The number of students in three sections of a class in a school are in the ratio 5:7:8. If 20, 25 and 40 more students are admitted in the three sections, respectively, the new ratio becomes 9:12:16. The total number of students after the new admissions is:

Let the initial number of students in the three sections be 5k, 7k, and 8k, where k is a constant multiplier.
After the new admissions, the number of students in each section becomes:

Section 1: 5k+20

Section 2: 7k+25

Section 3: 8k+40

We are given that the new ratio is 9:12:16. We can set up a proportion using any two sections, say Section 1 and Section 2:

7k+25
5k+20
​
=
12
9
​

Simplify the fraction on the right side:

12
9
​
=
4
3
​

Now, cross-multiply to solve for k:

4(5k+20)=3(7k+25)
20k+80=21k+75
21k−20k=80−75⟹k=5
Let's verify with Section 3 using k=5:

New Section 1 = 5(5)+20=45

New Section 2 = 7(5)+25=60

New Section 3 = 8(5)+40=80

The new ratio is 45:60:80, which reduces to 9:12:16 when divided by 5. This confirms our value of k is correct.

Calculate the total number of students after the new admissions:

Total Students=45+60+80=185