The difference between Charles’ and Shriya’s ages is 6 years. When they married each other 30 years ago, 4 times Cahrle’s age was the same as 5 times the age of Shriya. What is the current sum of their ages?
114 years
Solving Charles and Shriya's Age Problem
This problem involves finding the current sum of ages of two people, Charles and Shriya, given their current age difference and a relationship between their ages 30 years ago. We can solve this by setting up a system of linear equations.
Understanding the Given Information
- The current difference between Charles's and Shriya's ages is 6 years.
- 30 years ago, 4 times Charles's age was equal to 5 times Shriya's age.
- We need to find the current sum of their ages.
Setting Up the Equations
Let:
- \(C\) be Charles's current age.
- \(S\) be Shriya's current age.
From the first piece of information, the difference in their current ages is 6 years. This gives us one equation:
\(C - S = 6\) (Assuming Charles is older, we will verify this later)
30 years ago, Charles's age was \(C - 30\) and Shriya's age was \(S - 30\). The second piece of information gives us the relationship between their ages at that time:
\(4 \times (C - 30) = 5 \times (S - 30)\)
Solving the Equations
Now we have a system of two linear equations:
- \(C - S = 6\)
- \(4(C - 30) = 5(S - 30)\)
Let's simplify the second equation:
\(4C - 120 = 5S - 150\)
\(4C - 5S = -150 + 120\)
\(4C - 5S = -30\)
From the first equation, we can express \(C\) in terms of \(S\):
\(C = S + 6\)
Substitute this expression for \(C\) into the simplified second equation:
\(4(S + 6) - 5S = -30\)
\(4S + 24 - 5S = -30\)
\(-S + 24 = -30\)
Subtract 24 from both sides:
\(-S = -30 - 24\)
\(-S = -54\)
Multiply by -1:
\(S = 54\)
Now that we have Shriya's current age (\(S = 54\)), we can find Charles's current age (\(C\)) using the equation \(C = S + 6\):
\(C = 54 + 6\)
\(C = 60\)
So, Charles's current age is 60 years, and Shriya's current age is 54 years. This confirms our initial assumption that Charles is older, as \(60 - 54 = 6\).
Verifying the Ages 30 Years Ago
30 years ago:
- Charles's age was \(60 - 30 = 30\) years.
- Shriya's age was \(54 - 30 = 24\) years.
Let's check the condition: 4 times Charles's age was 5 times Shriya's age:
\(4 \times 30 = 120\)
\(5 \times 24 = 120\)
The condition \(120 = 120\) holds true, so our calculated ages are correct.
Calculating the Current Sum of Ages
The question asks for the current sum of their ages. This is \(C + S\):
Current Sum = \(60 + 54 = 114\)
The current sum of Charles's and Shriya's ages is 114 years.
Revision Table: Key Steps in Age Problems
| Step | Description |
|---|---|
| 1 | Define variables for current ages. |
| 2 | Translate the age differences and past/future age relationships into algebraic equations. |
| 3 | Solve the system of equations to find the unknown ages. |
| 4 | Verify the calculated ages using the original conditions. |
| 5 | Calculate the required value (e.g., sum, difference, specific age) based on the question. |
Additional Information on Age Word Problems
Age word problems are common in mathematics and often require setting up and solving linear equations. The key is to carefully read the problem and correctly represent the ages at different points in time using variables.
- Representing Ages: If a person's current age is \(x\), their age \(n\) years ago was \(x - n\), and their age \(n\) years from now will be \(x + n\).
- Relationships: Phrases like "twice as old," "half the age," "difference," and "sum" translate directly into algebraic expressions.
- Multiple People: When dealing with multiple people, use different variables (e.g., \(C\) and \(S\)) for each person's current age. Remember that the time elapsed (like 30 years in this problem) applies equally to everyone.
Practice with various types of age problems helps build confidence in setting up the correct equations and solving them efficiently.
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