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 e hours before. At what rate per hour should he have driven in order to reach the station just in time for the train? c= b Discuss the problem (1) when c = 0; (2) when c = ; (3) when b 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? 3. A wine merchant has two kinds of wine which he sells, one at a dollars, and the other at 6 dollars per gallon. He wishes to make a mixture of I gallons, that shall cost him on the average m dollars a gallon. How many gallons must he take of each? 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 3 x + 4y = 22, in positive integers. No other value of m gives positive integers for both x and y. Let 1+4y x=2+2y+ 1+4y x-2y-2= 1+4y 1+4 y must be integral. 5m-1 = m, then y = a fraction in form. To avoid this difficulty, it is necessary to make the coefficient of y 1+4y equal to unity. Since 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. is integral, any multiple of 1 + 4y is 5 (2) Since x = (11 + 14 y), from the original equation, Here it is obvious that m may have any positive value. If m = 1, If m = 2, If m = 3, and so on. x=14m- 9. x = 5, y = 1; x = 19, y = 6; x = 33, y = 11 ; 3. Solve 5x + 6 y = 30, so that x may be a multiple of y, 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 = 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? CHAPTER XIV. INEQUALITIES. 201. If a - b is positive, a is said to be greater than b; b is negative, a is said to be less than b. if a NOTE. Letters in this chapter are understood to stand for positive numbers, unless the contrary is expressly stated. 202. An Inequality is a statement in symbols that one of two numbers is greater than or less than the other. 203. The Sign of an Inequality is >, which always points toward the smaller number. Thus, a > b is read a is greater than b; c <dis read e is less than d. 204. The expressions that precede and follow the sign of an inequality are called, respectively, the first and second members of the inequality. 205. Two inequalities are said to subsist in the same sense if the signs of the inequalities point in the same direction; and two inequalities are said to be the reverse of each other if the signs point in opposite directions. Thus, ab and c>d subsist in the same sense, but a band c are the reverse of each other. d 206. If the signs of all the terms of an inequality are changed, the inequality is reversed. Thus, if a > b, then - a <- b. 207. If the members of an inequality are interchanged, the inequality is reversed. Thus, if a > b, then b<a. |