y is inversely proportional to x and y = 4, when x = $\frac{1}{2}$, find x when y = 10

1. Analyze the question

We are given that $y$ is inversely proportional to $x$. We are provided with an initial pair of values ($y=4, x=\frac{1}{2}$) to find the constant of proportionality, and then asked to find the new value of $x$ when $y=10$.

2. Explain the concept

When two variables are inversely proportional, it means that as one increases, the other decreases in such a way that their product remains constant. Mathematically, this is expressed as:

Where $k$ is the constant of proportionality. To solve this, we first find $k$ using the given values, then use that $k$ to find the unknown variable.

3. Show the solution

Step 1: Find the constant $k$ Using the formula $y = \frac{k}{x}$:

To solve for $k$, multiply both sides by $\frac{1}{2}$:

Step 2: Find $x$ when $y = 10$ Now that we know $k = 2$, our equation is $y = \frac{2}{x}$. Substitute $y = 10$ into the equation:

Multiply both sides by $x$:

Divide both sides by 10:

Simplify the fraction:

4. Identify the answer

Comparing our result to the options provided: A. 2 B. 10 C. $\frac{1}{5}$ D. $\frac{1}{10}$

The correct option is C.

5. Tip

In inverse proportion problems, always remember the "Product Rule": $x_1y_1 = x_2y_2$. Using this shortcut: $(4) \times (\frac{1}{2}) = (10) \times (x)$ $2 = 10x$ $x = \frac{2}{10} = \frac{1}{5}$ This method is much faster for JAMB exams where time management is key!