Seats for Mathematics, Physics and Biology in a school are in the ratio 5 : 7 : 8. If the seats for these three subjects 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 checks understanding of how percentage increases affect an existing ratio. Instead of working with actual numbers of seats, we can treat the ratio terms as basic units and apply the percentage changes directly to them. This technique is useful whenever proportional scaling is involved.

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 new ratio of seats.

Concept / Approach:
We can assume the original number of seats as 5k, 7k and 8k respectively. A percentage increase changes each term by a multiplicative factor. For example, a 40% increase corresponds to multiplying by 1.40. After updating each term, we obtain new numbers that can be simplified to their smallest integer ratio by dividing by a common factor.

Step-by-Step Solution:
Step 1: Let original seats be 5k, 7k and 8k. Step 2: Increase Mathematics seats by 40%. New Mathematics seats = 5k * 1.40 = 7k. Step 3: Increase Physics seats by 50%. New Physics seats = 7k * 1.50 = 10.5k. Step 4: Increase Biology seats by 75%. New Biology seats = 8k * 1.75 = 14k. Step 5: The new ratio before simplification is 7k : 10.5k : 14k. Step 6: Remove the common factor k to get 7 : 10.5 : 14. Step 7: To avoid decimals, multiply all terms by 2 to get 14 : 21 : 28. Step 8: Divide all terms by 7 to simplify: 14 ÷ 7 = 2, 21 ÷ 7 = 3 and 28 ÷ 7 = 4. Step 9: So the simplified new ratio is 2 : 3 : 4.

Verification / Alternative check:
We can select a convenient value for k, for example k = 10. Then original seats are 50, 70 and 80. After the respective increases, Mathematics has 70 seats, Physics has 105 seats and Biology has 140 seats. The ratio 70 : 105 : 140 simplifies by dividing all terms by 35 to 2 : 3 : 4, which confirms the earlier result.

Why Other Options Are Wrong:
Ratios such as 1 : 2 : 3 or 3 : 4 : 5 and 4 : 5 : 6 are not equivalent to 2 : 3 : 4 when you compare the relative increases implied. They do not match the actual scaled seat counts after applying 40%, 50% and 75% increases to the original ratio 5 : 7 : 8.

Common Pitfalls:
Learners sometimes add the percentage values directly to the ratio numbers rather than treating the percentages as multipliers. For example, adding 40 to 5 instead of doing 5 * 1.40 is incorrect. Another mistake is to forget to simplify the final numbers to their lowest integer ratio, leaving answers with decimals instead of neat whole numbers.

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