Price of sugar rises by 20% By how much percent should the consumption of sugar be reduced so that the expenditure does not change?

Options

A. 20
B. 10
C. `16(2)/(3)`
D. 15

Answer

To solve the problem of how much percent the consumption of sugar should be reduced to keep the expenditure constant after a 20% increase in price, we can follow these steps:

### Step 1: Understand the relationship between price, consumption, and expenditure.
Expenditure is defined as:
\[ \text{Expenditure} = \text{Price} \times \text{Consumption} \]
To keep the expenditure constant, if the price increases, the consumption must decrease.

### Step 2: Calculate the new price after a 20% increase.
Let the original price of sugar be \( P \). After a 20% increase, the new price \( P' \) is:
\[ P' = P + 0.2P = 1.2P \]

### Step 3: Set up the ratio of original and new prices.
The ratio of the original price to the new price is:
\[ \text{Ratio} = \frac{P}{P'} = \frac{P}{1.2P} = \frac{1}{1.2} = \frac{5}{6} \]

### Step 4: Use the inverse relationship to find the ratio of consumption.
Since expenditure remains constant, the ratio of consumption must be the inverse of the price ratio:
\[ \text{Consumption Ratio} = \frac{C'}{C} = \frac{1.2}{1} = \frac{6}{5} \]
Where \( C \) is the original consumption and \( C' \) is the new consumption.

### Step 5: Express the new consumption in terms of the original consumption.
Let the original consumption be \( C \). Then the new consumption \( C' \) can be expressed as:
\[ C' = \frac{5}{6}C \]

### Step 6: Calculate the reduction in consumption.
The reduction in consumption is given by:
\[ \text{Reduction} = C - C' = C - \frac{5}{6}C = \frac{1}{6}C \]

### Step 7: Calculate the percentage reduction in consumption.
To find the percentage reduction, we use the formula:
\[ \text{Percentage Reduction} = \left( \frac{\text{Reduction}}{C} \right) \times 100 = \left( \frac{\frac{1}{6}C}{C} \right) \times 100 = \frac{1}{6} \times 100 = \frac{100}{6} \approx 16.67\% \]

### Conclusion:
The percentage by which the consumption of sugar should be reduced to keep the expenditure constant is approximately \( \frac{100}{6} \) or \( 16 \frac{2}{3} \% \).