JAMB 2019 Mathematics · Question 65

Make q the subject of the formula in the equation $\frac{mn}{a^2} - \frac{pq}{b^2} = 1$

Options

A. $q = \frac{b^2(mn - a^2)}{a^2 p}$
B. $q = \frac{m^2 n - a^2}{p^2}$
C. $q = \frac{mn - 2b^2}{a^2}$
D. $q = \frac{b^2 (n^2 - ma^2)}{n}$

Answer: A. $q = \frac{b^2(mn - a^2)}{a^2 p}$

Explanation written with AI and checked against the answer key

Analysis

We are given the equation:

\frac{mn}{a^2} - \frac{pq}{b^2} = 1

Our goal is to isolate $q$ on one side of the equation.

Concept

To make a variable the subject of a formula, we use algebraic manipulation (addition, subtraction, multiplication, and division) to move all other terms to the opposite side of the equation until the desired variable stands alone.

Step-by-Step Solution

Step 1: Isolate the term containing $q$ . Subtract $\frac{mn}{a^2}$ from both sides:

-\frac{pq}{b^2} = 1 - \frac{mn}{a^2}

Step 2: Simplify the right side. Find a common denominator ( $a^2$ ) for the right side:

-\frac{pq}{b^2} = \frac{a^2 - mn}{a^2}

Step 3: Eliminate the negative sign. Multiply both sides by $-1$ to make the $q$ term positive:

\frac{pq}{b^2} = -\left(\frac{a^2 - mn}{a^2}\right)

Distributing the negative sign inside the parenthesis:

\frac{pq}{b^2} = \frac{mn - a^2}{a^2}

Step 4: Solve for $q$ . Multiply both sides by $b^2$ :

pq = \frac{b^2(mn - a^2)}{a^2}

Finally, divide both sides by $p$ :

q = \frac{b^2(mn - a^2)}{a^2 p}

Conclusion

Comparing our result to the given options, we find that it matches Option A.

Exam Tip

When dealing with fractions in algebraic equations, always try to clear the fractions by multiplying through by the Least Common Multiple (LCM) or by grouping terms with the same denominator. This makes the equation much easier to handle and reduces the chance of sign errors!

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Make q the subject of the formula in the equation mna2−pqb2=1\frac{mn}{a^2} - \frac{pq}{b^2} = 1a2mn​−b2pq​=1