Systems of Equation with Fraction Word Problems

Question

How would I solve a problem such as "The denominator of a fraction is 8 more than the numerator. If 3 is added to both the numerator and the denominator, the value of the resulting fraction is 1/2. What is the original fraction?" using systems of equations? I think the two equations are d=n+8 and 3+n/3+d=1/2, but I just don't know what to do from there. Also I'm not sure if the equations are correct.

Answer 1

Hi Avery,

I'm not sure that I would consider this a "systems of equations" problem, but let's take a look at what they tell us in the problem.

"The denominator of a fraction is 8 more than the numerator."

So let's describe what they've told us as an algebraic expression. I know I have a fraction, where the denominator is 8 more than the numerator. We don't know what either of these values are yet, so let's describe it as:

x

(x + 8)

Then they also tell us that "if 3 is added to both the numerator and the denominator, the value of the resulting fraction is 1/2".

So that tells us that the following must be true:

x + 3 = 1

x + 8 + 3 2

x + 3 = 1

x + 11 2

We can "cross multiply" to solve for the value of x, which we can then plug back into the original fraction.

2 * (x + 3) = 1 * (x + 11)

2x + 6 = x + 11

-x -x

x + 6 = 11

-6 -6

x = 5

Now that we know that x = 5, we can plug that into our original fraction:

5

5 + 8

5

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