Ram is 20 years younger than Shyam. 5 years ago, the ratio of their ages was 3:5. The sum of their present ages is :

Options

A. 75 years
B. 80 years
C. 90 years
D. 95 years

Answer

To solve the problem step by step, we can follow these steps:

### Step 1: Define the variables
Let Shyam's present age be \( S \) years. Since Ram is 20 years younger than Shyam, we can express Ram's present age as:
\[ R = S - 20 \]

### Step 2: Set up the equation based on the ratio of their ages 5 years ago
Five years ago, Shyam's age would have been \( S - 5 \) and Ram's age would have been \( R - 5 \). According to the problem, the ratio of their ages 5 years ago was 3:5. Therefore, we can write the equation:
\[ \frac{R - 5}{S - 5} = \frac{3}{5} \]

### Step 3: Substitute Ram's age in the equation
Substituting \( R = S - 20 \) into the equation gives:
\[ \frac{(S - 20) - 5}{S - 5} = \frac{3}{5} \]
This simplifies to:
\[ \frac{S - 25}{S - 5} = \frac{3}{5} \]

### Step 4: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ 5(S - 25) = 3(S - 5) \]

### Step 5: Expand both sides
Expanding both sides results in:
\[ 5S - 125 = 3S - 15 \]

### Step 6: Rearrange the equation to solve for \( S \)
Rearranging gives:
\[ 5S - 3S = 125 - 15 \]
\[ 2S = 110 \]
\[ S = 55 \]

### Step 7: Find Ram's age
Now that we have Shyam's age, we can find Ram's age:
\[ R = S - 20 = 55 - 20 = 35 \]

### Step 8: Calculate the sum of their present ages
The sum of their present ages is:
\[ S + R = 55 + 35 = 90 \]

### Final Answer
The sum of their present ages is **90 years**.

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