Question

Give three equivalent ratios of $ 4:9 $

Answer 1

Hint: Equivalent ratios means equal ratios. And we have to find out the three different ratios which are equal to $ 4:9 $ . So we will multiply the number in numerator and denominator simultaneously to get the different ratios.

Complete step by step answer:
Moving ahead with the question, as we know that ratios are the division of two numbers in the simplest form i.e. in $ \dfrac{a}{b} $ where ‘a’ and ‘b’ are the shorter values which cannot be further reduced. As there are an infinite number of cases possible which can have the same ratios. As ratios of two large and small values can get reduced to the smallest possible ratio which can be the same. For example $ \dfrac{10}{20} $ and $ \dfrac{20}{40} $ are the same ratio, but they are different numbers, which is $ \dfrac{1}{2} $ .
These numbers differ from each other by some multiple, these numbers are real numbers.
That means if we multiply any number in the ratio in numerator and denominator simultaneously then we will get the different number in the $ \dfrac{a}{b} $ form whose ratio is the same.
So for our question, we want three different numbers in the $ \dfrac{a}{b} $ form whose ratios are equal to $ \dfrac{4}{9} $ . So let us pick any three numbers, let it be 2, -3 and 10. Now to get the ratios multiply these three numbers, in numerator and denominator of given ratios to get the other equivalent ratios.
So let us first multiply 2, we will get;
 $ \dfrac{4\times 2}{9\times 2}=\dfrac{8}{18} $
Hence the $1^{st}$ equivalent ratio is $ \dfrac{8}{18} $ .
Now for second equivalent ratio, let us multiply -3, so we will get;
 $ \dfrac{4\times \left( -3 \right)}{9\times \left( -3 \right)}=\dfrac{-12}{-27}=\dfrac{12}{27} $
Hence the $2^{nd}$ equivalent ratio is $ \dfrac{12}{27} $ .
Going for the $3^{rd}$ equivalent ratio, let us multiply by 10, so we will get;
 $ \dfrac{4\times 10}{9\times 10}=\dfrac{40}{90} $
Hence the $3^{rd}$ equivalent ratio is $ \dfrac{40}{90} $ .
Hence answer is $ \dfrac{40}{90},\dfrac{12}{27},\dfrac{8}{18} $ , i.e. these are the three equivalent ratio of $ 4:9 $ .

Note: Rather than writing ratios in $ \dfrac{a}{b} $ form we can write it as $ a:b $ both are the same. Moreover, for the type of question there is no particular answer, as already mentioned there are infinite possible cases, meaning there are an infinite number of answers for this, three of them I had taken out.