Which of the following represents $ab = 64$? (Bank P.O., 2010)

### Concept & Proportionality A ratio equation (proportion) written as $x : y = p : q$ can be rewritten as fractions $\frac{x}{y} = \frac{p}{q}$. Cross-multiplying yields the product equation $xq = yp$. We must evaluate which option's cross-multiplication yields $ab = 64$. $$ \frac{x}{y} = \frac{p}{q} \implies x \cdot q = y \cdot p $$ ### Step-by-Step Solution * Test Option (a): $8 : a = 8 : b$ $$ \frac{8}{a} = \frac{8}{b} \implies 8b = 8a \implies a = b $$ (Incorrect) * Test Option (b): $a : 16 = b : 4$ $$ \frac{a}{16} = \frac{b}{4} \implies 4a = 16b \implies a = 4b $$ (Incorrect) * Test Option (c): $a : 8 = b : 8$ $$ \frac{a}{8} = \frac{b}{8} \implies 8a = 8b \implies a = b $$ (Incorrect) * Test Option (d): $32 : a = b : 2$ $$ \frac{32}{a} = \frac{b}{2} $$ * Cross-multiply Option (d): $$ 32 \times 2 = a \times b $$ $$ 64 = ab $$ (This matches the target equation!) ### Exam Strategy & Shortcut In a proportion $W : X = Y : Z$, the product of the extremes equals the product of the means ($W \times Z = X \times Y$). We need $a \times b = 64$. Look for an option where $a$ and $b$ are the inner terms (means) and two numbers multiplying to $64$ are the outer terms (extremes), or vice-versa. In option (d), the means are $a$ and $b$ (product $= ab$) and the extremes are $32$ and $2$ (product $= 64$). Thus, $ab = 64$. ### Common Pitfall Trying to assign arbitrary numerical values to $a$ and $b$ (like $a=8, b=8$) and testing options. While it might sometimes point to a clue, it does not mathematically prove the equation holds universally for all valid pairs. ### Final Answer Therefore, the correct answer is **$32 : a = b : 2$**.