The price of pomegranates is decreased by `16(2)/(3)%` . By how much percent must a customer increase his consumption so that his expenditure remains the same ?

Options

A. `20%`
B. `22%`
C. `26%`
D. `24%`

Answer

To solve the problem, we need to determine how much a customer must increase their consumption of pomegranates to keep their expenditure the same after a price decrease of \(16\frac{2}{3}\%\).

### Step-by-Step Solution:

1. **Convert the Percentage to a Fraction:**
The percentage decrease is \(16\frac{2}{3}\%\). We can convert this to an improper fraction:
\[ 16\frac{2}{3} = \frac{50}{3}\% \]
To convert this percentage to a decimal, we divide by 100:
\[ \frac{50}{3} \div 100 = \frac{50}{300} = \frac{1}{6} \]

2. **Calculate the New Price:**
Let's assume the original price of pomegranates is \(P\). After a decrease of \(\frac{1}{6}\) of the price, the new price \(P'\) can be calculated as:
\[ P' = P - \frac{1}{6}P = \frac{5}{6}P \]

3. **Set Up the Expenditure Equation:**
Let \(C\) be the original consumption. The original expenditure \(E\) can be expressed as:
\[ E = P \times C \]
After the price decrease, if the consumption increases to \(C'\), the new expenditure remains the same:
\[ E = P' \times C' = \frac{5}{6}P \times C' \]

4. **Equate the Expenditures:**
Since the expenditures are equal:
\[ P \times C = \frac{5}{6}P \times C' \]
We can cancel \(P\) from both sides (assuming \(P \neq 0\)):
\[ C = \frac{5}{6}C' \]

5. **Solve for the New Consumption:**
Rearranging gives:
\[ C' = \frac{6}{5}C \]

6. **Calculate the Increase in Consumption:**
The increase in consumption is:
\[ \text{Increase} = C' - C = \frac{6}{5}C - C = \left(\frac{6}{5} - 1\right)C = \frac{1}{5}C \]

7. **Calculate the Percentage Increase:**
The percentage increase in consumption is given by:
\[ \text{Percentage Increase} = \left(\frac{\text{Increase}}{C}\right) \times 100 = \left(\frac{\frac{1}{5}C}{C}\right) \times 100 = 20\% \]

### Final Answer:
The customer must increase their consumption by **20%** to keep their expenditure the same.

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