Question

What is Sonia’s present age?
I. Sonia’s present age is five times Deepak’s present age.
II. Five years ago her age was twenty-five times Deepak’s age at that time.
A. I alone sufficient while II alone not sufficient to answer
B. II alone sufficient while I alone not sufficient to answer
C. Either I or II alone sufficient to answer
D. Both I and II are not sufficient to answer
E. Both I and II are necessary to answer

Answer 1

Hint: In this problem, we have to find Sonia’s present age by using given information in two statements. Also we have to decide which statement is sufficient or necessary to find Sonia’s present age. For this, we will assume the present age as variable. Then, by using given information we will get linear equations. In solving those equations, we can find the present age of the given persons.

Complete step-by-step answer:
Let us assume that Sonia’s present age is $ x $ years and Deepak’s present age is $ y $ years. The following statements are given:
I. Sonia’s present age is five times Deepak’s present age.
II. Five years ago her age was twenty-five times Deepak’s age at that time.
By using information given in statement I, we can write
 $ x = 5y \Rightarrow x - 5y = 0 \cdots \cdots \left( 1 \right) $
Note that in equation $ \left( 1 \right) $ , there are two variables $ x $ and $ y $ . To find two unknowns, we need at least two equations. So, we can say that statement I alone is not sufficient to answer. Hence, option A is wrong.
Now if Sonia’s present age is $ x $ years then we can say that five years ago her age was $ x - 5 $ years. Similarly, if Deepak’s present age is $ y $ years then we can say that five years ago his age was $ y - 5 $ years. So, now by using information given in statement II, we can write
 $ x - 5 = 25\left( {y - 5} \right) $
 $ \Rightarrow x - 5 = 25y - 125 $
 $ \Rightarrow x - 25y = 5 - 125 $
 $ \Rightarrow x - 25y = - 120 \cdots \cdots \left( 2 \right) $
Note that in equation $ \left( 2 \right) $ , there are two variables $ x $ and $ y $ . To find two unknowns, we need at least two equations. So, we can say that statement II alone is not sufficient to answer. Hence, option B is wrong. Here option A and option B are wrong so we can say that option C is also wrong. Also note that we can find the values of $ x $ and $ y $ (that is, present age of the given persons) from equations $ \left( 1 \right) $ and $ \left( 2 \right) $ . Hence, option D is wrong.
Let us solve equations $ \left( 1 \right) $ and $ \left( 2 \right) $ . Let us subtract equation $ \left( 2 \right) $ from equation $ \left( 1 \right) $ . So, we get
 $ \left( {x - 5y} \right) - \left( {x - 25y} \right) = 0 - \left( { - 120} \right) $
 $ \Rightarrow x - 5y - x + 25y = 120 $
 $ \Rightarrow 20y = 120 $
 $ \Rightarrow y = \dfrac{{120}}{{20}} $
 $ \Rightarrow y = 6 $
Let us substitute $ y = 6 $ in equation $ \left( 1 \right) $ . So, we get
 $ x - 5\left( 6 \right) = 0 $
 $ \Rightarrow x = 30 $
Hence, Sonia’s present age is $ 30 $ years. Here we find Sonia’s present age by using both statements. So, option E is correct.

So, the correct answer is “Option E”.

Note: In the given problem, we need to find values of two unknown variables. So, we need two equations. To find values of $ n $ unknowns, we need $ n $ equations. Two linear equations can be solved by using a simple elimination method.