The price of coffee rose by 20% following shortange in availability in the market. As a result , Carlose decided to mitigate increasing expenses by reducing his coffee intake. By what percentage should Carlos reduce his intake so that there is no effect on the expenditure on coffe ?

Options

A. 0.05
B. 0.1667
C. 0.2
D. 0.25

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Define the Initial Variables
Let the initial price of coffee per pack be \( X_1 \) and the initial number of packs Carlos buys be \( Y_1 \).

### Step 2: Calculate the New Price After the Increase
The price of coffee rose by 20%. Therefore, the new price \( X_2 \) can be calculated as:
\[ X_2 = X_1 + \left( \frac{20}{100} \times X_1 \right) = X_1 + 0.2X_1 = 1.2X_1 \]

### Step 3: Set Up the Expenditure Equation
Carlos wants to keep his expenditure the same before and after the price increase. The initial expenditure is:
\[ \text{Initial Expenditure} = X_1 \times Y_1 \]
The final expenditure after the price increase will be:
\[ \text{Final Expenditure} = X_2 \times Y_2 \]
Setting these equal gives us:
\[ X_1 \times Y_1 = X_2 \times Y_2 \]

### Step 4: Substitute the New Price into the Expenditure Equation
Substituting \( X_2 = 1.2X_1 \) into the expenditure equation:
\[ X_1 \times Y_1 = 1.2X_1 \times Y_2 \]

### Step 5: Cancel \( X_1 \) from Both Sides
Assuming \( X_1 \) is not zero, we can divide both sides by \( X_1 \):
\[ Y_1 = 1.2 \times Y_2 \]

### Step 6: Solve for \( Y_2 \)
Rearranging gives:
\[ Y_2 = \frac{Y_1}{1.2} = \frac{Y_1}{\frac{6}{5}} = \frac{5Y_1}{6} \]

### Step 7: Calculate the Reduction in Intake
The reduction in intake is:
\[ \text{Reduction} = Y_1 - Y_2 = Y_1 - \frac{5Y_1}{6} = \frac{Y_1}{6} \]

### Step 8: Find the Percentage Reduction
The percentage reduction in intake is given by:
\[ \text{Percentage Reduction} = \frac{\text{Reduction}}{Y_1} \times 100 = \frac{\frac{Y_1}{6}}{Y_1} \times 100 = \frac{1}{6} \times 100 \approx 16.67\% \]

### Conclusion
Carlos should reduce his coffee intake by approximately **16.67%** to maintain the same expenditure.

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