Question

Find the value of $x$ in the following proportions :
(i) 7 : 15 :: $x$ : 30
(ii) $x$ : 3 :: 12 : 18
(iii) 30 : $x$ :: 45 : 24
(iv) 8 : 16 :: 25 : $x$

Answer 1

Hint: We know that for a proportion a : b :: c : d, we have the following definition $\dfrac{a}{b}=\dfrac{c}{d}$. So, by using this definition, we can find the value of $x$ in each of the above cases with the help of basic arithmetic operations.

Complete step by step solution:
We know that when four numbers are in proportion such that a : b :: c : d, then what we mean to say is that the two ratios on each side of the proportion sign are equal. We can write this mathematically as, $\dfrac{a}{b}=\dfrac{c}{d}$.
(i) 7 : 15 :: $x$ : 30
Here, by the definition of proportions, we can write,
$\dfrac{7}{15}=\dfrac{x}{30}$
Now, we can rearrange the terms and bring the variable $x$ on the left hand side. Thus, we get,
$x=\dfrac{7\times 30}{15}$
Now, by cancelling the common factors in the numerator and denominator, we get
$x=14$.
Hence, the value of $x$ is 14.
(ii) $x$ : 3 :: 12 : 18
Here, by the definition of proportions, we can write,
$\dfrac{x}{3}=\dfrac{12}{18}$
Now, we can rearrange the terms and keep the variable $x$ on the left hand side, and rest all on the right hand side. Thus, we get,
$x=\dfrac{12\times 3}{18}$
Now, by cancelling the common factors in the numerator and denominator, we get
$x=2$.
Hence, the value of $x$ is 2.
(iii) 30 : $x$ :: 45 : 24
Here, by the definition of proportions, we can write,
$\dfrac{30}{x}=\dfrac{45}{24}$
Now, we can rearrange the terms and keep the variable $x$ on the left hand side, and rest all on the right hand side. Thus, we get,
$x=\dfrac{30\times 24}{45}$
Now, by cancelling the common factors in the numerator and denominator, we get
$x=16$.
Hence, the value of $x$ is 16.
(iv) 8 : 16 :: 25 : $x$
Here, by the definition of proportions, we can write,
$\dfrac{8}{16}=\dfrac{25}{x}$
Now, we can rearrange the terms and bring the variable $x$ on the left hand side, and rest all on the right hand side. Thus, we get,
$x=\dfrac{25\times 16}{8}$
Now, by cancelling the common factors in the numerator and denominator, we get
$x=50$.

Hence, the value of $x$ is 50.

Note: We must understand that for the proportion a : b :: c : d, some texts use the following definition, $\dfrac{a}{c}=\dfrac{b}{d}$, which is basically the same as $\dfrac{a}{b}=\dfrac{c}{d}$. Hence, we should not get confused by such variations.