Question

Find the value of x in each of the following proportions:
A) \[\;36:81::x:63\]
B) $27:x::63:84$
C) $x:92::87:116$
D) $45:x::25:35$

Answer 1

Hint:
A proportion is a statement which implies that the two ratios are equal.
i.e. If it is given: $a:b::c:d$
Then, it means: $\dfrac{a}{b} = \dfrac{c}{d}$.
Write all the proportions given in the question in the form of an equation as shown above.
Cross-multiply and then solve for x.

Complete step by step solution:
As already told any proportion $a:b::c:d$ can be written as:
$\dfrac{a}{b} = \dfrac{c}{d}$
And then they can be cross-multiplied:
i.e. $ad = bc$
Applying the same steps in all the questions:
(i) \[\;36:81::x:63\]
It can be written as:
$\dfrac{{36}}{{81}} = \dfrac{x}{{63}}$
Now, cross-multiplying:
$
  36 \times 63 = 81 \times x \\
   \Rightarrow 81x = 36 \times 63 = (9 \times 4) \times \left( {9 \times 7} \right) \\
 $
Dividing both sides by 81, we get:
$
   \Rightarrow x = \dfrac{{9 \times 4 \times 9 \times 7}}{{81}} \\
   \Rightarrow x = \dfrac{{9 \times 9 \times 4 \times 7}}{{9 \times 9}} \\
 $
Cancelling two 9’s from numerator and denominator, we get:
$
   \Rightarrow x = 4 \times 7 \\
   \Rightarrow x = 28 \\
 $
(ii) $27:x::63:84$
It can be written as:
$\dfrac{{27}}{x} = \dfrac{{63}}{{84}}$
Now, cross-multiplying:
$
  27 \times 84 = 63 \times x \\
   \Rightarrow 63x = 27 \times 84 = \left( {9 \times 3} \right) \times \left( {12 \times 7} \right) \\
 $
Dividing both sides by 63, we get:
$
   \Rightarrow x = \dfrac{{(9 \times 7) \times (12 \times 3)}}{{63}} \\
   \Rightarrow x = \dfrac{{63 \times 36}}{{63}} \\
 $
Cancelling 63 from both numerator and denominator, we get:
$ \Rightarrow x = 36$
(iii) $x:92::87:116$
It can be written as:
$\dfrac{x}{{92}} = \dfrac{{87}}{{116}}$
Now, cross-multiplying:
$
  x \times 116 = 87 \times 92 \\
   \Rightarrow 116x = 87 \times 92 = (29 \times 3) \times \left( {23 \times 4} \right) \\
 $
Dividing both sides by 116, we get:
$
   \Rightarrow x = \dfrac{{(29 \times 4) \times (23 \times 3)}}{{116}} \\
   \Rightarrow x = \dfrac{{116 \times 69}}{{116}} \\
 $
Cancelling 116 from both numerator and denominator, we get:
$ \Rightarrow x = 69$
(iv) $45:x::25:35$
It can be written as:
$\dfrac{{45}}{x} = \dfrac{{25}}{{35}}$
Now, cross-multiplying:
$
  45 \times 35 = 25 \times x \\
   \Rightarrow 25x = 45 \times 35 = (9 \times 5) \times \left( {7 \times 5} \right) \\
 $
Dividing both sides by 116, we get:
$
   \Rightarrow x = \dfrac{{(5 \times 5) \times (9 \times 7)}}{{25}} \\
   \Rightarrow x = \dfrac{{25 \times 63}}{{25}} \\
 $
Cancelling 25 from both numerator and denominator, we get:
$ \Rightarrow x = 63$

Note:
Whenever upon simplification you have to divide the numerator and denominator, try to split the numerator into factors, so that they can be easily cancelled out with the denominator, as shown in all of the questions above.