3. I exchanged apples at $1.50 per bushel for 25 days' labor at $1.20 per day. How many bushels of apples did it take? 4. Three pieces of cloth containing 20 yards each, worth $5 a yard, were exchanged for 5 pieces of cloth containing 40 yards each. What was the second kind of cloth per yard ? 5. How many pounds of coffee at 24 cents per pound are required to pay for 3 hogsheads of sugar, each weighing 1464 pounds, and worth 15 cents per pound? 6. Four farms containing 80 acres each, worth $65 per acre, were exchanged for 5 farms containing 95 acres each. What was the value per acre of the farms received in exchange ? 7. How many firkins of butter, each containing 50 pounds, at 18 cents a pound, must be given for 3 barrels of sugar, each containing 200 pounds, at 9 cents a pound? 8. If 25 Jersey cows each give 8 quarts of milk a day, at 5 cents a quart, how many pieces of matting of 40 yards each, at 50 cents a yard, will pay for the milk of 12 days ? 9. A tailor bought 5 pieces of cloth, each piece containing 5 24 yards, at 3 dollars a yard. How many suits of clothes, at 18 dollars a suit, must be made from the cloth to pay for it? 10. A grocer bought 7 chests of Souchong tea, containing 24 pounds each, at $1.05 per pound. How many firkins of butter, at $.35 a pound, will be required to pay for the tea, each firkin containing 56 pounds ? 11. I bought 24 barrels of apples, each containing 2 bushels, at the rate of 75 cents a bushel. Find the number of cheeses, each weighing 30 pounds, at 15 cents a pound, that will pay for the apples. 12. How many days' work, at $1.80 a day, will pay for 84 bushels of corn, at $.45 a bushel ? 13. If 52 men can dig a trench in 15 days, working 10 hours a day, in how many days will 25 men dig a similar trench, working 12 hours a day? COMMON DIVISORS. a } INDUCTIVE STEPS. 2. Two, three, and four are in this case called Common factors or Common divisors. 6 2 X 3 3. Numbers 8 2 X 2 X 2 prime factors. 20 2 X 2 X 5 What single prime factor is common to all the numbers ? What, then, is the common divisor of 6, 8, and 20? What prime factor is common to 8 and 20 only ? Is it the same 2 that is common to all the numbers, or is it a different 2? Is there a 2 that is not common to any two of the numbers ? What two other factors are not common to any two of the numbers ? 15 3 x 5 4. Numbers 30 2 X 3 X 5 prime factors. 45 3 X 3 X 5 What two prime factors are common to all the numbers ? The numbers have what two common divisors ? Is the product of 3 and 5 a common divisor ? State the principle. Is 15 the greatest common divisor of 15, 30 and 45? Why? a DEFINITIONS. 1. A Common Divisor of two or more numbers is a number that exactly divides each of them. 2. The Greatest Common Divisor (G. C. D.) of two or more numbers is the greatest number that exactly divides each of them. 3. Numbers that have no Common Divisor are said to be prime to each other. PRINCIPLE. The G. C. D. of two or more numbers is the product of all their common factors. EXERCISES. 1. What is the G. C. D. of 42, 56 and 70 ? Process. Explanation. 42 2 X 3 X 7 Resolving the given numbers into 56 = 2 X 2 X 2 X 7 their prime factors, we have 42 = 2 X 3 X 7, 56 = 2 X 2 X 2 X 7, 70 2 X 5 70 = 2 X 5 X 7 X 7. By inspecting these prime factors, 2 x 7 = 14, G. C. D. we find that 2 and 7 are the only prime factors common to all the numbers. Hence their product, 14, is the G. C. D. of 42, 56 and 70. 2. Find the G..C. D. of: 1. 21, 35, 56. 5. 6, 12, 30. 2. 12, 18, 24. 6. 15, 25, 30. 7. 12, 18, 72. 8. 105, 35, 70. An abridgment of the above method is as follows: Process. Explanation. 5/105, 35, 70 Dividing by the common prime 721, 7, 14 factors 5 and 7, the quotients 3, 1, 2 are seen to be prime to one another. 3, 1, 2 Hence 5 and 7 are all the factors comG. C.D.= 5 x 7= 35. 5 mon to all the numbers, and 5 x 7 or 35 is the G. C. D. 5. 16, 48, 80. 9. 51, 105, 243. 17. 180, 300, 900. 20. 17, 27, 36. 16. 14, 98, 112. 24. 1326, 3044, 4520. Nos. 22, 23 and 24 may be reserved and factored by the next process. When the numbers are not readily factored, a method founded on principle 6, page 72, is adopted. 1. What is the G. C. D. of 169 and 195? Process. Explanation. 169 ) 195 (1 By Principle 6, a common divisor of 169 169 and 195, divides the remainder, 26, When it is required to find the G. C. D. of more than two numbers, first find the G. C. D. of two of them, then of that G. C. D., and one of the remaining numbers, and so on for all the numbers. The last G. C. D. will be the G. C. D. of all the numbers. 3. Find the G. C. D. of: 1. 492, 744, 1044. and consequently 156 and the remainder 26 ) 169 ( 6 13. Since 13 exactly divides itself and 156 26, it is a common divisor of 169 and 195. G. C. D. = 13) 26 ( 2 The G. C. D must also divide 26 and 26 13, and since it must divide 13 it cannot exceed 13. Therefore 13 is the G. C. D. 0 of 169 and 195. 2. Find the G. C. D. of: 1. 187 and 209. 11. 1215 and 1878. 18. 901 and 1060. 6. 121, 181, 221, 241. 8. 630, 1134, 1386. PROBLEMS. What is the length of the longest chain that will measure exactly the length and the width of a field 484 rods long and 120 rods wide ? 2. Three fields containing 24 acres, 18 acres and 42 acres are to be cut each into the least number of smaller fields of equal size. Find the size of the fields. 3. Two vats contain respectively 7875 and 16,128 gallons. Find the cask of greatest capacity that will exactly measure both vats. 4. What is the length of the longest pole with which you can measure the three lengths, 132, 156, and 168 ? 5. In a village some of the walks are 56 inches wide, some 70 inches, and others 84 inches. What is the width of the widest flagging that will suit all the walks? 6. What is the greatest length of board that can be used without cutting in fencing a triangular field whose sides are 80, 112 and 144 feet? 7. The Erie Railroad has 3 side-tracks of the following lengths : 3013, 2231, and 2047 feet. Find the length of the longest rail that will exactly lay each side-track. |