Question

Express the following ratios in simplest form: $ 20:40 $ ?

Answer 1

Hint: In this question, we are given a ratio that is not in the simplest form. The ratio doesn’t comprise co-prime numbers and hence the numbers have common factors between them. Any ratio can also be converted into a fraction with numerator and denominator. Then, we can simplify the fraction obtained by cancelling the common factors in the numerator and denominator.

Complete step by step solution:
So, the ratio given to us in the problem is $ 20:40 $ .
The ratio can be converted into fraction as $ \dfrac{{20}}{{40}} $ .
Now, to convert the fraction in simplest terms, we have to cancel the common factor in numerator and denominator. So, we first find out the prime factors of numerator and denominator. So, we get,
Prime factorization of $ 20 $ is:
 $ 20 = 2 \times 2 \times 5 $
Prime factorization of $ 40 $ is:
 $ 40 = 2 \times 2 \times 2 \times 5 $
We see that $ 2 $ and $ 5 $ are the prime factors of both the numerator and the denominator. So, we have $ 2 $ and $ 5 $ as common factors, and thus we divide the numerator and the denominator by $ 20 $ .
So, we get, $ \left( {\dfrac{{20}}{{40}}} \right) = \dfrac{{\left( {\dfrac{{20}}{{20}}} \right)}}{{\left( {\dfrac{{40}}{{20}}} \right)}} = \left( {\dfrac{1}{2}} \right) $
Now, the numerator and the denominator are both co-prime numbers and don’t have any common factor, so it cannot be simplified further.
Hence, the simplified form of $ \dfrac{{20}}{{40}} $ is $ \left( {\dfrac{1}{2}} \right) $ .
Now, converting the obtained fraction into ratio, we get,
 $ \Rightarrow 20:40 = 1:2 $
So, $ 20:40 $ can be written in the simplest terms as $ 1:2 $ .
So, the correct answer is “ $ 1:2 $ ”.

Note: One must have knowledge of how to deal with ratios and fractions in order to deal with such types of questions. We must know to convert the ratio into fraction and to simplify a fraction into its lowest terms. Following the information and the steps mentioned in the above solution, we can solve similar questions.