Ratio Calculation for Increased School Seats

The question asks for the new ratio of seats for Mathematics, Physics, and Biology after specific percentage increases.

Initial Seat Ratio

The initial ratio of seats for Mathematics, Physics, and Biology is given as 5:7:8.

Let the initial number of seats be represented by $5x$, $7x$, and $8x$ for Mathematics, Physics, and Biology, respectively, where $x$ is a common multiplier.

Percentage Increases

• Mathematics seats are proposed to increase by 40%.
• Physics seats are proposed to increase by 50%.
• Biology seats are proposed to increase by 75%.

Calculating Increased Seats

We need to calculate the new number of seats for each subject after applying the respective percentage increases.

Mathematics Seat Increase

Initial seats: $5x$

Increase amount: $40\%$ of $5x = \frac{40}{100} \times 5x = 0.40 \times 5x = 2x$

New Mathematics seats = Initial seats + Increase amount

New Mathematics seats = $5x + 2x = 7x$

Physics Seat Increase

Initial seats: $7x$

Increase amount: $50\%$ of $7x = \frac{50}{100} \times 7x = 0.50 \times 7x = 3.5x$

New Physics seats = Initial seats + Increase amount

New Physics seats = $7x + 3.5x = 10.5x$

Biology Seat Increase

Initial seats: $8x$

Increase amount: $75\%$ of $8x = \frac{75}{100} \times 8x = 0.75 \times 8x = 6x$

New Biology seats = Initial seats + Increase amount

New Biology seats = $8x + 6x = 14x$

Determining the New Ratio

The new numbers of seats are $7x$, $10.5x$, and $14x$. The new ratio is:

$7x : 10.5x : 14x$

We can cancel out the common multiplier $x$:

$7 : 10.5 : 14$

To eliminate the decimal, we can multiply each part of the ratio by 2:

$7 \times 2 : 10.5 \times 2 : 14 \times 2$

$14 : 21 : 28$

Now, we simplify this ratio by dividing each number by their greatest common divisor, which is 7:

$\frac{14}{7} : \frac{21}{7} : \frac{28}{7}$

$2 : 3 : 4$

Final Ratio of Increased Seats

The ratio of the increased seats for Mathematics, Physics, and Biology is 2:3:4.