If the price of a commodity is increased by 50% by what fraction must its consumption be reduced so as to keep the same expenditure on its consumption ?

Options

A. `(1)/(4)`
B. `(1)/(3)`
C. `(1)/(2)`
D. `(2)/(3)`

Answer

To solve the problem, we need to determine by what fraction the consumption of a commodity must be reduced to maintain the same expenditure after a 50% increase in its price.

### Step-by-Step Solution:

1. **Understand the Initial Situation**:
Let the initial price of the commodity be \( P \) and the initial consumption be \( C \). Therefore, the initial expenditure \( E \) can be expressed as:
\[ E = P \times C \]

2. **Calculate the New Price After Increase**:
If the price of the commodity is increased by 50%, the new price \( P' \) becomes:
\[ P' = P + 0.5P = 1.5P \]

3. **Set Up the Equation for Constant Expenditure**:
To keep the expenditure the same after the price increase, the new expenditure \( E' \) must equal the original expenditure \( E \). If the new consumption is \( C' \), we have:
\[ E' = P' \times C' = 1.5P \times C' \]
Setting this equal to the original expenditure:
\[ 1.5P \times C' = P \times C \]

4. **Simplify the Equation**:
Dividing both sides of the equation by \( P \) (assuming \( P \neq 0 \)):
\[ 1.5C' = C \]
Now, solve for \( C' \):
\[ C' = \frac{C}{1.5} = \frac{C}{\frac{3}{2}} = \frac{2C}{3} \]

5. **Determine the Reduction in Consumption**:
The reduction in consumption can be calculated as:
\[ \text{Reduction} = C - C' = C - \frac{2C}{3} = \frac{3C}{3} - \frac{2C}{3} = \frac{C}{3} \]

6. **Calculate the Fraction of Reduction**:
The fraction by which the consumption must be reduced is:
\[ \text{Fraction of reduction} = \frac{\text{Reduction}}{C} = \frac{\frac{C}{3}}{C} = \frac{1}{3} \]

### Final Answer:
The consumption must be reduced by \( \frac{1}{3} \) to keep the same expenditure after a 50% increase in price.

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