If two taps fill a tank together, one taking $x$ hours and the other $x+3$, the fraction of the tank they fill together in one hour is $\frac{1}{x} + \frac{1}{x+3}$. That sum, with the unknown in the denominator, is an algebraic fraction: it is handled in exactly the same way as a numerical fraction, but with polynomials on the top and the bottom.
To simplify, you factorise the numerator and the denominator and cancel the factors that appear on both the top and the bottom. It is the same as with numerical fractions: $\frac{12}{18} = \frac{2\cdot2\cdot3}{2\cdot3\cdot3} = \frac{2}{3}$.
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Simplify the following algebraic fraction by taking out common factors in the numerator and in the denominator: $\frac{4x^3 - 2x^2}{6x^2 - 2x}$
Simplify the following algebraic fraction by taking out common factors in the numerator and in the denominator: $\frac{4x^3 - 2x^2}{6x^2 - 2x}$
A fraction does not exist if its denominator equals $0$. These restrictions are read from the original denominator, before simplifying, and they remain even if some factor disappears when simplifying.
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The base of a rectangular solar panel is $x+5$ metres and its height is $x-5$ metres. The side of a square solar panel is $x$ metres. Write down the algebraic fraction…
The base of a rectangular solar panel is $x+5$ metres and its height is $x-5$ metres. The side of a square solar panel is $x$ metres. Write down the algebraic fraction representing the quotient of the area of the rectangular panel and the area of the square panel. Calculate the value of this fraction if $x = 10$ metres.
As with numbers: to multiply, multiply numerator by numerator and denominator by denominator; to divide, multiply by the reciprocal fraction. It is a good idea to factorise before multiplying, so that you can simplify along the way.
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Multiply the following algebraic fractions and give the result in its simplest form: $\frac{x}{x+2} \cdot \frac{x^2-4}{x^2}$
Multiply the following algebraic fractions and give the result in its simplest form: $\frac{x}{x+2} \cdot \frac{x^2-4}{x^2}$
A hot water tap takes $x$ hours to fill a bath, while the cold water tap takes $x+3$ hours. Write a simplified algebraic fraction representing the fraction of the bath…
A hot water tap takes $x$ hours to fill a bath, while the cold water tap takes $x+3$ hours. Write a simplified algebraic fraction representing the fraction of the bath that fills in one hour if both taps are turned on together.
You need a common denominator, found by factorising each denominator first: the lowest common multiple takes each factor with its highest power, without repeating it.
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Perform the following addition of algebraic fractions and simplify the result: $\frac{3x+2}{x-2} + \frac{-x-6}{x-2}$
Perform the following addition of algebraic fractions and simplify the result: $\frac{3x+2}{x-2} + \frac{-x-6}{x-2}$
Back to the beginning: With the common denominator $x(x+3)$, the sum for the two taps is $\dfrac{x+3}{x(x+3)} + \dfrac{x}{x(x+3)} = \dfrac{2x+3}{x^2+3x}$ of the tank per hour.
Summary
Let's practice! 10 medium exercises in this section