x is inversely proportional to y. If x is increased by 37.5% then find % change in the value of y?

Options

A. `27.27%`
B. `33.33%`
C. 0.45
D. `31.25%`

Answer

To solve the problem step by step, we will follow the mathematical principles of inverse proportionality and percentage change.

### Step 1: Understand the relationship
Given that \( x \) is inversely proportional to \( y \), we can express this relationship mathematically as:
\[ x \cdot y = k \]
where \( k \) is a constant.

### Step 2: Determine the initial value of \( x \)
Let the initial value of \( x \) be \( x_0 \) and the initial value of \( y \) be \( y_0 \). Therefore, we have:
\[ x_0 \cdot y_0 = k \]

### Step 3: Calculate the new value of \( x \)
If \( x \) is increased by 37.5%, the new value of \( x \) (denoted as \( x_1 \)) can be calculated as:
\[ x_1 = x_0 + 0.375 \cdot x_0 = 1.375 \cdot x_0 \]

### Step 4: Find the new value of \( y \)
Using the inverse proportionality, we can find the new value of \( y \) (denoted as \( y_1 \)):
\[ x_1 \cdot y_1 = k \]
Substituting \( x_1 \):
\[ 1.375 \cdot x_0 \cdot y_1 = k \]
Since \( k = x_0 \cdot y_0 \), we can substitute \( k \) into the equation:
\[ 1.375 \cdot x_0 \cdot y_1 = x_0 \cdot y_0 \]

### Step 5: Simplify to find \( y_1 \)
Dividing both sides by \( x_0 \) (assuming \( x_0 \neq 0 \)):
\[ 1.375 \cdot y_1 = y_0 \]
Now, solving for \( y_1 \):
\[ y_1 = \frac{y_0}{1.375} \]

### Step 6: Calculate the percentage change in \( y \)
The percentage change in \( y \) can be calculated using the formula:
\[ \text{Percentage Change} = \frac{y_1 - y_0}{y_0} \times 100 \]
Substituting \( y_1 \):
\[ \text{Percentage Change} = \frac{\frac{y_0}{1.375} - y_0}{y_0} \times 100 \]
This simplifies to:
\[ \text{Percentage Change} = \left(\frac{1}{1.375} - 1\right) \times 100 \]

### Step 7: Calculate the numerical value
Calculating \( \frac{1}{1.375} \):
\[ \frac{1}{1.375} \approx 0.7273 \]
Thus,
\[ \text{Percentage Change} = (0.7273 - 1) \times 100 \approx -27.27\% \]

### Final Answer
The percentage change in the value of \( y \) is approximately **-27.27%**.

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