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Linear Equations and Functions


Date: 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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