17.* To what must you add the difference between 8 and 36, that the amount may be 50% ? 18. If 7§ X-23 is the minuend, and the remainder, what is the subtrahend? 40 146* ILL. EX. GREATEST COMMON DIVISOR OF FRACTIONS. * OPERATION. Ans. We find the G. C. D. of the numerators 6, 8, and 4 to be G. C. D. of 6, 8, and 4 = 2 L. C. M. of 7, 9, and 5 = 315' 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 8, 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 numer ators 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 24 or . 3. Find the G. C. D. of 34, 76, %, and 1g. 4. Find the G. C. D. of 4, §, and 4. NOTE. 5. Find the G. C. D. of 27, 2, §, and 2. 6. What is the size of the largest cup which is an exact measчre of 12, 1, 81, and 21⁄2 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? 4 can be regarded as 4. Ans. Ans. 3. For Dictation Exercises, see Key. 147* LEAST COMMON MULTIPLE OF FRACTIONS.* ILL. Ex. Find the least common multiple of 1, 2, and §. We find the L. C. M. of the numerators, 1, 3, and 5, to be 15. But we do not wish to ascertain the least number that OPERATION. L. C. M. of 1, 3, and 5 = 15 Ans. * 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. EXAMPLES. 1. Find the L. C. M. of, 13, and 73. 13 2. Find the L. C. M. of of 31, and 6. 21 Ans. 4763. 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., 11⁄2 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., 1 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 2, 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? What are fractional numbers? mixed numbers? What is a prime number? a composite number? What are the factors of a number? What is a prime factor? A composite number equals what product? When are numbers prime to each other? What is a power of a number? What is 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 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. C. 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 7. 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 fractions 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 do you reduce compound fractions to simple ones? Can you give one general rule for multiplying fractions, whole or mixed numbers, by fractions? DIVISION OF FRACTIONS. How do you divide a fraction by a whole number? a mixed number by a whole number? a whole number by a fraction? Explain, by an example, the method of dividing a fraction by a fraction, and give the rule. Give one general rule for dividing a fraction, a whole or mixed number by a fraction. How do you reduce complex fractions to simple ones? How do you find what part of one number another is ? LEAST COMMON MULTIPLE. - Define a multiple; a common multiple of two or more numbers; the least common multiple. When do you make use of the L. C. M.? Give and explain the first method of finding it; the second. What does the L. C. M. of prime numbers equal? COMMON DENOMINATOR. When are fractions said to have a common denominator? In what operations upon fractions do we first reduce them to those having the same denominator ? Can we change fractions to those of any denominator? How? (Ans. By dividing or multiplying the numerator by the same number by which we divide or multiply the denominator to produce the denominator required.) What denominator is generally chosen? Reduce a simple, a compound, and a complex fraction to those of the same denominator, explain the process, and give the rule. ADDITION AND SUBTRACTION OF FRACTIONS.-How do you add fractions of different denominators ? How do you subtract one fraction from another? How do you add mixed numbers? In subtraction of one mixed number from another, how do you proceed when the fraction in the subtrahend exceeds that in the minuend? 148. MISCELLANEOUS EXAMPLES. Into strips of what widths may I cut cloth which is 36 inches wide, that none may be wasted, the width of the strips to be expressed in inches? *Optional. 2. How many gallons in the largest vessel which will exactly measure 3 hogsheads, containing severally 128, 94, and 158 gallons? 3. What will 16 yards of cloth cost at $.53 a yard? 4. What cost 93 bushels of corn, at $.87 a bushel ? 5. What cost 271 acres of land, at $313 per acre? 6 of 2 of 56 times what number equals 18? 7. I paid $.65 for 2 boxes of strawberries; what will be the cost of 45 boxes at the same rate? 8. What is my bill for 7 pear trees, $ apiece for the trees, and $2 a dozen for setting? 9. What do I receive per pound by selling 15 pounds of coffee for $31? 10. of a man's property is in land, and is valued at $2324§; what is the value of his whole property? 11. Bought of an acre of land for $40.75; what would 1 acre cost at the same rate? 12. What costs 3 pieces of calico, 37 yards in a piece, at 194 cents per yard? 13. If 32 acres of land cost $1100, what costs 1 acre? 14. Sold my house and farm of 47% acres for $6150; allowing $3500 for the house, what did I receive per acre for the land? 15. How long will a barrel of flour last a family of 8 persons, if it lasts 3 persons 4 months? 16* What number is that from which if you take, the remainder will be ? 17. What number is that to which if you add 97, the sum will be 124g? 18. What is that number to which if you add of 261, the sum will be 147? yards of 19. Bought 7 yards broadcloth at $5 per yard, 14 kerseymere at $14 per yard, 43 yards of silk at $ per yard, and yards of doeskin at $44 per yard, for which I gave in payment a $100 bill. What balance is due me? 20. I have paving stones 12 inches long and 10 inches wide; what must be the width of a walk which will just receive these stones, laid either lengthwise or widthwise? 21* What is the smallest sum of money which can be exactly paid in pieces of money worth either $.163 or $.121? |