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# What number must be added to each term of the ratio $9:16$ to make the ratio $2:3$?

Last updated date: 13th Jun 2024
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Hint:To find the term needed to add to $9:16$ to make $2:3$, we assume that the terms given are $x$ in both the denominator and the numerator with the ratio $9:16$ added to which is equal to $2:3$ . After placing the values in the ratio we find the value of $x$ by cross multiplying the ratios.

Complete step by step solution:
Let us assume that the number needed to be added is taken as $x$.
Now to find the term\ratio needed i.e. $2:3$ to add the unknown term of $x$ to the numerator and denominator of the previous ratio of $9:16$.
The value of the ratio on the LHS of the equation down below is equal to the ratio on the RHS when $x$ is added in both the numerator and denominator of the LHS given.
$\Rightarrow \dfrac{9+x}{16+x}=\dfrac{2}{3}$
Cross multiplying the value of the ratio of the LHS and RHS, we get the value of the unknown variable of $x$ as:
$\Rightarrow 3\left( 9+x \right)=2\left( 16+x \right)$
$\Rightarrow 27+3x=32+2x$
$\Rightarrow x=5$
Therefore, the value needed to be added with the ratio of $9:16$ to get $2:3$ is $5$.

Note: Student may go wrong if they try to add a single number instead of adding two number in both the denominator and numerator as given below:
$\Rightarrow \dfrac{9+x}{16+x}=\dfrac{2}{3}$ correct form
$\Rightarrow \dfrac{9}{16}+x=\dfrac{2}{3}$ Incorrect form