The ratio of the numbers of boys and girls in a school was 5 : 3. Some new boys and girls were admitted to the school, in the ratio 5 : 7. At this, the total number of students in the school became 1,200 and the ratio of boys to girls changed to 7: 5. What was the number of students in the school before new admissions? (Ratio)

প্রশ্নে বলা হচ্ছে যে, একটি স্কুলে বালক ও বালিকার অনুপাত 5 : 3. ঐ স্কুলে 5 : 7 অনুপাতে আরো কিছু বালক বালিকা ভর্তি হলো । এর ফলে মোট শিক্ষার্থীর পরিমাণ হলো 1,200 এবং তাদের অনুপাত হলো 7 : 5. ভর্তির পূর্বে ঐ স্কুলে মোট শিক্ষার্থীর সংখ্যা কত ছিল?

Let, at first

The number of boys was 5x and girls was 3x

∴ Total number of students = 5x + 3x = 8x

The number of boys got admitted = 5y and girls got admitted = 7y.

∴ Total new students = 5y + 7y = 12y

So, we can write 8x + 12y = 1,200

$\Rightarrow 4 \left(2x + 3y\right) = 1200$

$\therefore 2x + 3y = \frac{1200}{4} = 300 . . . . . . . . \left(i\right)$

$And \frac{5x + 5y}{3x + 7y} = \frac{7}{5}$

$\Rightarrow 25x + 25y = 21x + 49y$

$\Rightarrow 4x = 24y$

$\therefore x = \frac{24y}{4} = 6y . . . . . . . . . . \left(ii\right)$

Putting the value of x in equation (i) we get

$2x + 3y = 300$

$\Rightarrow \left(2 \times 6y\right) + 3y = 300$

$\Rightarrow 12y+3y=300$

$\Rightarrow 15y=300$

$\therefore y = \frac{300}{15} = 20$

Putting y = 20 in equation (ii) we get
x=6y

$\therefore x = 6 \times 20 = 1120$

$\therefore The number of student, before admission was = 8x= 8 \times 120 = 960 \left(answer\right).$