Then the number of minute-divisions passed over by the hour-hand is equal to the number of minute-divisions passed over by the minute-hand, minus 5 n; that is, to x − 5 n. But since the minute-hand moves 12 times as fast as the hour-hand, we have Etc. If n = 11, it was 660 or 60 minutes past 11; i.e., 12 o'clock. 119 If n = 12, it was 120, or 655 minutes past 12; i.e., 5 minutes past 1. Notice that the two hands coincide at 12 o'clock, but not between 12 and 1. EXERCISES II. Find the general solution of each of the following problems, and from this solution obtain the particular solution for the numerical values assigned to the literal numbers in the problem. 1. Find a number, such that the result of adding it to n shall be equal to n times the number. Let n = 2; 5. 1 2. Divide a into two parts, such that of the first, plus of the Let a = 100, b = 30, m = = 3, n = = 5. second, shall be equal to b. 3. Find a number, such that the sum of the results of subtracting it from a and from b shall be equal to c. Let a 3, b = 6, c = 5. = 4. One boy said to another: "Think of some number, add 3 to it, multiply the sum by 2, add 4 to the product, divide the result by 2, subtract 1 from the quotient, multiply the difference by 4, add 4 to the product, divide the result by 4, tell me the result, and I will tell you the number you have in mind." If the result be d, what number does the boy think of? 5. A sum of d dollars is divided between A and B. B receives b dollars as often as A receives a dollars. How much does each receive? Let d = 7000, a = 3, b = 2. 20, 6. A father's age exceeds his son's age by m years, and the sum of their ages is n times the son's age. What are their ages? Let m = n = 4; m = 25, n = = 7. 7. If two trains start together and run in the same direction, one at the rate of my miles an hour, and the other at the rate of m2 miles an hour, after how many hours will they be d miles apart? Let d = 200, m1 = = 30. m2 = = 35, 8. A farmer can plow a field in a days, and his son in b days; in how many days can they plow the field, working together? Let a=10, b=15. 9. A pupil was told to add m to a certain number, and to divide the sum by n. But he misunderstood the problem, and subtracted n from the number and multiplied the remainder by m. Nevertheless he obtained the correct result. What was the number? Let m = = 12, n = 13. 10. What time is it, if the number of hours which have elapsed since noon is m times the number of hours to midnight? Let m = }. 11. A starts from P and walks to Q, a distance of d miles. At the same time B starts from Q and walks to P. If A walk at the rate of m miles a day and B at the rate of n miles a day, at what distance from P do they meet, and how many days after they start? Let m = 20, n = 30, d = 600. 12. Two friends, A and B, each intending to visit the other, start from their houses at the same time. A could reach B's house in m minutes, and B could reach A's house in n minutes. After how many minutes do they meet? Let m = 121, n = 10. 13. Two couriers start at the same time and move in the same direction, the first from a place d miles ahead of the second. The first courier travels at the rate of my miles an hour, and the second at the rate of m2 miles an hour. After how many hours will the second courier overtake the first? Let d = 15, m1 = 17, m2 = 20. From the result of the preceding example find the results of Exx. 14–16. 14. At what rate must the second courier travel in order to overtake the first after h hours? Let d = 18, mi 15, h = 3. = 15. At what rate must the first courier travel in order that the second may overtake him after h hours? Let d = 12, m2 = 22, h = 3. 16. How many miles behind the first courier must the second start in order to overtake the first after h hours? Let m1 = 18, m2 = 21, h = 4. 17. In a company are a men and b women; and to every m unmarried men there are n unmarried women. How many married couples are in the company? Let a = 13, b = 17, m = 3, n = = 5. 18. The annual dues of a certain club are at first a dollars. Subsequently the yearly expenses increased by d dollars, while the number of members decreased by n. In consequence the annual dues were increased by b dollars. How many members were originally in the club? Let a = 25, d = 315, n = 7, and b = 2. 19. A father divided his property equally among his sons. To the oldest he gave d dollars and 1 of what remained; to the second son he gave 2 d dollars and 1 of what was then left; to the third son he gave 3 d 1 n n dollars and of the remainder; and so on. What was the amount of his 20. Two couriers start from the same place and move in the same direction, one h hours after the other. The first one travels at the rate of m1 miles an hour, and the second at the rate of m2 miles an hour. After how many hours will the second courier overtake the first? Let h = 2, m1 = 15, m2 = : 20. From the result of the preceding example, find the results of Exx. 21-23. 