Harsha is 40 yr old and Rith is 60 yr old. How many years ago was the ratio of their ages `3 : 5`?

Options

A. 10 yr
B. 20 yr
C. 37 yr
D. 5 yr

Answer

To solve the problem of finding out how many years ago the ratio of Harsha's and Rith's ages was 3:5, we can follow these steps:

### Step 1: Define the current ages
- Harsha's current age = 40 years
- Rith's current age = 60 years

### Step 2: Define the variable for years ago
Let \( x \) be the number of years ago we are looking for.

### Step 3: Write the ages in terms of \( x \)
- Harsha's age \( x \) years ago = \( 40 - x \)
- Rith's age \( x \) years ago = \( 60 - x \)

### Step 4: Set up the ratio equation
According to the problem, the ratio of their ages \( x \) years ago was \( 3:5 \). Therefore, we can write the equation:
\[ \frac{40 - x}{60 - x} = \frac{3}{5} \]

### Step 5: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 5(40 - x) = 3(60 - x) \]

### Step 6: Distribute both sides
Distributing the numbers gives:
\[ 200 - 5x = 180 - 3x \]

### Step 7: Rearrange the equation
Now, we will rearrange the equation to isolate \( x \):
\[ 200 - 180 = 5x - 3x \]
\[ 20 = 2x \]

### Step 8: Solve for \( x \)
Dividing both sides by 2 gives:
\[ x = 10 \]

### Conclusion
Thus, the number of years ago when the ratio of their ages was \( 3:5 \) is **10 years**.

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