Seats for Mathematics, Physics and Biology in a school are in the ratio 5:7:8. There is a proposal to increase these seats by 40%, 50% and 75% respectively. What will be the ratio of increased seats

Options

A. A) 1:2:3
B. B) 2:3:4
C. C) 3:4:5
D. D) 4:5:6

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Define the initial number of seats
The seats for Mathematics, Physics, and Biology are in the ratio of 5:7:8. Let the number of seats for Mathematics be \(5x\), for Physics be \(7x\), and for Biology be \(8x\).

### Step 2: Calculate the increased seats
We need to calculate the increased number of seats after the proposed percentage increases:
- For Mathematics, the increase is 40%. The new number of seats will be:
\[ \text{New Mathematics seats} = 5x + 0.40 \times 5x = 5x \times (1 + 0.40) = 5x \times 1.40 = 7x \]

- For Physics, the increase is 50%. The new number of seats will be:
\[ \text{New Physics seats} = 7x + 0.50 \times 7x = 7x \times (1 + 0.50) = 7x \times 1.50 = 10.5x \]

- For Biology, the increase is 75%. The new number of seats will be:
\[ \text{New Biology seats} = 8x + 0.75 \times 8x = 8x \times (1 + 0.75) = 8x \times 1.75 = 14x \]

### Step 3: Write the increased seats in ratio form
Now we have the new number of seats:
- Mathematics: \(7x\)
- Physics: \(10.5x\)
- Biology: \(14x\)

We can express these in ratio form:
\[ \text{Ratio} = 7x : 10.5x : 14x \]

### Step 4: Simplify the ratio
To simplify the ratio, we can divide each term by \(x\):
\[ 7 : 10.5 : 14 \]

Next, we can convert \(10.5\) to a fraction:
\[ 10.5 = \frac{21}{2} \]
Thus, the ratio becomes:
\[ 7 : \frac{21}{2} : 14 \]

To eliminate the fraction, we can multiply the entire ratio by \(2\):
\[ 7 \times 2 : \frac{21}{2} \times 2 : 14 \times 2 = 14 : 21 : 28 \]

### Step 5: Final ratio
The final ratio of the increased seats for Mathematics, Physics, and Biology is:
\[ 14 : 21 : 28 \]

### Step 6: Simplify the final ratio
We can simplify this ratio further by dividing each term by 7:
\[ \frac{14}{7} : \frac{21}{7} : \frac{28}{7} = 2 : 3 : 4 \]

Thus, the final answer is:
\[ \text{The ratio of increased seats is } 2 : 3 : 4. \]

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