The Sum of Two Numbers is 20...

Date: 11/03/97 at 17:40:34
From: Beth Hurley
Subject: Math

The sum of two numbers is 20. Twice one number is 4 more than four 
times the other. Find the numbers.

I tried at school with the smartest girl in the grade, then at home, 
and I still don't get it!

Date: 11/06/97 at 10:49:54
From: Doctor Allan
Subject: Re: Math

Hi Beth,

This is a difficult question and I am not sure that you have the
mathematical background needed to solve the problem, but I'll show you 
one way you could think about it, and then we will find out if that 
helps you understand. If you don't, please tell me and I'll find 
another way of explaining.

We are told that we have two numbers. Let us call these numbers 
x and y. Then we are told that the sum of the two numbers is 20. 
That means that

  x + y = 20

If one of the numbers is x, then twice that number must be 2x. And if 
the second number is y, then 4 times that number must be 4y. Okay?

Well - we are told that twice one number is 4 more than four times the 
other number. But what does that mean? It means that when we take the 
bigger number and subtract the smaller number we will get the number 4 
(it's just like 9 is 4 more than 5 because 9 - 5 = 4). So we can write 
this in the following way:

  2x - 4y = 4

What we just did was to translate the word problem into math. Now we 
have to do the math part.

We know that x + y = 20, which means that x = 20 - y 

This means that every time you see x you can replace it with 20 - y. 
Do you understand this part? If you do, then look at the second 
equation. It says:

  2x - 4y = 4

But here you see x, so you replace it with 20 - y, and therefore you 
get

  2(20 - y) - 4y = 4

Now you have an equation with only one variable, and I assume you know 
how to solve this. I will leave the calculations to you.

What you need to do after you find the value of y is exactly the same
thing. Every time you see y, you can substitute for it the value of y, 
which you just found. Looking at the first equation you can see that 
you can find the value for x because you know the value of y.

Now try to do this part yourself.

Once you have found the values for x and y we can try to translate 
your results back into word problem language. We started by calling 
our two numbers x and y, and found values for x and y. These values 
must correspond to our original numbers. If you take the sum of the 
values you should find that it is 20. Likewise you should find that 
twice one value is four more than four times the second value.

So by solving the math part and finding values for x and y, you found
exactly the numbers that are asked for in the problem. You could 
finish your answer in word problem language something like this:

"If the sum of two numbers is 20 and twice one number is four more 
than four times the other number, then the numbers are x and y."

Substituting in the values you found doing the math will make the 
above sentence true. Try it out and let me know how you did.

What we did here is called solving two equations with two unknowns.

I hope I was able to help you. If not please write again.

-Doctor Allan,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/