In a college the students in Arts and Commerce faculties were in the ratio of 4 : 5 respectively. When 65 more students joined Commerce faculty the ratio becomes 8 : 11 respectively. How many students are there in Arts faculty? (Bank P.O., 2009)

### Concept & Single Variable Transformation When a specific quantity is added to only one side of a ratio, the quantity on the other side remains constant. You can set up an algebraic equation based on the addition to the specific variable. ### Step-by-Step Solution * Let the initial number of students in Arts and Commerce be $4x$ and $5x$ respectively. * $65$ more students join the Commerce faculty. * New number of students in Commerce = $5x + 65$. * The number of students in Arts remains the same ($4x$). * The new ratio of Arts to Commerce is $8 : 11$. * Set up the equation: $$ \frac{4x}{5x + 65} = \frac{8}{11} $$ * Cross-multiply: $$ 11(4x) = 8(5x + 65) $$ $$ 44x = 40x + 520 $$ * Solve for $x$: $$ 44x - 40x = 520 $$ $$ 4x = 520 $$ $$ x = 130 $$ * The question asks for the number of students in the Arts faculty. * Arts students = $4x = 4 \times 130 = 520$. ### Exam Strategy & Shortcut Look at the ratios: Arts is initially $4$ parts and finally $8$ parts. To compare them easily, make the Arts parts equal. Initial ratio $A:C = 4:5 = 8:10$. Final ratio $A:C = 8:11$. Notice that Arts remained $8$ parts, while Commerce went from $10$ parts to $11$ parts. This is an increase of $1$ part. We are told $65$ students joined Commerce, so $1 \text{ part} = 65$. The Arts faculty has $8$ parts, so total Arts students = $8 \times 65 = 520$. ### Common Pitfall A common mistake is finding the value of $x$ (which is $130$) and stopping there, assuming it is the final answer, or calculating the total number of students instead of just the Arts faculty students. ### Final Answer Therefore, the correct answer is **520**.