21. At what rate must the second courier travel in order to overtake the first after H hours? Let H=6, h = 2, m1 = 12. 22. At what rate must the first courier travel in order that the second may overtake him after H hours? Let H = 4, h = 1, m2 = 20. 23. How many hours after the first courier starts must the second start in order to overtake the first after H hours? Let H 6, m1 = 14, m2 = 22. = 24. Two boys run a race from A to B, a distance of d yards. The first runs a yards a second; after reaching B, he turns and runs back at the same rate to meet the other boy, who runs b yards a second. How many seconds after they start does the faster runner meet the other? Let d = 253, a = 2.5, b = 2.1. 25. An accommodation train leaves A every h hours, and runs to B at the rate of m miles an hour. At the same time an express train leaves B and runs to A at the rate of n miles an hour. What time elapses after an express train meets an accommodation train until it meets the next accommodation train? Let h=3, m = 20, n = €40. 26. At what time between n and n + 1 o'clock will the hands of a clock be in a straight line? Let n = 1; 2; 3; to 12. 27. At what time between n and n + 1 o'clock are the minute-hand and the hour-hand of a clock at right angles to each other? Let n = 1; 2; 3;. to 12. 28. At what time between n and n + 1 o'clock does the second-hand bisect the angle between the hour-hand and the minute-hand? Let n = 1; 2; 3;. to 12. ... CHAPTER XII. INTERPRETATION OF THE SOLUTIONS OF PROBLEMS. 1. In solving equations we do not concern ourselves with the meaning of the results. When, however, an equation has arisen in connection with a problem, the interpretation of the result becomes important. In this chapter we shall interpret the solutions of some linear equations in connection with the problems from which they arise. Positive Solutions. 2. Pr. A company of 20 people, men and women, proposed to arrange a fair for the benefit of a poor family. Each man contributed $3, and each woman $1. If $55 were contributed, how many men and how many women were in the company? Let x stand for the number of men; then the number of women was 20x. The amount contributed by the men was 3 x dollars, that by the women 20. x dollars. By the condition of the problem, we have The result, 171, satisfies the equation, but not the problem. For the number of men must be an integer. This implied condition could not be introduced into the equation. The conditions stated in the problem are impossible, since they are inconsistent with the implied condition. If the problem be generalized, its solution will show how the given ́ data can be modified so that all the conditions, expressed and implied, shall be consistent. The generalized problem may be stated thus: A company of m people, men and women, proposed to arrange a fair for the benefit of a poor family. Each man contributed a dollars, and each woman b dollars. If n dollars were contributed, how many men and how many women were in the company? The solution of the equation of this problem is In order that x may be an integer, n - bm must be exactly divisible by a b. Thus, if, in the given problem, the number of people were 21 instead of 20, the other data being the same, we should have If all the conditions of a problem, expressed and implied, be consistent, a positive solution will satisfy these conditions and therefore give the solution of the problem. Negative Solutions. 3. Pr. A father is 40 years old, and his son 10 years old. After how many years will the father be seven times as old as his son ? Let x stand for the required number of years. Then after x years the father will be 40+x years old, and the son 10+ years old. By the condition of the problem, we have 40+x=7(10+x), whence x = - 5. (1) This result satisfies the equation, but not the condition of the problem. For since the question of the problem is "after how many years?" the result, if added to the number of years in the ages of father and son, should increase them, and therefore be positive. Consequently, at no time in the future will the father be seven times as old as his son. But since to add 5 is equivalent to subtracting 5, we conclude that the question of the problem should have been, "How many years ago?" The equation of the problem, with this modified question, is: 40 − x = 7 (10 - x); whence x = = 5. (2) Notice that equation (2) could have been obtained from equation (1) by changing x into X. 4. The interpretation of a negative result in a given problem is often facilitated by the following principle: If ·x be substituted for x in an equation which has a negative root, the resulting equation will have a positive root of the same absolute value; and vice versa. E.g., the equation x + 1 =-x 3 has the root while the equation x+1=x -3 has the root 2. In general, the equation ax = b has the root And the equation b α ax b α a α is positive; and vice versa. |