Hema’s age is 5 years more than twice of Hari’s age. Suresh’s age is 13 years less than 10 times Hari’s age. If Suresh is 3 times as old as Hema, how old is Hema?
19
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:
| Relationship Description | Equation |
|---|---|
| Hema’s age is 5 years more than twice of Hari’s age. | \( E = 2H + 5 \quad \text{(Equation 1)} \) |
| Suresh’s age is 13 years less than 10 times Hari’s age. | \( S = 10H - 13 \quad \text{(Equation 2)} \) |
| Suresh is 3 times as old as Hema. | \( S = 3E \quad \text{(Equation 3)} \) |
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.
- 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:
\[ S = 6H + 15 \quad \text{(Equation 4)} \] - 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. - 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.
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