198. The discussion of a problem consists in making various suppositions as to the relative values of the given numbers, and explaining the results. We will illustrate by the following example: Two couriers are traveling along the same road, in the same direction. A travels m miles an hour, and B travels n miles an hour. At 12 o'clock B is d miles in advance of A. When will the couriers be together? Suppose they will be together x hours after 12. Then A has trav eled mx miles, and B has travelec na miles, and as A has traveled d miles more than B Discussion. 1. If m is greater than n, the value of x is positive, and A will overtake B after 12 o'clock. 2. If m is less than n, the value of x is negative. In this case B travels faster than A, and as he is d miles ahead of A at 12 o'clock, A cannot overtake B after 12 o'clock, but B passed A before 12 o'clock. The supposition, therefore, that the couriers are together after 12 o'clock is incorrect, and the negative value of x points to an error in the supposition. 3. If m equais n, then the value of x assumes the form d. Now, if the couriers are d miles apart at 12 o'clock, and if they travel at the same rates, it is obvious that they never will be together, so that the d symbol may be regarded as the symbol of impossibility. d m- n becomes Now, if the 4. If m equals n and d is 0, then couriers are together at 12 o'clock, and if they travel at the same rates, it is obvious that they will be together all the time, so that x may have an indefinite number of values. Hence, the symbol may be regarded as the symbol of indetermination. EXERCISE 78. 1. A train traveling 6 miles per hour is m hours in advance of a second train that travels a miles per hour. In how many hours will the second train overtake the first? Discuss the problem (1) when a is greater than b; (2) when a is equal to b; (3) when a is less than b. 2. A man setting out on a journey drove at the rate of a miles an hour to the nearest railway station, distant b miles from his house. On arriving at the station he found that the train left c hours before. At what rate per hour should he have driven in order to reach the station just in time for the train? ; (3) when Discuss the problem (1) when c = 0; (2) when c = b c = a In case (2), how many hours did the man have to drive from his house to the station? In case (3), what is the meaning of the negative value of c? He 3. A wine merchant has two kinds of wine which he sells, one at a dollars, and the other at b dollars per gallon. wishes to make a mixture of 7 gallons, that shall cost him on the average m dollars a gallon. must he take of each? How many gallons Discuss the problem (1) when a = b; (2) when a or b=m; (3) when a = b=m; (4) when a is greater than b and less than m; (5) when a is greater than b and b is greater than m. CHAPTER XIII. SIMPLE INDETERMINATE EQUATIONS. 199. If a single equation is given with two unknown numbers, and no other condition is imposed, the number of its solutions is unlimited; for, if any value is assigned to one of the unknown numbers a corresponding value may be found for the other. An equation that has an indefinite number of solutions is said to be indeterminate. 200. The values of the unknown numbers in an indeterminate equation are dependent upon each other; so that they are confined to a particular range. This range may be still further limited by requiring these values to satisfy some given condition; as, for instance, that they shall be positive integers. 1. Solve 3x+4y= 22, in positive integers. No other value of m gives positive integers for both x and y. x = 6 + 4 m. (2) then y = 1 and x = 6. then 4 and x = 2. To avoid this difficulty, it is necessary to make the coefficient of y is equal to unity. Since 1+ 4y 1+ 4y is integral, any multiple of 5 integral. Multiply the numerator of the fraction, then, by a number that will make the division of the coefficient of y give a remainder of 1. In this case, multiply by 4. Here it is obvious that m may have any positive value. (1) (2) If m = = 3, and so on. 3. Solve 5x+6y= 30, so that x may be a multiple of y, and both positive. 15. A man spent $114 in buying calves at $5 apiece, and pigs at $3 apiece. How many did he buy of each? 16. In how many ways can a man pay a debt of $87 with five-dollar bills and two-dollar bills? 17. Find the smallest number that, when divided by 5 or when divided by 7, gives 4 for a remainder. Let n the number, then n 4 n 4 = x, and = y. 18. A farmer sold 15 calves, 14 lambs, and 13 pigs for $200. Some days after, at the same price for each kind, he sold 7 calves, 11 lambs, and 16 pigs, and received $141. What was the price of each? First eliminate one of the unknowns from the two equations. 19. A number is expressed by three digits. The sum of the digits is 20. If 16 is subtracted from the number and the remainder divided by 2, the digits will be reversed. Find the number. 20. In how many ways may 100 be divided into two parts, one of which shall be a multiple of 7 and the other a multiple of 9? |