If the price of tea falls down by 6%, by how much per cent must a householder increase its consumption, so as not to decrease expenditure?

Options

A. `5(16)/(47)`
B. `4(18)/(67)`
C. `6(18)/(17)`
D. `6(17)/(47)`

Answer

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

### Step 1: Understand the Problem
The price of tea has decreased by 6%, and we need to find out by what percentage the householder must increase their consumption to keep their expenditure the same.

### Step 2: Set Up the Original Price
Let the original price of tea be \( P = 100 \) (this makes calculations easier).

### Step 3: Calculate the New Price
Since the price falls by 6%, we can calculate the new price as follows:
\[ \text{New Price} = P - 6\% \text{ of } P = 100 - 6\% \text{ of } 100 \]
Calculating \( 6\% \text{ of } 100 \):
\[ 6\% \text{ of } 100 = \frac{6}{100} \times 100 = 6 \]
Thus, the new price is:
\[ \text{New Price} = 100 - 6 = 94 \]

### Step 4: Determine the Change in Price
Now we need to find out how much the consumption must increase. The original price was \( 100 \) and the new price is \( 94 \).

### Step 5: Set Up the Consumption Equation
Let the original consumption be \( C \). The expenditure before the price drop is:
\[ \text{Expenditure} = \text{Price} \times \text{Consumption} = 100 \times C \]
After the price drop, if the consumption increases to \( C' \), the expenditure must remain the same:
\[ \text{Expenditure} = \text{New Price} \times \text{New Consumption} = 94 \times C' \]
Setting the two expenditures equal gives:
\[ 100C = 94C' \]

### Step 6: Solve for New Consumption
Rearranging the equation to find \( C' \):
\[ C' = \frac{100C}{94} \]

### Step 7: Calculate the Increase in Consumption
The increase in consumption is:
\[ \text{Increase} = C' - C = \frac{100C}{94} - C \]
To simplify:
\[ \text{Increase} = \left(\frac{100C - 94C}{94}\right) = \frac{6C}{94} \]

### Step 8: Calculate the Percentage Increase
To find the percentage increase in consumption:
\[ \text{Percentage Increase} = \left(\frac{\text{Increase}}{C}\right) \times 100 = \left(\frac{6C/94}{C}\right) \times 100 = \frac{6 \times 100}{94} \approx 6.38\% \]

### Final Answer
The householder must increase their consumption by approximately \( 6.38\% \) to keep their expenditure the same.

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