Seats for Mathematics, Physics and Biology in a school are in the ratio 5 : 7 : 8. If the seats are increased by 40%, 50% and 75% respectively, what will be the new ratio of seats for Mathematics, Physics and Biology?

Introduction / Context:
This question examines the combined effect of percentage increases on a given ratio. It is a classic aptitude problem where initial ratios are scaled by different percentage factors, and then the final ratio must be simplified. Such questions are useful for learning how proportional changes affect multiple related quantities.

Given Data / Assumptions:
• Initial ratio of seats for Mathematics : Physics : Biology = 5 : 7 : 8.
• Mathematics seats are increased by 40%.
• Physics seats are increased by 50%.
• Biology seats are increased by 75%.
• We must find the ratio of the new numbers of seats after these increases.

Concept / Approach:
First, represent the initial numbers of seats as 5x, 7x and 8x for Mathematics, Physics and Biology. Then, apply the respective percentage increases by multiplying each by (1 + percentage increase). After calculating the new effective quantities, we find their ratio and simplify by dividing by the greatest common factor or scaling to small integers.

Step-by-Step Solution:
Step 1: Let original seats be M = 5x, P = 7x, B = 8x.Step 2: Increase Mathematics by 40%: new M = 5x * 1.40 = 7x.Step 3: Increase Physics by 50%: new P = 7x * 1.50 = 10.5x.Step 4: Increase Biology by 75%: new B = 8x * 1.75 = 14x.Step 5: The new ratio is 7x : 10.5x : 14x.Step 6: Cancel x to get 7 : 10.5 : 14. Multiply each term by 2 to clear the decimal: 14 : 21 : 28.Step 7: Divide by 7: 14 / 7 = 2, 21 / 7 = 3, 28 / 7 = 4. So the simplified ratio is 2 : 3 : 4.

Verification / Alternative check:
One can take actual numbers instead of 5x, 7x and 8x, for example, 5, 7 and 8. Applying the same percentage increases gives new values: 5 * 1.4 = 7, 7 * 1.5 = 10.5, 8 * 1.75 = 14. These three numbers again simplify to the ratio 2 : 3 : 4 when scaled appropriately, confirming the result is independent of the specific value of x.

Why Other Options Are Wrong:
Ratios 1 : 2 : 3, 3 : 4 : 5 and 4 : 5 : 6 do not arise from multiplying 5 : 7 : 8 by 1.4, 1.5 and 1.75 respectively. If you substitute these ratios back into the percentage changes, they do not match the proportional increases given in the problem.

Common Pitfalls:
Students sometimes incorrectly add the percentages directly to the ratio numbers (for example, adding 40 to 5 instead of increasing 5 by 40%), or apply the same percentage increase to all terms. Another mistake is to forget to clear decimals properly when simplifying ratios. Always multiply all terms by a suitable common factor to avoid decimal confusion.

Final Answer:
The new ratio of seats for Mathematics, Physics and Biology is 2 : 3 : 4.