Here's the method in general:

If A:B = 2:3
& B:C = 3:4
then A:B = 2:4 .....easy?

Now say B:D = 5:6

Then A:D = ??

The common one between them is B, so take such a value for B that the value is same for both A and D and gives integers.

ie, B is 3x wrt to A, and 5y wrt to D, so take lcm of 3 and 5 so that you get 15x for A and 15y for D.

make A:B = 2:3 or 10:15
make B:D = 5:6 or 15:18

so A:D = 10:18.... or 5:9

Ans to your question will be 12:15 or 4:5

Based on the same principle as what rashminet84 has explained

Its simple

Step 1 ) write down the question as it states
a) 2nd grade:4th grade = 8:5
b) 1st grade:2nd grade = 3:4
c) 3rd grade:4th grade = 3:2

We need to find out 1st grade:3rd grade

Step 2) The only way to tie 1st grade and 3rd grade is by getting a common grade between the ratios 1st grade:4th grade and 4th grade:3rd grade

So lets find out what is 1st grade:4th grade

Reverse ratio a) and b) as 4th grade:2nd grade = 5:8 and 2nd grade:1st grade = 4:3

Now to find the ratio between 4th grade and 1st grade, the right hand side of the above two ratios should have the same value for the 2nd grade position as of now it is 8 and 4

So 4th grade:2nd grade::2nd grade:1st grade = 5:8::4*2:3*2
= 5:8::8:6

So we have 8 on both sides so canceling them we have

4th grade:1st grade = 5:6



Step 3) Similar to above step we need to get a common grade between the ratios 1st grade:4th grade and 4th grade and 3rd grade

So 4th grade:1st grade = 5:6 Reversing them we have 1st grade:4th grade = 6:5


and 3rd grade:4th grade = 3:2 Reversing them we have 4th grade:3rd grade = 2:3

Combining them we have

1st grade:4th grade::4th grade:3rd grade = 6:5::2:3

two get the common value for 4th grade multiple the first ratio by 2 and second ratio by 5

==> 1st grade:4th grade::4th grade:3rd grade = 6*2:5*2::2*5:3*5
==> 1st grade:4th grade::4th grade:3rd grade = 12:10::10:15

Canceling the 10's out we have

1st grade:4th grade::4th grade:3rd grade = 12:15


Hope this helps, I tried to explain in real detail.

And yeah it looks long but I had to do it to give clear explanation. But it doesn't take more than 30-60 seconds if you know the concept

N1,N2,N3,N4 be the number of students in 1st,2nd,3rd and 4th grade.

we have N2/N4 = 8/5, N1/N2 = 3/4, N3/N4 = 3/2. We need to find N1/N3
N1/N3 = N1/N2 * N2/N4 * N4/N3 = 3/4 * 8/5 * 2/3 = 4/5

Before you use an approach, ensure you know why it works. This approach is exactly what has been done in posts above.

Say,
A:B = 3:4 = 6:8 (to make Bs equal in A:B and B:C)
B:C = 8:5
You get A:B:C = 6:8:5

Instead, if you have A:B = 3:4 and B:C = 5:6, how do you make Bs equal?
A:B = 3:4 = 15:20
B:C = 5:6 = 20:24
You multiply the B's to get the LCM.

That's what is done in the video too. He just multiplies the Bs to get a common value (even though it is not the least common value, it doesn't matter to us since the ratio is unchanged) and represents the whole thing in a table format. The method is no different from what is done above.
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Karishma
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Here is my method: The key here is to cross cancel.

1) B/D = 8/5 or 16/10
2) A/B = 3/4
3) C/D = 3/2 or 15/10
A/C = ?

1) & 2) can be multiplied to get A/D
B/D X A/B = A/D
8/5 *3/4 = 6/5

2) Now we know A/D & C/D. Fine the LCM so that you have:

A/D = 6/5 or 12/10
C/D = 15/10
so A/C = 12/15 or 4/5