The respective ratio of students studying in school P, Q and Ris 8:6:7. What would be the respective ratio of students if the ratio increases by 15%, 20% and 25% respectively?

Options

A. `189:140:175`
B. `149:125:165`
C. `1:2:3`
D. `184:144:175`

Answer

To solve the problem step by step, we will first determine the initial number of students in schools P, Q, and R based on the given ratio of 8:6:7. Then we will apply the respective percentage increases to find the new number of students and finally calculate the new ratio.

### Step 1: Assign values based on the ratio
The ratio of students in schools P, Q, and R is given as 8:6:7. We can assign a base value to these ratios for easier calculations. Let's assume:
- Students in school P = 8x
- Students in school Q = 6x
- Students in school R = 7x

### Step 2: Calculate the increased number of students for each school
Now, we will calculate the increased number of students based on the given percentage increases.

1. **For school P (15% increase)**:
\[ \text{Increase} = 15\% \text{ of } 8x = \frac{15}{100} \times 8x = 1.2x \]
\[ \text{New number of students in P} = 8x + 1.2x = 9.2x \]

2. **For school Q (20% increase)**:
\[ \text{Increase} = 20\% \text{ of } 6x = \frac{20}{100} \times 6x = 1.2x \]
\[ \text{New number of students in Q} = 6x + 1.2x = 7.2x \]

3. **For school R (25% increase)**:
\[ \text{Increase} = 25\% \text{ of } 7x = \frac{25}{100} \times 7x = 1.75x \]
\[ \text{New number of students in R} = 7x + 1.75x = 8.75x \]

### Step 3: Write the new ratio
Now, we have the new number of students as:
- School P: \(9.2x\)
- School Q: \(7.2x\)
- School R: \(8.75x\)

Thus, the new ratio of students in schools P, Q, and R is:
\[ 9.2x : 7.2x : 8.75x \]

### Step 4: Simplify the ratio
To simplify the ratio, we can eliminate \(x\) from all terms:
\[ 9.2 : 7.2 : 8.75 \]

To simplify further, we can convert these to whole numbers by multiplying each term by 100:
\[ 920 : 720 : 875 \]

Now, we can find the greatest common divisor (GCD) to simplify the ratio further. The GCD of 920, 720, and 875 is 5. Therefore, we divide each term by 5:
\[ \frac{920}{5} : \frac{720}{5} : \frac{875}{5} = 184 : 144 : 175 \]

### Final Result
The respective ratio of students studying in schools P, Q, and R after the increases is:
\[ \text{184 : 144 : 175} \]