What number has to be added to the terms of 3 : 5 to make the ratio 5 : 6? (R.R.B., 2006)

### Concept & Ratio Modification When the same number is added to both terms of a ratio, express the initial terms using a common variable (or just the raw numbers if we are looking for absolute addition to the simplest form) and equate to the new fraction. ### Step-by-Step Solution * Let the number to be added to both terms be $x$. * The original terms are $3$ and $5$. * After adding $x$ to both, the new terms become $(3 + x)$ and $(5 + x)$. * The new ratio is $5 : 6$. Set up the equation: $$ \frac{3 + x}{5 + x} = \frac{5}{6} $$ * Cross-multiply to solve for $x$: $$ 6(3 + x) = 5(5 + x) $$ $$ 18 + 6x = 25 + 5x $$ * Rearrange the equation to isolate $x$: $$ 6x - 5x = 25 - 18 $$ $$ x = 7 $$ * The number that must be added is $7$. ### Exam Strategy & Shortcut Use Option Elimination. Just add the options to $3$ and $5$ and see which one yields the fraction $\frac{5}{6}$. Test (a) $6$: $\frac{3+6}{5+6} = \frac{9}{11}$. Incorrect. Test (b) $7$: $\frac{3+7}{5+7} = \frac{10}{12} = \frac{5}{6}$. Correct! This avoids algebra entirely. ### Common Pitfall A common mistake is adding the variable to the numerator and denominator separately without treating them as a cohesive binomial during cross-multiplication. ### Final Answer Therefore, the correct answer is **7**.