Two numbers are in the ratio of 3 : 5. If 9 is subtracted from each, they are in the ratio of 12 : 23. What is the larger number?

### Concept & Ratio Adjustment When a constant is subtracted from two numbers in a given ratio, you can set up a linear equation by expressing the original numbers in terms of a variable ($x$) and equating the modified numbers to the new ratio fraction. ### Step-by-Step Solution * Let the two numbers be $3x$ and $5x$. The larger number is $5x$. * According to the problem, $9$ is subtracted from both numbers. The new numbers become $(3x - 9)$ and $(5x - 9)$. * The new ratio is $12 : 23$. Set up the equation: $$ \frac{3x - 9}{5x - 9} = \frac{12}{23} $$ * Cross-multiply to solve for $x$: $$ 23(3x - 9) = 12(5x - 9) $$ $$ 69x - 207 = 60x - 108 $$ * Rearrange terms to isolate $x$: $$ 69x - 60x = 207 - 108 $$ $$ 9x = 99 $$ $$ x = 11 $$ * The question asks for the larger number, which is $5x$. * Calculate the larger number: $5 \times 11 = 55$. ### Exam Strategy & Shortcut Use the Cross-Product method for initial and final ratios. Initial ratio = $3 : 5$ Final ratio = $12 : 23$ Change = $-9, -9$ Cross-multiply ratios: $(3 \times 23) - (5 \times 12) = 69 - 60 = 9$ parts. Cross-multiply final ratio with change: $(12 \times 9) - (23 \times 9) = 108 - 207 = -99$ (magnitude is $99$). Equate parts to magnitude: $9 \text{ parts} = 99 \implies 1 \text{ part} = 11$. Larger number is $5 \text{ parts} = 5 \times 11 = 55$. ### Common Pitfall A common pitfall is calculating the value of $x$ (which is $11$) but forgetting to multiply it back by $5$ to find the larger number, or accidentally multiplying it by $3$ to find the smaller number instead. ### Final Answer Therefore, the correct answer is **55**.