Study the information carefully to answer the questions that follow:
A school consisting of 1560 students has boys and girls in the ratio of 7:5. All the students are enrolled for different countries’ tours, viz Russia, Switzerland and Japan. One-fifth of the boys are enrolled for only Switzerland tour. Twenty per cent of the girls are enrolled for only Japan tour. Ten, per cent of the boys are enrolled for only Russia tour. Twenty four per cent of the girls are enrolled for both Russia and Switzerland tour together but not for Japan.. The number of the girls enrolled for only Russia tour is two hundred per cent of the boys enrolled for the same tour. One-thirteenth of the boys are enrolled for all the three tours together. The ratio of the boys enrolled for Switzerland and Japan tour together but not for Russia to the,girls enrolled for the same tour is 2 : 1. Ten per cent of the girls are enrolled for only Switzerland tour whereas eight per cent of the girls are enrolled for both Switzerland and Japan tour together but not for Russia tour. The remaining girls are enrolled for all the three tours together. The number of boys enrolled for Russia and Switzerland tour together but not for Japan tour is fifty per cent of the number of girls enrolled for the same. The remaining boys are enrolled for only Japan tour.
What is the ratio ofthe number of girls enrolled for only Japan tour and the number of boys' enrolled for only Switzerland tour together to the number of boys enrolled for only Japan tour and the number of girls enrolled for only Switzerland tour together?

Options

A. `77:52`
B. `75:52`
C. `52:75`
D. `52:77`

Answer

To solve the problem step by step, we need to extract the relevant information and calculate accordingly.

### Step 1: Determine the number of boys and girls in the school.
- The total number of students is 1560.
- The ratio of boys to girls is 7:5.

Let the number of boys be \( 7x \) and the number of girls be \( 5x \).

Setting up the equation:
\[ 7x + 5x = 1560 \\ 12x = 1560 \\ x = 130 \]

Calculating the number of boys and girls:
\[ \text{Boys} = 7x = 7 \times 130 = 910 \\ \text{Girls} = 5x = 5 \times 130 = 650 \]

### Step 2: Calculate the enrollment for specific tours.
1. **Boys enrolled for only Switzerland tour:**
\[ \text{Boys for Switzerland} = \frac{1}{5} \times 910 = 182 \]

2. **Girls enrolled for only Japan tour:**
\[ \text{Girls for Japan} = 20\% \times 650 = 0.2 \times 650 = 130 \]

3. **Boys enrolled for only Russia tour:**
\[ \text{Boys for Russia} = 10\% \times 910 = 0.1 \times 910 = 91 \]

4. **Girls enrolled for both Russia and Switzerland tour (not Japan):**
\[ \text{Girls for Russia and Switzerland} = 24\% \times 650 = 0.24 \times 650 = 156 \]

5. **Girls enrolled for only Russia tour:**
\[ \text{Girls for only Russia} = 200\% \times 91 = 2 \times 91 = 182 \]

6. **Boys enrolled for all three tours:**
\[ \text{Boys for all three} = \frac{1}{13} \times 910 = 70 \]

### Step 3: Calculate remaining enrollments.
1. **Girls for only Switzerland tour:**
\[ \text{Girls for only Switzerland} = 10\% \times 650 = 65 \]

2. **Girls for both Switzerland and Japan (not Russia):**
\[ \text{Girls for Switzerland and Japan} = 8\% \times 650 = 52 \]

3. **Remaining girls for all three tours:**
\[ \text{Remaining girls} = 650 - (65 + 130 + 182 + 156 + 52) = 65 \]

### Step 4: Calculate boys and girls for tours.
1. **Boys for Russia and Switzerland (not Japan):**
\[ \text{Boys for Russia and Switzerland} = 50\% \times 156 = 78 \]

2. **Remaining boys for only Japan tour:**
\[ \text{Remaining boys} = 910 - (182 + 91 + 70 + 78) = 489 \]

### Step 5: Calculate the required ratio.
- **Girls enrolled for only Japan tour:** 130
- **Boys enrolled for only Switzerland tour:** 182
- **Boys enrolled for only Japan tour:** 489
- **Girls enrolled for only Switzerland tour:** 65

Now, we need to find the ratio:
\[ \text{Ratio} = \frac{130 + 182}{489 + 65} = \frac{312}{554} \]

### Step 6: Simplify the ratio.
To simplify \( \frac{312}{554} \):
1. Find the GCD of 312 and 554.
2. Divide both the numerator and denominator by their GCD.

The GCD is 2:
\[ \frac{312 \div 2}{554 \div 2} = \frac{156}{277} \]

### Final Answer:
The ratio of the number of girls enrolled for only Japan tour and the number of boys enrolled for only Switzerland tour together to the number of boys enrolled for only Japan tour and the number of girls enrolled for only Switzerland tour together is \( 156:277 \).