The above examples are easily performed, as the quantities to be operated upon are like quantities, that is, have the same denominator. In such cases, we have only to add the numerators. When fractions of different denominators are to be added, they must first be reduced to fractions having a common denominator. ILL. EX. · Add and. 7.1+1+3=? Ans. 1. 14. A+ 15 +18=? 8. 70+8+ =? Ans. 24.15.1+8+1=? 18. 12+1+{+}}=? 12. 1+1+8=? 13. NOTE. T2 19+2+2+8=? 20+1+1+3=? Add the whole numbers and fractions of the following, and similar examplės, separately. 21. 3+2+7 = ? 22. 183+16+283=? 23. 272161+18§? † What operation should first be performed upon these fractions? NOTE. -The denominators in Example 7 being unlike, the fractions must be reduced to fractions having the same denominator. Ans. . 15. 8-32 ? Ans. 5. 18. 181-15=? 16. 74-2? Ans. 5. 19. 17-12? Ans. 43. 17. 10-518 ? NOTE. As = cannot be taken from, it will be necessary to reduce 1 of the 17 to halves, making the minuend 163, when subtraction can be zasily performed. 16 24. 19-21=? Ans. 163. 28. 18÷4-1of2=what? For Dictation Exercises, see Key. ADDITION AND SUBTRACTION OF FRACTIONS. 145. ADDITION AND SUBTRACTION OF FRACTIONS COMBINED. Give a rule for the addition of fractions; for subtraction. 87. Ans. 188. 12. A man receives 43 per cent. commission for selling goods; he pays per cent. for storage; what per cent. does he retain ? 13. If he receives 63 per cent. for selling goods, and 13 per cent. for insuring their sale, and pays 13 per cent. for storage, and per cent. for auctioneering; what per cent. does he retain? 14. How much will be left of a piece of cloth containing 7 yards, after cutting from it 2 vests and a coat, allowing yard for a vest and 44 yards for a coat? of a 15. Bought of Mrs. Frye 1 bonnet for $4.373, 2 hats at $2.12 apiece, 4 yards ribbon at $.163 per yard, 2 yards ribbon at 33 cents a yard, and gave in payment a ten dollar bill; what should she give me in return? 16. From 8 apple trees I gathered as follows: 2 barrels, 5 barrels, 5 barrels, 44 barrels, 3 barrels, 13 barrels, 34 barrels, and 2 barrels. I sold 15 barrels to one man, and 2 barrels to another, how many barrels had I left? 17.* To what must you add the difference between 8 and 36, that the amount may be 503? 18. If 7 X-2136 is the minuend, and the remainder, what is the subtrahend? 146* GREATEST COMMON DIVISOR OF FRACTIONS. * ILL. EX. Find the greatest common divisor of §, &, and §. OPERATION. Ans. G. C. D. of 6, 8, and 4 = 2 We find the G. C. D. of the L. C. M. of 7, 9, and 5 = 315' numerators 6, 8, and 4 to be 2. 2 is a divisor of 6, but must be divided by 7 to be a divisor of . It must also be divided by 9 to be a divisor of §, and by 5 to be a divisor of . To be at the same time a divisor of these fractions, it must therefore be divided by 7, and 9, and 5, or by their least common multiple. Hence the RULE. To find the G. C. D. of fractions: Reduce the frac tions to their lowest terms; then divide the G. C. D. of the numerators by the L. C. M. of the denominators. EXAMPLES. 1. Find the G. C. D. of 3, 3, and §. 2. Find the G. C. D. of, 1, and 23 or §. NOTE. -4 can be regarded as 4. 5. Find the G. C. D. of %, 2%, §, and 2. Ans. Ans. J' 6. What is the size of the largest cup which is an exact measчre of 11, 12, 81, and 2§ pints? 7. What is the width of the widest carpeting that will fit 4 rooms of the following widths: 131 feet, 21 feet, 311 feet, 361 feet? For Dictation Exercises, see Key. 147* LEAST COMMON MULTIPLE OF FRACTIONS. ILL. EX. Find the least common multiple of 1, 2, and §. OPERATION. L. C. M. of 1, 3, and 5 = 15 We find the L. C. M. of the numerators, 1, 3, and 5, to be G. C. D. of 2, 4, and 6 — Ans. 2 15. But we do not wish to ascertain the least number that * Articles 146 and 147 can be omitted by younger pupils will contain 1, 3, and 5, but one that will contain,, and . To contain each of these fractions separately, it might be divided by 2, by 4, or by 6; but to contain them at once, it can be divided only by their G. C. D. Hence the RULE. To find the L. C. M. of fractions: Reduce the fractions to their lowest terms, then divide the L. C. M. of the numerators by the G. C. D. of the denominators. 3. What is the width of the narrowest cloth that can be cu into strips either 2, 14, or 4 inches wide? 4. What will be the length of the shortest court that can be paved with stones of either of the following lengths, viz., 1 ft. 2 ft., 4 ft., or 23 ft.? Ans. 24 ft. 5. What must be the width of the narrowest court that will receive either of the same stones widthwise, their widths being 1 ft., 14 ft., 3 ft., and 2 ft. ? 6. On a stringed instrument in perfect tune, while C makes 1 vibration, D makes , E, F, G, A, B 15, and C' 2. If all are struck at once, in how many vibrations of C will they all again coincide? 7. In how many vibrations of C will C, E, G, and C' coincide? will C and D coincide? C and E? B and C'? C and C'? For Dictation Exercises, see Key. QUESTIONS FOR REVIEW. DEFINITIONS AND PROPERTIES OF NUMBERS.-What is the sign for plus? for minus? for greater than? less than? equal to? multiplied by? divided by? therefore? What does a parenthesis or vinculum signify? What are integral numbers? tional numbers? mixed numbers? What is a prime posite number? What are the factors of a number? factor? A composite number equals what product? bers prime to each other? What is a power of a number? the square or second power of a number? the fifth power? What is a root of a number? the square root? the cube root? the sixth root? What are fracnumber? a comWhat is a prime When are numWhat is |