Find Two Numbers

Date: 11/14/2001 at 21:04:30
From: Cristina Jones
Subject: having trouble with math

I have a middle school math problem that I am having trouble with. 

The problem is: The sum of two numbers is 16 and their product is 48. 
What are the numbers?

Date: 11/16/2001 at 16:41:28
From: Doctor Ian
Subject: Re: having trouble with math

Hi Cristina,

Suppose I just told you the two numbers.  ould you be able to check 
them? How would you do that? 

First, you'd have to add the numbers together, and see if they add up 
to 16:

  (first number) + (second number) = 16

Then, you'd have to multiply the two numbers to see if the product is 
48:

  (first number) * (second number) = 48

Okay, now that we know how to _check_ our work, we can set out to find 
the numbers we're looking for.  


There are a couple of different approaches you could take to finding 
out what the numbers are. For example, you might guess...

  10 and 6 add up to 16... but 10*6 = 60, which is too big. 

  9 and 7 add up to 16...  but  9*7 = 63, which is even bigger. 
                                          Let's make the first number
                                          bigger, not smaller.

  11 and 5 add up to 16... but 11*5 = 55, which is still too big,
                                          but closer.

When you don't know what else to do, guessing can be a pretty good way 
to proceed, as long as you're smart about your guesses. 

In this case, there are lots of pairs of numbers that add up to 16.  
However, there aren't many pairs of numbers whose product is 48.

  1 * 48
  2 * 24
  3 * 16
  4 * 12
  6 *  8

Only one of these pairs will add up to 16.  Can you find it? 

Does this help? Write back if you'd like to see some other ways to 
find the answer with a little less work, or if you have any other 
questions. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/