The sum of the present ages of Ram and Shyam is 75 years. Ten years ago, Ram's age was 4 times the age of Shyam. Find the difference between their present ages (in years)

Options

A. 22
B. 23
C. 33
D. 30

Explanation

Let's solve the problem step by step.

### Step 1: Define the variables
Let Ram's present age be \( R \) and Shyam's present age be \( S \).

### Step 2: Set up the equations
From the problem, we know:
1. The sum of their ages is 75 years:
\[ R + S = 75 \]

2. Ten years ago, Ram's age was 4 times Shyam's age:
\[ R - 10 = 4(S - 10) \]

### Step 3: Simplify the second equation
Let's simplify the second equation:
\[ R - 10 = 4(S - 10) \]
Expanding the right side:
\[ R - 10 = 4S - 40 \]
Now, rearranging gives:
\[ R = 4S - 30 \]

### Step 4: Substitute the value of \( R \) in the first equation
Now we will substitute \( R \) from the second equation into the first equation:
\[ (4S - 30) + S = 75 \]
Combining like terms:
\[ 5S - 30 = 75 \]
Adding 30 to both sides:
\[ 5S = 105 \]
Dividing both sides by 5:
\[ S = 21 \]

### Step 5: Find Ram's age
Now we can find Ram's age using the first equation:
\[ R + 21 = 75 \]
Subtracting 21 from both sides:
\[ R = 54 \]

### Step 6: Find the difference between their ages
Now we find the difference between their present ages:
\[ R - S = 54 - 21 = 33 \]

### Final Answer
The difference between their present ages is **33 years**.

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