Algebraic Fractions: A reader asks

Oo, a question to answer! This one’s from Deborah.

How would you solve:
$\frac{1}{x-2} + \frac{1}{x} = \frac{3}{4}$ and $\frac{1}{x}+\frac{1}{2x+1}=\frac{7}{10}$?

When I substitute my answer back in, I can’t get it to work out!

Forgive me if my working is a bit pedestrian — I figure it’s better to err on the side of too helpful rather than not helpful enough.

To solve $\frac{1}{x-2} + \frac{1}{x} = \frac{3}{4}$, I’d combine the left hand side into a single fraction, multiplying the first one by $\frac{x}{x}$ and the second by $\frac{(x-2)}{(x-2)}$ to get:

$\frac{x}{x(x-2)} + \frac{(x-2)}{x(x-2)} = \frac{2x-2}{x(x-2)} = \frac{3}{4}$

Cross-multiplying:

$4(2x - 2) = 3x(x-2) \\
8x - 8 = 3x^2 - 6x\\
0 = 3x^2 - 14x +8 \\
0 = (3x-2)(x-4) $

Then either $x = \frac{2}{3}$ or $x = 4$. Substituting either value back in as $x$ on the left hand side does indeed give $\frac{3}{4}$.

The second one is similar:

$\frac{1}{x} + \frac{1}{2x+1} = \frac{7}{10} \\
\frac{2x+1}{x(2x+1)} + \frac{x}{x(2x+1)} = \frac{7}{10} \\
\frac{3x+1}{x(2x+1)} = \frac{7}{10} \\
10(3x+1) = 7x(2x+1) \\
30x + 10 = 14x^2 + 7x \\
0 = 14x^2 - 23x - 10 \\
0 = (14x+5)(x-2)$

(Yikes!) So $x = 2$ or $x = -\frac{5}{14}$. It’s clear enough that 2 works, but I’ll need to work it out for $-\frac{5}{14}$:

$\frac{1}{-5/14} + \frac{1}{-5/7 + 1} = -\frac{14}{5} + \frac{7}{2} = -\frac{28}{10} + \frac{35}{10} = \frac{7}{10}$

Whew!

If you'd like me to answer any of your questions, you can improve your chances by showing me that you've made a decent stab at it first, like Deborah did.

Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

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2 comments on “Algebraic Fractions: A reader asks

  • Joshua Zucker

    Rather than making common denominators and then cross-multiplying, I’d multiply by all the denominators at the very beginning. For example for the second one I’d start with
    $$\frac{1}{x}(x)(2x+1)(10) + \frac{1}{2x+1}(x)(2x+1)(10) = \frac{7}{10}(x)(2x+1)(10)$$
    which then simplifies to
    $$(2x+1)(10) + (x)(10) = 7(x)(2x+1)$$

    It is rather similar to your approach, for sure, but somehow I seem to make fewer mistakes in doing it this way.

    • Colin

      I like that approach, too – thanks, Joshua!

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