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The problem states:

- Radhika spends \( \frac{1}{2} \) of her salary on installments, \( \frac{1}{3} \) on conveyance, and donates 500 rupees.
- After all expenses, she is left with 10,000 rupees.
- We need to determine her total salary and the amount she spent on installments.

### Solution

Let her salary be \( x \) rupees.

1. **Amount spent on installments**: \( \frac{1}{2}x \)
2. **Amount spent on conveyance**: \( \frac{1}{3}x \)
3. **Amount donated**: 500 rupees
4. **Amount left after all expenses**: 10,000 rupees

So, we have:
\[
x - \left(\frac{1}{2}x + \frac{1}{3}x + 500\right) = 10,000
\]

#### Step 1: Simplify the Equation

Combine terms:
\[
x - \left(\frac{1}{2}x + \frac{1}{3}x\right) - 500 = 10,000
\]

#### Step 2: Find a Common Denominator

The terms inside the parentheses can be combined by finding a common denominator:
\[
\frac{1}{2}x + \frac{1}{3}x = \frac{3}{6}x + \frac{2}{6}x = \frac{5}{6}x
\]

So the equation becomes:
\[
x - \frac{5}{6}x - 500 = 10,000
\]

#### Step 3: Simplify Further

Now, \( x - \frac{5}{6}x = \frac{1}{6}x \), so:
\[
\frac{1}{6}x - 500 = 10,000
\]

#### Step 4: Isolate \( x \)

Add 500 to both sides:
\[
\frac{1}{6}x = 10,500
\]

Multiply both sides by 6:
\[
x = 10,500 \times 6
\]
\[
x = 63,000
\]

### Answer
- Radhika's salary is 63,000 rupees.
- The amount spent on installments is \( \frac{1}{2} \times 63,000 = 31,500 \) rupees.

Would you like further details or have any questions?

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#### Related Questions

1. What amount did she spend on conveyance?
2. How much did she donate as a percentage of her salary?
3. If her salary increased by 10%, how much would she spend on installments now?
4. What fraction of her salary was left after conveyance, installment, and donation?
5. How would the remaining balance change if the donation was increased to 1,000 rupees?

#### Tip
When dealing with fraction-based expenses, always combine terms under a common denominator to simplify.

Radhika spent 1/2 of her salary paying installments, 1/3 on conveyance, and gave ₹500 as a donation. If she is left with ₹10,000, how much is her salary and how much did she spend on installments?

Learn how to calculate Radhika's total salary given her spending fractions on installments, conveyance, and a donation. This problem involves algebraic equations with fractions and is suitable for Grades 7-9 students.

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