 17 40 [Take 1 of the 4 units, and reduce it to thirds. Then subtract the fraction. from leaves. Ans. 3}. 20. From 5 subtract . [54] Ans. 3. 4 = 33 Min. Sub. 31 Rem.] Ans. 44. Ans. 165. 22. Take from 3. Take from 3. Take from 2. 10 1 [Take from 3, and annex the result to 5.] 1. 24. Subtract from 21. From 22. Take 25. Subtract from 67. 8 26. Subtract 21 from 4%. Ans. 512. from 63. Ans. 61. Ans. 138. 30 [is greater than . Hence reduce both mixed numbers to improper fractions. The sum then becomes, Subtract from 25. Now proceed as before.] 27. Subtract 13 from 375. 28. Subtract 201 from 241. 29. Subtract from 12. 100 30. Subtract 3 from 13. |