Arnav and Prerna are aged 8 years and 10 years respectively. Their mother divides Rs 90 in the ratio of their ages How much does each get ?

Answer

To solve the problem of how much Arnav and Prerna receive from the Rs 90 divided in the ratio of their ages, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Ages**:
- Arnav's age = 8 years
- Prerna's age = 10 years

2. **Find the Ratio of Their Ages**:
- The ratio of Arnav's age to Prerna's age is 8:10.
- To simplify this ratio, divide both numbers by their greatest common divisor (GCD), which is 2.
- Simplified ratio = 8 ÷ 2 : 10 ÷ 2 = 4:5.

3. **Set Up the Equation for the Total Amount**:
- Let the common multiple be represented as \( x \).
- The amount Arnav receives = \( 4x \).
- The amount Prerna receives = \( 5x \).
- Therefore, the total amount can be expressed as:
\[ 4x + 5x = 90 \]

4. **Combine Like Terms**:
- Combine the terms on the left side:
\[ 9x = 90 \]

5. **Solve for \( x \)**:
- To find \( x \), divide both sides by 9:
\[ x = \frac{90}{9} = 10 \]

6. **Calculate the Amounts for Arnav and Prerna**:
- Amount received by Arnav:
\[ 4x = 4 \times 10 = 40 \text{ rupees} \]
- Amount received by Prerna:
\[ 5x = 5 \times 10 = 50 \text{ rupees} \]

7. **Verify the Total Amount**:
- Total amount = Amount received by Arnav + Amount received by Prerna:
\[ 40 + 50 = 90 \text{ rupees} \]
- This confirms that our calculations are correct.

### Final Answer:
- Arnav receives **40 rupees**.
- Prerna receives **50 rupees**.

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