The price of a certain item is increased by 15%. If a consumer wants to keep his expenditure on the item the same as before, how much percent must he reduce his consumption of that item?

Options

A. 0.15
B. 13 `1/3` %
C. 16 `2/3` %
D. `20/23` %

Answer

To solve the problem step by step, we will follow these instructions:

### Step-by-Step Solution:

1. **Understand the Problem**:
- The price of an item increases by 15%.
- We need to find out how much the consumer must reduce their consumption to keep their expenditure the same.

**Hint**: Remember that expenditure is the product of price and quantity consumed.

2. **Define Variables**:
- Let the original price of the item be \( P \).
- Let the original quantity consumed be \( Q \).
- The original expenditure \( E \) can be expressed as:
\[ E = P \times Q \]

**Hint**: Write down the formula for expenditure to see how it relates to price and quantity.

3. **Calculate the New Price**:
- The new price after a 15% increase is:
\[ \text{New Price} = P + 0.15P = 1.15P \]

**Hint**: A 15% increase means you multiply the original price by 1.15.

4. **Set Up the Equation for Constant Expenditure**:
- Let the new quantity consumed be \( Q' \). Since the expenditure remains constant:
\[ E = \text{New Price} \times Q' \]
- Therefore, we can write:
\[ P \times Q = 1.15P \times Q' \]

**Hint**: Since expenditure is constant, set the two expressions for expenditure equal to each other.

5. **Simplify the Equation**:
- Dividing both sides by \( P \) (assuming \( P \neq 0 \)):
\[ Q = 1.15 \times Q' \]
- Rearranging gives:
\[ Q' = \frac{Q}{1.15} \]

**Hint**: Isolate \( Q' \) to find the new quantity in terms of the original quantity.

6. **Calculate the Reduction in Consumption**:
- The reduction in consumption is:
\[ \text{Reduction} = Q - Q' = Q - \frac{Q}{1.15} \]
- This can be simplified to:
\[ \text{Reduction} = Q \left(1 - \frac{1}{1.15}\right) = Q \left(\frac{1.15 - 1}{1.15}\right) = Q \left(\frac{0.15}{1.15}\right) \]

**Hint**: Factor out \( Q \) to simplify the reduction calculation.

7. **Calculate the Percentage Reduction**:
- The percentage reduction in consumption is given by:
\[ \text{Percentage Reduction} = \left(\frac{\text{Reduction}}{Q}\right) \times 100 = \left(\frac{0.15}{1.15}\right) \times 100 \]
- Calculating this gives:
\[ \text{Percentage Reduction} \approx 13.04\% \]

**Hint**: To find the percentage, divide the reduction by the original quantity and multiply by 100.

### Final Answer:
The consumer must reduce their consumption by approximately **13.04%** to keep their expenditure the same.