Age Problem Solution

Let $A$ represent Ayushi's present age and $N$ represent Nidhi's present age.

Setting Up the Equations

• The difference between their present ages is 14 years. Since Ayushi is older:

  $A - N = 14$ (Equation 1)

• Five years ago, Ayushi's age was $A - 5$ and Nidhi's age was $N - 5$.
• The sum of their ages five years ago was 60:

  $(A - 5) + (N - 5) = 60$

• Simplifying the second equation:

  $A + N - 10 = 60$

  $A + N = 70$ (Equation 2)

Solving for Nidhi's Age

We now have a system of two linear equations:

1. $A - N = 14$
2. $A + N = 70$

To find Nidhi's age ($N$), we can subtract Equation 1 from Equation 2:

$(A + N) - (A - N) = 70 - 14$

$A + N - A + N = 56$

$2N = 56$

$N = \frac{56}{2}$

$N = 28$

Verification

Nidhi's present age is 28 years.

Using Equation 1: $A - 28 = 14 \implies A = 14 + 28 = 42$. Ayushi's age is 42.

Check ages 5 years ago: Ayushi was $42 - 5 = 37$, Nidhi was $28 - 5 = 23$.

Sum of ages 5 years ago: $37 + 23 = 60$. The conditions are satisfied.

Nidhi's present age is 28 years.