Five years ago, Ram was three times as old as Shyam. Four years from now, Ram will be only twice as old as Shyam. What is the present age of Ram?
32 years
Understanding the Age Word Problem
This question is a classic example of an age word problem, which can be solved using linear equations. We are given two conditions relating the ages of Ram and Shyam at different points in time. Our goal is to find Ram's current age.
Setting Up the Equations
Let's define variables for their present ages:
- Let Ram's present age be \( R \) years.
- Let Shyam's present age be \( S \) years.
Now, we will translate the given conditions into algebraic equations:
- Condition 1: Five years ago
- Ram's age five years ago was \( R - 5 \).
- Shyam's age five years ago was \( S - 5 \).
- The condition states that Ram was three times as old as Shyam five years ago.
- Equation 1: \( R - 5 = 3(S - 5) \)
- Condition 2: Four years from now
- Ram's age four years from now will be \( R + 4 \).
- Shyam's age four years from now will be \( S + 4 \).
- The condition states that Ram will be twice as old as Shyam four years from now.
- Equation 2: \( R + 4 = 2(S + 4) \)
Solving the System of Equations
We now have a system of two linear equations with two variables:
- \( R - 5 = 3(S - 5) \)
- \( R + 4 = 2(S + 4) \)
Let's simplify both equations:
- From Equation 1:
\( R - 5 = 3S - 15 \)
\( R = 3S - 15 + 5 \)
\( R = 3S - 10 \) (Equation 1 Simplified)
- From Equation 2:
\( R + 4 = 2S + 8 \)
\( R = 2S + 8 - 4 \)
\( R = 2S + 4 \) (Equation 2 Simplified)
Now we can use the substitution method or elimination method. Since both simplified equations give an expression for \( R \), we can set them equal to each other:
\( 3S - 10 = 2S + 4 \)
Now, solve for \( S \):
\( 3S - 2S = 4 + 10 \)
\( S = 14 \)
So, Shyam's present age is 14 years.
Now substitute the value of \( S \) (14) into either of the simplified equations to find \( R \). Let's use Equation 2 Simplified (\( R = 2S + 4 \)):
\( R = 2(14) + 4 \)
\( R = 28 + 4 \)
\( R = 32 \)
So, Ram's present age is 32 years.
Verifying the Present Ages
Let's check if these present ages satisfy the original conditions:
| Ram's Age | Shyam's Age | Check Condition | |
|---|---|---|---|
| Present | 32 | 14 | - |
| 5 years ago | \( 32 - 5 = 27 \) | \( 14 - 5 = 9 \) | Is \( 27 = 3 \times 9 \)? Yes, \( 27 = 27 \). (Condition 1 satisfied) |
| 4 years from now | \( 32 + 4 = 36 \) | \( 14 + 4 = 18 \) | Is \( 36 = 2 \times 18 \)? Yes, \( 36 = 36 \). (Condition 2 satisfied) |
Both conditions are satisfied with Ram's present age as 32 and Shyam's present age as 14. Therefore, the present age of Ram is 32 years.
Revision Table: Ram Shyam Age Problem
| Concept | Description | Application in Problem |
|---|---|---|
| Representing Ages | Using variables for unknown quantities (present ages). | Using \( R \) for Ram's present age and \( S \) for Shyam's present age. |
| Ages in Past/Future | Age changes by adding/subtracting years from the present age. | \( R-5 \), \( S-5 \) (5 years ago); \( R+4 \), \( S+4 \) (4 years from now). |
| Forming Equations | Translating word statements into algebraic relationships. | \( R-5 = 3(S-5) \), \( R+4 = 2(S+4) \). |
| Solving System of Equations | Finding the values of variables that satisfy all equations. | Using substitution or elimination to find \( R \) and \( S \). |
| Verification | Checking if the calculated values satisfy the original conditions. | Plugging \( R=32 \) and \( S=14 \) back into the initial statements. |
Additional Information: Solving Age Problems
Age problems are common in algebra and often involve setting up and solving linear equations. Here are some tips:
- Always start by defining variables for the present ages of the individuals involved.
- Carefully read the conditions and determine if they refer to past ages (subtract years) or future ages (add years).
- Write down each condition as an algebraic equation. Pay close attention to phrases like "times as old as", "older than", "younger than", etc.
- If you have two variables, you will typically need two independent equations to find a unique solution.
- Solve the system of equations using methods like substitution or elimination.
- After finding the values, always check your answer by plugging the calculated ages back into the original word problem statements to ensure they hold true. This helps catch calculation errors.
Understanding how to translate words into mathematical expressions is key to solving age problems effectively.
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