NOTE. - The denominators in Example 7 being unlike, the fractions must be reduced to fractions having the same denominator. NOTE. - Subtract without changing the mixed numbers to improper fractions. 15. 8-38? Ans. 5. 18. 181-15=? 16. 74-2? Ans. 5. 19. 17-12? Ans. 43. 17. 103-518? NOTE. As cannot be taken from, it will be necessary to reduce 1 of the 17 to halves, making the minuend 168, when subtraction can be sasily performed. 24. 19-21: = ? Ans. 163. 28. 183÷÷÷-1}of 2=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. 13. Ans. 1893. Ans. 613 Ans. 410. 12. A man receives 4 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 1 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 of a yard for a vest and 44 yards for a coat? 15. Bought of Mrs. Frye 1 bonnet for $4.37, 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, 32 barrels, 12 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 8g and 36, that the amount may be 50% ? 18 If 7 X-236 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 . We find the G. C. D. of the numerators 6, 8, and 4 to be 1 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 fractions 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, 2, and §. 2. Find the G. C. D. of,, and 2 or §. 3. Find the G. C. D. of 34, 76, §, and 1g. 4. Find the G. C. D. of 4, §, and 4. Ans. 12. Ans. 3. 6. What is the size of the largest cup which is an exact measure of 1, 1, 8, and 2 pints? 7. What is the width of the widest carpeting that will fit 4 rooms of the following widths: 13 feet, 21 feet, 311 feet, 362 feet? For Dictation Exercises, see Key. 147 LEAST COMMON MULTIPLE OF FRACTIONS. OPERATION. We find the L. C. M. of the numerators, 1, 3, and 5, to be ILL. EX. L. C. M. of 1, 3, and 5 = 15 Ans. 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 frac tions to their lowest terms, then divide the L. C. M. of the numer. ators by the G. C. D. of the denominators. EXAMPLES. 1. Find the L. C. M. of, 13, and 7. 2. Find the L. C. M. of 2 of 31, and 6. Ans. 4763. Ans. 390 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 g, 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? a root of a number? the square root? the cube root? the sixth root? What are frac number? a com What is a prime When are num、 What is What is What is the sign for a power? for a root? What indicates the degree of root? What is an even number? an odd? DIVISIBILITY OF NUMBERS. When are numbers divisible by 2? by 3 by 4 by 5? by 6? by 8? by 9? by 10? by 11? by any composite number? How shall we ascertain whether any given number is prime? Describe Eratosthenes' sieve? FACTORING OF NUMBERS. What is the simplest way of resolving numbers into their prime factors? What other method can you describe, and when would you use it? Find the factors of 180 by first method, and explain the process. Find the factors of 10296 by second method, and explain the process. GREATEST COMMON DIVISOR.-What is a divisor of a number? a common divisor of two or more numbers? the greatest common divisor? Find the G. D. of three numbers by the first method given. Explain and give the rule. Find the G. C. D. of three numbers by second method. Explain and give the rule. In what cases is the second method the better? When is it necessary to find the G. C. D. of numbers? FRACTIONS. What is a fraction? Name and describe its terms. Name the different kinds of fractions of which you have learned. Define a common fraction; a decimal fraction; a proper fraction; an improper fraction; a mixed number; a compound fraction; a complex fraction. Give an example of each. Explain the expression . Upon what does the value of a fraction depend? Which of the fundamental rules is indicated by a fraction ? What effect does multiplying the numerator of a fraction have upon that fraction? Why? In what other way could you produce the same effect, and why? What effect does dividing the numerator have upon a fraction? Why? In what other way could you produce the same effect, and why? What effect does multiplying both terms of a fraction by the same number have upon it? Why? What effect does dividing both terms of a fraction have upon it? Why? REDUCTION OF FRACTIONS.— How do you reduce fractions to lower terms? What is cancellation? How do you reduce whole or mixed numbers to improper fractions? How do you reduce improper frac tions to whole or mixed numbers? MULTIPLICATION OF FRACTIONS. How do you multiply a fraction by a whole number? a mixed number by a whole number? Explain, by an example, the method of multiplying a whole number by a fraction. Multiply a fraction by a fraction; explain and give the rule. How do you multiply a mixed number by a mixed number or a fraction? How |