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 find the sum of the present ages of Ram and Shyam, we can follow these steps:

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

### Step 2: Set up the equation based on the information given
According to the problem, 5 years ago, the ratio of their ages was 3:5. Therefore, we can express their ages 5 years ago as:
- Ram's age 5 years ago: \( R - 5 = (S - 20) - 5 = S - 25 \)
- Shyam's age 5 years ago: \( S - 5 \)

The ratio of their ages 5 years ago can be written as:
\[ \frac{R - 5}{S - 5} = \frac{3}{5} \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 5(R - 5) = 3(S - 5) \]
Substituting \( R = S - 20 \) into the equation:
\[ 5((S - 20) - 5) = 3(S - 5) \]
This simplifies to:
\[ 5(S - 25) = 3(S - 5) \]

### Step 4: Expand both sides of the equation
Expanding both sides, we get:
\[ 5S - 125 = 3S - 15 \]

### Step 5: Rearrange the equation to solve for \( S \)
Now, we can rearrange the equation:
\[ 5S - 3S = 125 - 15 \]
\[ 2S = 110 \]
\[ S = 55 \]

### Step 6: Find Ram's age
Now that we have Shyam's age, we can find Ram's age:
\[ R = S - 20 = 55 - 20 = 35 \]

### Step 7: Calculate the sum of their present ages
Finally, we can find the sum of their present ages:
\[ \text{Sum} = R + S = 35 + 55 = 90 \]

Thus, the sum of their present ages is **90 years**.

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