When the price of sugar gets raised by 20%, a person increases his expenditure on sugar only by 12%. By what percentage (correct up to two decimal places) should he reduce his consumption of sugar so as to be able to maintain the same level of expenditure?

Options

A. `6.25%`
B. `6.67%`
C. `7.05%`
D. `5.25%`

Answer

To solve the problem, we need to determine the percentage by which a person should reduce his consumption of sugar after a price increase. Let's break down the solution step by step.

### Step 1: Understand the Initial Situation
Let the initial price of sugar be \( P \) per kg and the initial quantity consumed be \( Q \) kg. Therefore, the initial expenditure on sugar is:
\[ \text{Initial Expenditure} = P \times Q \]

### Step 2: Calculate the New Price
When the price of sugar is raised by 20%, the new price \( P' \) becomes:
\[ P' = P + 0.20P = 1.20P \]

### Step 3: Determine the New Expenditure
The person increases his expenditure on sugar by 12%. Therefore, the new expenditure \( E' \) is:
\[ E' = \text{Initial Expenditure} + 0.12 \times \text{Initial Expenditure} = 1.12 \times (P \times Q) \]

### Step 4: Set Up the Equation for New Quantity
Let the new quantity of sugar consumed be \( Q' \) kg. The new expenditure can also be expressed as:
\[ E' = P' \times Q' = 1.20P \times Q' \]

### Step 5: Equate the Two Expressions for New Expenditure
Now we can set the two expressions for the new expenditure equal to each other:
\[ 1.12 \times (P \times Q) = 1.20P \times Q' \]

### Step 6: Simplify the Equation
Dividing both sides by \( P \) (assuming \( P \neq 0 \)):
\[ 1.12Q = 1.20Q' \]

### Step 7: Solve for New Quantity \( Q' \)
Rearranging gives:
\[ Q' = \frac{1.12Q}{1.20} \]
\[ Q' = \frac{1.12}{1.20}Q \]
\[ Q' = \frac{112}{120}Q = \frac{28}{30}Q = \frac{14}{15}Q \]

### Step 8: Calculate the Reduction in Consumption
The reduction in consumption is:
\[ \text{Reduction} = Q - Q' = Q - \frac{14}{15}Q = \frac{1}{15}Q \]

### Step 9: Calculate the Percentage Reduction
The percentage reduction in consumption is given by:
\[ \text{Percentage Reduction} = \left( \frac{\text{Reduction}}{Q} \right) \times 100 = \left( \frac{\frac{1}{15}Q}{Q} \right) \times 100 = \frac{1}{15} \times 100 = \frac{100}{15} \approx 6.67\% \]

Thus, the person should reduce his consumption of sugar by approximately **6.67%**.