Question

What number shall be added to each of the numbers 4, 7, 14, 22 to form the terms in a proportion?
$
  {\text{A}}{\text{. 1}} \\
  {\text{B}}{\text{. 2}} \\
  {\text{C}}{\text{. 3}} \\
  {\text{D}}{\text{. 4}} \\
$

Answer 1

Hint – In order to find the number to be added, let us consider it to be variable x, add it to all the numbers and find their ratios. We now equate their ratios and solve for x.

Complete step-by-step answer::
Given Data,
4, 7, 14, 22
Let the number to be added be x, hence the number become
∴4+x, 7+x, 14+x, 22+x are in proportion.
$
   \Rightarrow \dfrac{{{\text{4 + x}}}}{{{\text{7 + x}}}} = \dfrac{{14 + {\text{x}}}}{{22 + {\text{x}}}} \\
   \Rightarrow \left( {{\text{4 + x}}} \right)\left( {{\text{22 + x}}} \right) = \left( {14 + {\text{x}}} \right)\left( {7 + {\text{x}}} \right) \\
   \Rightarrow 88 + 4{\text{x + 22x + }}{{\text{x}}^2} = 98 + 7{\text{x + 14x + }}{{\text{x}}^2} \\
   \Rightarrow 88 + 26{\text{x = 98 + 21x}} \\
   \Rightarrow {\text{26x - 21x = 98 - 88}} \\
   \Rightarrow {\text{5x = 10}} \\
   \Rightarrow {\text{x = 2}} \\
$
Hence the number to be added for the numbers 4, 7, 14, and 22 to be in proportion is 2.
Option B is the correct answer.

Note – In order to solve questions of this type the key is to know the definition of proportionality. Proportion is the fact of a variable quantity having a constant ratio to another quantity. We can apply this rule in any preferred order using the given numbers, it still gives us the same answer, 2.