The difference between Peter and Preeti’s ages is 5 years. When they married each other 35 years ago, 4 times Peter’s age was the same as 5 times the age of Preeti’s. What is the current sum of their ages?
115 years
Solving the Peter and Preeti Age Problem
This problem involves finding the current ages of Peter and Preeti and then calculating the sum of their ages. We are given two pieces of information relating their ages at different points in time.
Setting up the Equations
Let Peter's current age be \(P\) years and Preeti's current age be \(R\) years.
From the first statement, "The difference between Peter and Preeti’s ages is 5 years", we can write the equation:
\(|P - R| = 5\)
This means either \(P - R = 5\) or \(R - P = 5\). One person is 5 years older than the other.
The second statement gives information about their ages 35 years ago. 35 years ago, Peter's age was \(P - 35\) and Preeti's age was \(R - 35\). The statement says, "4 times Peter’s age was the same as 5 times the age of Preeti’s" at that time. So, we can write the second equation:
\(4(P - 35) = 5(R - 35)\)
Solving the System of Equations
Now we need to solve the system of equations. Let's first simplify the second equation:
\(4P - 4 \times 35 = 5R - 5 \times 35\)
\(4P - 140 = 5R - 175\)
Rearranging the terms to group P and R:
\(4P - 5R = -175 + 140\)
\(4P - 5R = -35\)
Now we consider the two cases from the first equation, \(|P - R| = 5\).
Case 1: Peter is 5 years older than Preeti
If Peter is older, then \(P - R = 5\), which means \(P = R + 5\).
Substitute \(P = R + 5\) into the simplified second equation \(4P - 5R = -35\):
\(4(R + 5) - 5R = -35\)
\(4R + 20 - 5R = -35\)
\(-R + 20 = -35\)
\(-R = -35 - 20\)
\(-R = -55\)
\(R = 55\)
Now find Peter's age using \(P = R + 5\):
\(P = 55 + 5\)
\(P = 60\)
Let's check if these ages are valid 35 years ago. Peter's age was \(60 - 35 = 25\). Preeti's age was \(55 - 35 = 20\). Both ages are positive, which is necessary for a past age. Let's check the condition from 35 years ago: \(4 \times 25 = 100\) and \(5 \times 20 = 100\). The condition \(4(P - 35) = 5(R - 35)\) is satisfied.
So, Peter's current age is 60 years and Preeti's current age is 55 years. The difference \(60 - 55 = 5\) is also correct.
Case 2: Preeti is 5 years older than Peter
If Preeti is older, then \(R - P = 5\), which means \(R = P + 5\).
Substitute \(R = P + 5\) into the simplified second equation \(4P - 5R = -35\):
\(4P - 5(P + 5) = -35\)
\(4P - 5P - 25 = -35\)
\(-P - 25 = -35\)
\(-P = -35 + 25\)
\(-P = -10\)
\(P = 10\)
Now find Preeti's age using \(R = P + 5\):
\(R = 10 + 5\)
\(R = 15\)
Let's check if these ages are valid 35 years ago. Peter's age was \(10 - 35 = -25\). This is not possible, as age cannot be negative. Therefore, this case is invalid.
Based on the valid case (Case 1), Peter's current age is 60 years and Preeti's current age is 55 years.
Calculating the Current Sum of Ages
The question asks for the current sum of their ages.
Current Sum = Peter's current age + Preeti's current age
Current Sum = \(P + R\)
Current Sum = \(60 + 55\)
Current Sum = \(115\) years.
The current sum of their ages is 115 years.
Conclusion
The current sum of Peter and Preeti's ages is 115 years. This matches one of the given options.
Revision Table: Key Information
| Information Given | Mathematical Representation |
|---|---|
| Current age difference is 5 years | \(|P - R| = 5\) (or \(P - R = 5\) or \(R - P = 5\)) |
| 35 years ago, 4 times Peter's age was 5 times Preeti's age | \(4(P - 35) = 5(R - 35)\) |
| Simplified relationship 35 years ago | \(4P - 5R = -35\) |
Additional Information: Solving Age Word Problems
Age word problems are common in mathematics and tests. They often involve setting up and solving systems of linear equations. Here are some tips:
- Identify Variables: Clearly define variables for the unknown ages (usually current ages).
- Translate Statements into Equations: Read each sentence carefully and translate the relationships between ages into algebraic equations. Pay attention to terms like "ago" (subtract from current age), "in x years" (add to current age), "times", "ratio", "sum", "difference".
- Handle Multiple Time Periods: If the problem refers to ages at different times (past, present, future), express the ages at those times in terms of the current age variables.
- Solve the System: Use methods like substitution or elimination to solve the system of equations you created.
- Check for Validity: After finding the potential ages, check if they make sense in the context of the problem (e.g., ages must be non-negative).
- Answer the Specific Question: Make sure you answer what the question asks for (e.g., sum of ages, individual age, difference in a few years).
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