Hema's Age Problem: A Step-by-Step Approach

This problem involves finding the ages of three individuals: Hema, Hari, and Suresh, based on several relationships between their ages. We will use algebraic equations to represent these relationships and then solve them step-by-step to find Hema's age.

Age Variables: Naming Unknowns

To make the problem easier to solve, let's assign variables to each person's age:

• Let \(H\) represent Hari's age in years.
• Let \(E\) represent Hema's age in years.
• Let \(S\) represent Suresh's age in years.

Equations for Age Relationships

We are given three pieces of information that can be translated into linear equations:

Solving for Ages: Step-by-Step Method

Our goal is to find Hema's age (\(E\)). We can solve this system of equations by substitution.

1. Substitute Equation 1 into Equation 3:
   Since \(S = 3E\) and \(E = 2H + 5\), we can replace \(E\) in Equation 3 with its expression from Equation 1:
   \[ S = 3(2H + 5) \]
   Distribute the 3:
   \[ S = 6H + 15 \quad \text{(Equation 4)} \]
2. Equate Equation 2 and Equation 4 to find Hari's age (\(H\)):
   Now we have two expressions for Suresh's age (\(S\)):
   From Equation 2: \(S = 10H - 13\)
   From Equation 4: \(S = 6H + 15\)
   Set these two expressions equal to each other:
   \[ 10H - 13 = 6H + 15 \]
   Subtract \(6H\) from both sides:
   \[ 10H - 6H - 13 = 15 \]
   \[ 4H - 13 = 15 \]
   Add 13 to both sides:
   \[ 4H = 15 + 13 \]
   \[ 4H = 28 \]
   Divide by 4:
   \[ H = \frac{28}{4} \]
   \[ H = 7 \]
   So, Hari's age is 7 years.
3. Calculate Hema's Age (\(E\)):
   Now that we know Hari's age, we can find Hema's age using Equation 1:
   \[ E = 2H + 5 \]
   Substitute \(H = 7\):
   \[ E = 2(7) + 5 \]
   \[ E = 14 + 5 \]
   \[ E = 19 \]
   Therefore, Hema's age is 19 years.

Verifying the Calculated Ages

Let's check if all conditions are met with the calculated ages:

• Hari's age \(H = 7\) years
• Hema's age \(E = 19\) years
• Suresh's age \(S = 57\) years (calculated using \(S = 10H - 13\) or \(S = 3E\))

1. Hema's age (19) is 5 years more than twice Hari's age (7):
\(2 \times 7 + 5 = 14 + 5 = 19\). This statement holds true.

2. Suresh's age (57) is 13 years less than 10 times Hari's age (7):
\(10 \times 7 - 13 = 70 - 13 = 57\). This statement holds true.

3. Suresh's age (57) is 3 times as old as Hema's age (19):
\(3 \times 19 = 57\). This statement holds true.

All conditions are satisfied, confirming Hema's age is 19.

Hema's Final Age

Based on the calculations, Hema's age is 19 years.