Two numbers are in the ratio $1\frac{1}{2}:2\frac{2}{3}$. When each of these is increased by 15, their ratio becomes $1\frac{2}{3}:2\frac{1}{2}$. The greater of the numbers is (S.S.C., 2005)

### Concept & Simplifying Mixed Fraction Ratios Before applying any ratio logic, always convert mixed fractions into improper fractions and cross-multiply to find the simplest integer ratio. Then apply the standard constant addition logic. ### Step-by-Step Solution * **Step 1: Simplify the initial ratio.** $$ 1\frac{1}{2} : 2\frac{2}{3} = \frac{3}{2} : \frac{8}{3} $$ * Multiply by the LCM of denominators ($6$) to clear fractions: $$ \left(\frac{3}{2} \times 6\right) : \left(\frac{8}{3} \times 6\right) = 9 : 16 $$ * Let the numbers be $9x$ and $16x$. The greater number is $16x$. * **Step 2: Simplify the final ratio.** $$ 1\frac{2}{3} : 2\frac{1}{2} = \frac{5}{3} : \frac{5}{2} $$ * Cross-multiply the numerators (since they both have a $5$, it cancels): $$ \frac{1}{3} : \frac{1}{2} = 2 : 3 $$ * **Step 3: Apply the condition.** * Both numbers are increased by $15$. $$ \frac{9x + 15}{16x + 15} = \frac{2}{3} $$ * Cross-multiply to solve: $$ 3(9x + 15) = 2(16x + 15) $$ $$ 27x + 45 = 32x + 30 $$ $$ 45 - 30 = 32x - 27x $$ $$ 15 = 5x $$ $$ x = 3 $$ * **Step 4: Find the greater number.** * Greater number = $16x = 16 \times 3 = 48$. ### Exam Strategy & Shortcut Instead of algebraic equations, balance the gaps in the integer ratios. Initial Ratio = $9 : 16$ (Gap is $16 - 9 = 7$) Final Ratio = $2 : 3$ (Gap is $3 - 2 = 1$) Since the same number ($15$) is added to both, the "gap" between the two numbers must remain constant. To make the gaps equal, multiply the final ratio by $7$. Final Ratio becomes = $14 : 21$ (Gap is $7$) Now look at the vertical progression: The first term goes from $9$ to $14$ (+$5$ parts). The second term goes from $16$ to $21$ (+$5$ parts). This $5$ part increase equals the $15$ added. So, $1 \text{ part} = 3$. The greater number is $16 \text{ parts} = 16 \times 3 = 48$. ### Common Pitfall Working with the fractions directly in the algebraic equation ($ \frac{3/2 x + 15}{8/3 x + 15} = ... $) often leads to severe calculation errors. Always convert ratios to standard integers first. ### Final Answer Therefore, the correct answer is **48**.