# Ex.3.7 Q1 Pair of Linear Equations in Two Variables Solution - NCERT Maths Class 10

## Question

The ages of two friends Ani and Biju differ by \(3\) years. Ani’s father Dharam is twice as old as Ani and Biju is twice as old as his sister Cathy. The ages of Cathy and Dharam differs by \(30\) years. Find the ages of Ani and Biju.

## Text Solution

**Reasoning:**

The difference between the ages of Biju and Ani is \(3\) years. Either Biju is \(3\) years older than Ani or Ani is \(3\) years older than Biju. However, it is obvious that in both cases, Ani’s father’s age will be \(30\) years more than that of Cathy’s age.

**Steps:**

Let the age of Ani and Biju be *\(x\)* and *\(y\)* years respectively.

Therefore, age of Ani’s father, Dharam be \(2x\) years

And age of Biju’s sister Cathy be \(\begin{align}\frac{y}{2}\end{align}\) years

**Case (I) When Ani is older than Biju**

The ages of Ani and Biju differ by \(3\) years,

\[x - y = 3 \qquad \left( 1 \right)\]

The ages of Cathy and Dharam differs by \(30\) years,

\[\begin{align}2x - \frac{y}{2} &= 30\\4x - y &= 60 \qquad \left( 2 \right)\end{align}\]

Subtracting \((1)\) from \((2),\) we obtain

\[\begin{align}3x &= 57\\x &= 19\end{align}\]

Substituting \(x = 19\) in equation \((1),\) we obtain

\[\begin{align}19 - y &= 3\\y& = 16\end{align}\]

Therefore, Ani is \(19\) years old and Biju is \(16\) years old

**Case (II) When Biju is older than Ani.**

The ages of Ani and Biju differ by \(3\) years,

\[\begin{align}y - x &= 3\\ - x + y &= 3 \qquad \left( 1 \right)\end{align}\]

The ages of Cathy and Dharam differs by \(30\) years,

\[\begin{align}2x - \frac{y}{2} &= 30\\4x - y &= 60 \qquad (2)\end{align}\]

Adding \((1)\) and \((2),\) we obtain

\[\begin{align}3x &= 63\\x &= 21\end{align}\]

Substituting \(x = 21\) in equation \((1),\) we obtain

\[\begin{align} - 21 + y &= 3\\y &= 24\end{align}\]

Therefore, Ani is \(21\) years old and Biju is \(24\) years old.

Hence, Ani is \(19\) years old and Biju is \(16\) years old or Ani is \(21\) years old and Biju is \(24\) years old.