Linear Equations and FunctionsDate: 10/11/97 at 21:49:35 From: Hillary Hartman Subject: Linear Equations and Functions Hi, I'm in 11th grade doing Algebra II. Here's the question. The digits of a positive two-digit integer N are interchanged to form an integer K. Find all possibilities for N under the conditions described. The average of N, K, and 35 is 30. I tried to figure it out but my class never did this and I really don't know where to start. Thanks, Hillary Date: 10/13/97 at 09:15:08 From: Doctor Chita Subject: Re: Linear Equations and Functions Hi Hillary: What an interesting problem! Let's see if I can give you a little push. First, read the problem carefully and be sure that you recognize the nature of the numbers you are looking for. N is a positive two-digit number. If you switch its digits, then K will also be a positive two- digit number. This narrows the search a little. You don't have to worry about negative numbers. Second, you need to know how to represent a two-digit number. The digits are the numbers {0, 1, 2, 3, 4, 5, 6, 7, 8, and 9}. All numbers, two-digit, three-digit, four-digit, etc., are combinations of these 10 digits. Consider the number 85, for example. The 8 is in the tens place, and the 5 is in the ones place. Therefore, you can write 85 as 8*10 + 5*1. Every two-digit number, then, consists of a digit multiplied by 10 and a digit multiplied by 1. To represent any two-digit number algebraically, you have to designate two variables to represent the number of 10s and the number of 1s. Let t and u be those variables. Now you can represent the number N in the following way: (1) N = 10*t + u If you switch the two digits in the number (say from 85 to 58), then K is the following: (2) K = 10*u + t The problem now is that you have two equations, but four variables: t, u, N, and K. So you have to use the last piece of information in the problem about the average. Write an equation representing the average as stated in the problem: (N + K + 35)/3 = 30 Simplify the equation by combining the numbers on the right side: (3) N + K = 55. Okay, now you have three equations. Look at them carefully. Can you see a way to put them together somehow? How about combining the first two equations to get another way to express N + K? (4) N + K = 10t + u + 10u + t = 11t + 11u Substitute equation (4) into equation (3) and simplify the result. Now, you're almost there. Look at the resulting equation and the answer should be obvious. I won't spoil it for you, but if you're still stuck, let us know. By the way, I've outlined an algebraic solution. However, you could always use trial-and-error, too. Use equation (3) and see what two positive integers have a sum of 55. Which two are mirror images, like 85 and 58? This isn't an elegant solution, but it works. Good luck! -Doctor Chita, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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