If the price of milk falls down by 20%, by how much per cent must a householder increase its consumption, so as not to decrease his expenditure on this item?

Answer

To solve the problem step by step, we need to determine how much a householder must increase their consumption of milk after a 20% decrease in price to maintain the same expenditure.

### Step 1: Understand the given information
- The price of milk has decreased by 20%.
- We need to find out by what percentage the consumption must increase to keep the expenditure constant.

**Hint:** Remember that expenditure is calculated as the product of price and quantity consumed.

### Step 2: Define the variables
Let:
- \( P \) = original price of milk
- \( C \) = original consumption of milk
- \( E \) = original expenditure on milk

The original expenditure can be expressed as:
\[ E = P \times C \]

**Hint:** Think about how the expenditure changes when the price changes.

### Step 3: Calculate the new price of milk
Since the price of milk falls by 20%, the new price \( P' \) can be calculated as:
\[ P' = P - (20\% \text{ of } P) = P - 0.2P = 0.8P \]

**Hint:** A decrease of 20% means you are left with 80% of the original price.

### Step 4: Set up the equation for new expenditure
Let the new consumption be \( C' \). The new expenditure must equal the original expenditure to keep it constant:
\[ E = P' \times C' \]

Substituting the new price:
\[ P \times C = (0.8P) \times C' \]

**Hint:** You can simplify this equation by canceling out \( P \) from both sides.

### Step 5: Simplify the equation
Cancel \( P \) from both sides (assuming \( P \neq 0 \)):
\[ C = 0.8 \times C' \]

Now, we can express \( C' \) in terms of \( C \):
\[ C' = \frac{C}{0.8} = 1.25C \]

**Hint:** This shows how much the new consumption \( C' \) is compared to the original consumption \( C \).

### Step 6: Calculate the percentage increase in consumption
To find the percentage increase:
\[ \text{Percentage Increase} = \left( \frac{C' - C}{C} \right) \times 100\% \]

Substituting \( C' \):
\[ \text{Percentage Increase} = \left( \frac{1.25C - C}{C} \right) \times 100\% = \left( \frac{0.25C}{C} \right) \times 100\% = 25\% \]

**Hint:** The increase is calculated based on the original consumption.

### Final Answer
The householder must increase their consumption of milk by **25%** to maintain the same expenditure after a 20% decrease in price.

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