Question

If 1 is added to the age of the elder sister the ratio of the ages of the two sisters becomes \[0.5:1\] but if 2 is subtracted from the age of the younger ones, the ratio becomes \[1:3\], the age of younger sister will be
A) 9 years
B) 5 years
C) 18 years
D) 15 years

Answer 1

Hint: We are given conditions of two sisters’ age in the form of a ratio. So forming two equations in two variables like x and y as sister’s age will help us to solve the problem.

Complete step by step answer:

Let younger sister’s age be x years and that of elder sister be y years.
From first statement,
If 1 is added to the age of the elder sister the ratio of the ages of the two sisters becomes \[0.5:1\]
Then ratio will become,
\[\begin{gathered}
  \dfrac{x}{{y + 1}} = \dfrac{{0.5}}{1} \\
  \dfrac{x}{{y + 1}} = \dfrac{1}{2} \\
\end{gathered} \]
On cross multiplying,
\[\begin{gathered}
  2x = y + 1 \\
  2x - y = 1 \\
  y = 2x - 1 \\
\end{gathered} \]
Now ,from second statement,
but if 2 is subtracted from the age of younger one , the ratio becomes \[1:3\]
then ratio will become,
\[\dfrac{{x - 2}}{y} = \dfrac{1}{3}\]
On cross multiplying,
\[y = 3x - 6\]
Now, since L.H.S. of these equations are same then comparing R.H.S.
\[\begin{gathered}
  2x - 1 = 3x - 6 \\
  6 - 1 = 3x - 2x \\
  x = 5 \\
\end{gathered} \]
Hence found the age of younger sister is 5 years.
So, option B is the correct answer.

Note: Students get confused in forming the ratios. But here ratio is between the age of younger sister is to the elder sister. We have equated the equations and found the value of one person’s age. Remember don’t reverse the conditions of the age.
Additional information: This problem is applications of ratio. In this, we are given two different conditions in the form of ratios.