The price of sugar is increased by 20%. If the expenditure on sugar has to be kept the same as eartler, the ratio between the reduction in consumption and the original consumption is

Options

A. `1:3`
B. `1:4`
C. `1:4`
D. `1:5`

Answer

To solve the problem step by step, we can follow these steps:

### Step 1: Understand the Problem
The price of sugar has increased by 20%. We need to find the ratio between the reduction in consumption and the original consumption while keeping the expenditure constant.

### Step 2: Define Variables
Let:
- \( P \) = Original price of sugar
- \( C \) = Original consumption of sugar
- \( E \) = Original expenditure on sugar

From the definition of expenditure, we have:
\[ E = P \times C \]

### Step 3: Calculate the New Price
Since the price of sugar increases by 20%, the new price \( P' \) can be calculated as:
\[ P' = P + 0.20P = 1.20P \]

### Step 4: Maintain the Same Expenditure
To keep the expenditure the same after the price increase, we can denote the new consumption as \( C' \). The new expenditure must equal the original expenditure:
\[ E = P' \times C' \]
Substituting for \( E \) and \( P' \):
\[ P \times C = 1.20P \times C' \]

### Step 5: Simplify the Equation
We can cancel \( P \) from both sides (assuming \( P \neq 0 \)):
\[ C = 1.20 \times C' \]

### Step 6: Solve for New Consumption
Rearranging gives us:
\[ C' = \frac{C}{1.20} \]

### Step 7: Calculate the Reduction in Consumption
The reduction in consumption \( R \) can be calculated as:
\[ R = C - C' = C - \frac{C}{1.20} \]
\[ R = C \left(1 - \frac{1}{1.20}\right) \]
\[ R = C \left(\frac{1.20 - 1}{1.20}\right) \]
\[ R = C \left(\frac{0.20}{1.20}\right) \]
\[ R = \frac{C}{6} \]

### Step 8: Find the Ratio
Now, we need to find the ratio of the reduction in consumption \( R \) to the original consumption \( C \):
\[ \text{Ratio} = \frac{R}{C} = \frac{\frac{C}{6}}{C} = \frac{1}{6} \]

### Final Answer
The ratio between the reduction in consumption and the original consumption is \( \frac{1}{6} \).

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