The sum of 2 numbers is 80. One fifth of one of them plus one quarter of the others is 17. Find the numbers.

Question

I need to find the value of both numbers using only one algebraic variable. 

Answer 1

I am not sure what you mean by only using one variable......Let's first solve it in the conventional way:

 

Call the numbers X and Y.

 

From the first sentence, we have:

  X + Y = 80

 

Then, from the second sentence:

  X/5 + Y/4 = 17

 

Rearrange the 1st equation to get this:

  X = 80 - Y

 

Now we can substitute for X in the 2nd equation:

   (80 - Y)/5 + Y/4 = 17   <<There's your equation using only one variable!>>

 

Expand and Simplify:

   16 - Y/5 + Y/4 = 17

 

    Y/4 - Y/5 = 1

 

    5Y - 4Y = 20 <<multiplied both sides by 20>>

 

   Y = 20

 

    X = 60

Answer 2

Usually this would be done with two variables, but it can easily be done with one.

 

Suppose x is one of the numbers.  Since they add up to 80, then the other number has to be 80 - x.  (You can prove that to yourself by adding the two together: x + (80 -x) = x - x + 80 = 80.)

 

So now use the other information to come up with an equation:

 

"One fifth of one of them" :  1/5 x, or x/5

"Plus" : +

"One quarter of the other" : 1/4 (80 - x), or (80 - x)/4

"Is 17" : = 17

 

So the equation is:

 

x/5 + (80 - x)/4 = 17

 

To solve, multiply both sides by 20 to get rid of all the fractions:

20 (x/5) + 20 (80 - x)/4 = 20*17

4x + 5(80 - x) = 340

4x + 400 - 5x = 340

-x + 400 = 340

-x = -60

x = 60

 

So the other number must be 20, in order for them to add to 80.