If p and q are two non zero numbers and 18(p+q) = (18+p)q, which of the following must be true?

Analysis of the Question

We are given the equation $18(p + q) = (18 + p)q$ where $p$ and $q$ are non-zero numbers. We need to simplify this expression to determine the relationship between $p$ and $q$ and identify which of the given options must be true.

Concept Explanation

This is an algebraic manipulation problem. The goal is to expand the brackets on both sides of the equation and rearrange the terms to isolate one variable or find a constant relationship.

Step-by-Step Solution

Step 1: Expand both sides of the equation. Given: $18(p + q) = (18 + p)q$

Expand the left side: $18p + 18q$

Expand the right side: $18q + pq$

Step 2: Set the expanded sides equal to each other. $18p + 18q = 18q + pq$

Step 3: Simplify the equation. Subtract $18q$ from both sides: $18p = pq$

Step 4: Solve for the variables. Since the problem states that $p$ and $q$ are non-zero numbers, we can safely divide both sides by $p$: $\frac{18p}{p} = \frac{pq}{p}$ $18 = q$

This shows that regardless of the value of $p$ (as long as it is not zero), $q$ must be equal to 18 for the equation to hold true.

Conclusion

Comparing our result ($q = 18$) with the given options, we can see that option A is the correct statement.

Exam Tip

When you see an equation with variables on both sides, always expand the brackets first. If you see the same term on both sides (like the $18q$ we had here), you can cancel them out immediately to save time and reduce the risk of calculation errors!