If the duty on an article is reduced by 40% of its present rate, by how much per cent must its consumption increase in order that the revenue remains unaltered?

Options

A. 0.6
B. 62 `1/3` %
C. 0.72
D. 66 `2/3` %

Answer

To solve the problem step by step, we will first define the variables and then calculate the required percentage increase in consumption to keep the revenue unaltered after a reduction in duty.

### Step 1: Define the Variables
Let:
- \( R \) = original rate of duty (in percentage)
- \( C \) = original consumption (in units)
- \( D \) = original revenue

The original revenue can be expressed as:
\[ D = R \times C \]

### Step 2: Calculate the New Rate of Duty
The duty is reduced by 40% of its present rate. Therefore, the new rate of duty (\( R' \)) will be:
\[ R' = R - 0.4R = 0.6R \]

### Step 3: Set Up the Equation for Unaltered Revenue
To keep the revenue unaltered, the new revenue (\( D' \)) must equal the original revenue (\( D \)). The new revenue can be expressed as:
\[ D' = R' \times C' \]
Where \( C' \) is the new consumption.

Setting the revenues equal gives us:
\[ R \times C = (0.6R) \times C' \]

### Step 4: Simplify the Equation
We can simplify the equation by dividing both sides by \( R \) (assuming \( R \neq 0 \)):
\[ C = 0.6 \times C' \]

### Step 5: Solve for New Consumption
Rearranging the equation to find \( C' \):
\[ C' = \frac{C}{0.6} \]
\[ C' = \frac{C}{\frac{3}{5}} = \frac{5C}{3} \]

### Step 6: Calculate the Increase in Consumption
The increase in consumption is:
\[ \text{Increase} = C' - C = \frac{5C}{3} - C \]
\[ = \frac{5C}{3} - \frac{3C}{3} = \frac{2C}{3} \]

### Step 7: Calculate the Percentage Increase
To find the percentage increase in consumption, we use the formula:
\[ \text{Percentage Increase} = \left( \frac{\text{Increase}}{C} \right) \times 100 \]
Substituting the increase:
\[ \text{Percentage Increase} = \left( \frac{\frac{2C}{3}}{C} \right) \times 100 \]
\[ = \frac{2}{3} \times 100 = \frac{200}{3} \approx 66.67\% \]

### Final Answer
Thus, the consumption must increase by approximately \( 66 \frac{2}{3}\% \) to keep the revenue unaltered.

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