Using Algebraic Fractions To Find Perimeter

Question

Make a formula to find the perimeter of the rectangle, the perimeter is 24 units.

The longer side is $\frac{5}{x+1}$ and the shorter side is $\frac{2}{x}$.

I know that the answer is $\frac{1}{4}$ but no idea how.

Any ideas how to solve it?

Answer 1

Well, for any rectangle, the perimeter is given by

$$P=2L+2W$$

where $L$ and $W$ are the lengths and widths. Plugging in the values you get, we get

$$P=2\left(\frac5{x+1}\right)+2\left(\frac2x\right)=\frac{10}{x+1}+\frac4x=\frac{14x+4}{x(x+1)}$$

Since the perimeter is given as $20$, then

$$24=\frac{14x+4}{x(x+1)}$$

$$24x(x+1)=14x+4$$

$$24x^2+24x=14x+4$$

$$24x^2+10x-4=0$$

$$12x^2+5x-2=0$$

$$x=\frac{-5\pm11}{20}=\frac14,-\frac23$$

Since $x$ can't be negative, or else we'd have negative side lengths, the correct answer is given as

$$x=\frac14$$

Answer 2

If the perimeter is supposed to be 24 (not 20 as originally stated), then using the setup from @SimpleArt, we get a quadratic equation of $12x^2+5x-2$. The quadratic formula yields $x=-\frac{2}{3}$ or $x=\frac{1}{4}$. Since negative lengths on a rectangle don't make sense, it seems the correct answer is indeed $\frac{1}{4}$.