Question

How do you change the fraction $ \dfrac{{w - 3}}{{w + 5}} $ into an equivalent fraction with the denominator $ {w^2} + w - 20 $ ?

Answer 1

Hint: In this question, the numerator of the given fraction is $ w - 3 $ and the denominator is equal to $ w + 5 $ and we have to find an equivalent fraction such that the denominator of the fraction is $ {w^2} + w - 20 $ . When the numerator and the denominator are in the form of prime factors and don’t have any common factor, the fraction is said to be simplified. We can find an equivalent fraction by multiplying or dividing the numerator and denominator of the fraction by the same number.

Complete step-by-step answer:
So, we have to find an equivalent fraction such that the denominator of the fraction $ \dfrac{{w - 3}}{{w + 5}} $ is $ {w^2} + w - 20 $ .
So, we have to multiply the given fraction $ \dfrac{{w - 3}}{{w + 5}} $ with an expression such that the the denominator of the fraction becomes $ {w^2} + w - 20 $ .
So, we will first factorise the expression $ {w^2} + w - 20 $ using the splitting the middle term method.
Hence, we have, $ {w^2} + w - 20 $
We split the middle term $ w $ of the expression $ {w^2} + w - 20 $ into $ 5w $ and $ - 4w $ .
 $ \Rightarrow {w^2} + \left( {5 - 4} \right)w - 20 $
 $ \Rightarrow {w^2} + 5w - 4w - 20 $
So, we take w common from the first two terms and $ - 4 $ common from the last two terms. We get,
 $ \Rightarrow w\left( {w + 5} \right) - 4\left( {w + 5} \right) $
Simplifying further, we get,
 $ \Rightarrow \left( {w - 4} \right)\left( {w + 5} \right) $
So, the denominator of the equivalent fraction should be $ \left( {w - 4} \right)\left( {w + 5} \right) $ . So, we have to multiply the numerator and denominator of the given fraction $ \dfrac{{w - 3}}{{w + 5}} $ by $ \left( {w - 4} \right) $ .
So, we get, $ \dfrac{{w - 3}}{{w + 5}} \times \dfrac{{w - 4}}{{w - 4}} $
 $ \Rightarrow \dfrac{{\left( {w - 3} \right)\left( {w - 4} \right)}}{{\left( {w + 5} \right)\left( {w - 4} \right)}} $
Simplifying further, we get,
 $ \Rightarrow \dfrac{{{w^2} - 7w + 12}}{{{w^2} + w - 20}} $
Hence, the equivalent fraction of the fraction $ \dfrac{{w - 3}}{{w + 5}} $ with denominator $ {w^2} + w - 20 $ is $ \dfrac{{{w^2} - 7w + 12}}{{{w^2} + w - 20}} $ .
So, the correct answer is “ $ \dfrac{{{w^2} - 7w + 12}}{{{w^2} + w - 20}} $ ”.

Note: Equivalent fractions are the fractions that have different numerator and denominator but are equal to the same value. Following the information and the steps mentioned in the above solution, we can solve similar questions.