Question

Which ratio is larger in the following pairs, $3:4$ or $9:16$?

Answer 1

Hint: In this problem, we will first convert these two given ratios in terms of fractions. Then we will compare the fractions by making the numerator or denominator in both of them equal, and then check which of them is smaller or larger. The ratios can be converted into fractions as: $x:y = \dfrac{x}{y}$.

Complete step by step answer:
We have been given the two ratios $3:4$ and $9:16$. We will first convert them into fractions, so that they can be easily compared to each other as,
$3:4 = \dfrac{3}{4}$
$\Rightarrow 9:16 = \dfrac{9}{{16}}$
Now, we have to convert these fractions such that they can be compared to each other. To do this, we can either make their denominators equal or their numerators equal. Here, we will try to make the numerators equal. So, we will multiply the numerator and denominator in the first ratio by $3$, which can be written as,
$\dfrac{3}{4} = \dfrac{{3 \times 3}}{{4 \times 3}} = \dfrac{9}{{12}}$

Now these two fractions are in their comparable form. They have the same numerator but different denominators. We can use the property that the larger the denominator the smaller is the number, that is the value of a fraction is inversely proportional to its denominator. We know that $16$ is greater than $12$, so we can write that: $12 < 16$.
$ \Rightarrow \dfrac{1}{{12}} > \dfrac{1}{{16}}$
On multiplication of $9$ to both sides, we get
$ \Rightarrow \dfrac{9}{{12}} > \dfrac{9}{{16}}$
$ \therefore \dfrac{3}{4} > \dfrac{9}{{16}}$

Hence, we can see that the ratio $3:4$ is greater than the ratio $9:16$.

Note: The most common mistake in such types of questions is students often reverse the sign in the comparison. There is a misconception that the larger number should have a larger denominator, the larger number is the number, as shown in the question above. We should take care of the calculations so as to be sure of our final answer.