The ratio of Boys and Girls is `5 : 3`. Some students took admission in the ratio `5 : 7.` Now total students are 1200 and ratio becomes `7 : 5`. Find the number of students before admission.

Answer

To solve the problem step by step, we will follow the information given in the question and use the ratios to find the number of students before admission.

### Step 1: Understand the initial ratio of boys and girls
The initial ratio of boys to girls is given as 5:3. This means for every 5 boys, there are 3 girls.

Let the number of boys be \( 5k \) and the number of girls be \( 3k \).

### Step 2: Calculate the total number of students before admission
The total number of students before admission can be expressed as:
\[ \text{Total students before admission} = 5k + 3k = 8k \]

### Step 3: Understand the admission ratio and total students after admission
Some students took admission in the ratio of 5:7. After admission, the total number of students is 1200, and the new ratio of boys to girls becomes 7:5.

### Step 4: Calculate the number of boys and girls after admission
From the new ratio of 7:5, we can find the number of boys and girls after admission:
- Total parts in the ratio = \( 7 + 5 = 12 \)
- Number of boys after admission = \( \frac{7}{12} \times 1200 = 700 \)
- Number of girls after admission = \( \frac{5}{12} \times 1200 = 500 \)

### Step 5: Set up equations based on the admission
Let \( x \) be the number of boys admitted and \( y \) be the number of girls admitted.
From the initial counts:
- Boys after admission: \( 5k + x = 700 \)
- Girls after admission: \( 3k + y = 500 \)

### Step 6: Express the number of students admitted in terms of the ratio
Since the students admitted are in the ratio of 5:7, we can express \( x \) and \( y \) as:
\[ \frac{x}{y} = \frac{5}{7} \implies 7x = 5y \implies y = \frac{7}{5}x \]

### Step 7: Substitute \( y \) in the girls' equation
Substituting \( y \) in the girls' equation:
\[ 3k + \frac{7}{5}x = 500 \]

### Step 8: Solve the equations
Now we have two equations:
1. \( 5k + x = 700 \) (1)
2. \( 3k + \frac{7}{5}x = 500 \) (2)

From equation (1), we can express \( x \):
\[ x = 700 - 5k \]

Substituting \( x \) in equation (2):
\[ 3k + \frac{7}{5}(700 - 5k) = 500 \]
Multiplying through by 5 to eliminate the fraction:
\[ 15k + 7(700 - 5k) = 2500 \]
Expanding:
\[ 15k + 4900 - 35k = 2500 \]
Combining like terms:
\[ -20k + 4900 = 2500 \]
Solving for \( k \):
\[ -20k = 2500 - 4900 \implies -20k = -2400 \implies k = \frac{2400}{20} = 120 \]

### Step 9: Find the number of boys and girls before admission
Now substituting \( k \) back to find the number of boys and girls:
- Number of boys = \( 5k = 5 \times 120 = 600 \)
- Number of girls = \( 3k = 3 \times 120 = 360 \)

### Step 10: Calculate the total number of students before admission
Total students before admission:
\[ 600 + 360 = 960 \]

Thus, the total number of students before admission is **960**.

---