# Thread: Isolating a variable in the numerator

1. ## Isolating a variable in the numerator

One homework problem I have asks me to find what age added to a set of 15 ages, would lead to a mean of 54.94. The total of the 15 ages given minus the missing age = 828.

I am not sure if I did this right, but I set up the following equation to locate the X variable (the missing age in the line of ages).

828+x divided by 16 = 54.94 or (828 + X)/16 = 54.94

I seem to forget how to isolate the X variable. No matter what I do, it doesn't seem to work out. Perhaps I didn't even set it up correctly.

I would appreciate any help on this.

2. Originally Posted by Ian
One homework problem I have asks me to find what age added to a set of 15 ages, would lead to a mean of 54.94. The total of the 15 ages given minus the missing age = 828.

I am not sure if I did this right, but I set up the following equation to locate the X variable (the missing age in the line of ages).

828+x divided by 16 = 54.94 or (828 + X)/16 = 54.94

I seem to forget how to isolate the X variable. No matter what I do, it doesn't seem to work out. Perhaps I didn't even set it up correctly.

I would appreciate any help on this.
Call y the sum of the 15 given ages and x the new age.

Then
$\displaystyle \frac{x + y}{16} = 54.94$

and

$\displaystyle y - x = 828$

So this isn't as simple as setting up one equation in one unknown.

So take the second equation and solve for y.
$\displaystyle y = 828 + x$

and insert this into the first equation:
$\displaystyle \frac{x + (828 + x)}{16} = 54.94$

$\displaystyle \frac{2x + 828}{16} = 54.94$ <-- Multiply both sides by 16

$\displaystyle 2x + 828 = 879.04$

$\displaystyle 2x = 879.04 - 828= 51.04$

$\displaystyle 2x = 51.04$

$\displaystyle x = \frac{51.04}{2} = 25.52$

-Dan

3. Thank you. I appreciate you help. As a follow up, so I know for the future, why is it that we used dual variables instead of simply plugging in the sum of the original figures? What principle?

And thanks again. I will make up another problem for myself and reverse engineer your answer.

4. Originally Posted by Ian
Thank you. I appreciate you help. As a follow up, so I know for the future, why is it that we used dual variables instead of simply plugging in the sum of the original figures? What principle?

And thanks again. I will make up another problem for myself and reverse engineer your answer.
Originally Posted by Ian
The total of the 15 ages given minus the missing age = 828.

828+x divided by 16 = 54.94 or (828 + X)/16 = 54.94
I'm looking at this again and wondering if I took the above sentence correctly. Literally speaking your statement implies that (sum of 15 ages) - (missing age) = 828, which is why my solution involved two variables because we don't know what the sum of the 15 ages is.

However from your followup post and re-reading the question I am wondering if this was a typo. In your original post were you trying to say that the sum of the 15 ages is 828? If so then the equation you posted is correct:
$\displaystyle \frac{828 + x}{16} = 54.94$<-- Multiply by 16

$\displaystyle 828 + x = 54.94 \cdot 16 = 879.04$

$\displaystyle x = 879.04 - 828 = 51.04$

-Dan

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