Ratio and Proportion: Question 3

Worked solution

Setting up the relationship

The force is inversely proportional to the square of the distance, so

$F = \frac{k}{d^{2}}$

where $k$ is a positive constant. The word “square” is essential: the distance appears as $d^2$, not $d$.

Step 1: Find the constant $k$

Substitute the known pair $d = 4$, $F = 18$:

$18 = \frac{k}{4^{2}} = \frac{k}{16}$

$k = 18 \times 16 = 288$

So the rule connecting the two quantities is

$F = \frac{288}{d^{2}}.$

Step 2: Use the rule at the new distance

Now substitute $d = 6$:

$F = \frac{288}{6^{2}} = \frac{288}{36} = 8.$

$\boxed{F = 8\text{ N}}$

Alternative method: ratio of values

For inverse-square proportion the forces are in the reciprocal ratio of the squared distances:

$\frac{F_{2}}{F_{1}} = \left(\frac{d_{1}}{d_{2}}\right)^{2} = \left(\frac{4}{6}\right)^{2} = \frac{16}{36} = \frac{4}{9}.$

Therefore

$F_{2} = 18 \times \frac{4}{9} = 8\text{ N}.$

Both methods agree.

Sense check

The magnets move further apart (from $4\text{ cm}$ to $6\text{ cm}$), and with inverse proportion the force must decrease. Since $8 < 18$, the answer is physically reasonable. Any answer larger than $18\text{ N}$ (such as $27$ or $40.5$) would correspond to direct proportion and can be rejected immediately.

The correct option is A.