Aptitude - Alligation or Mixture - Discussion

I don't understand the answer please explain clearly.

15(x)+20(y)/x+y = 16.50.

(Here x and y are the ratios in which they are to be mixed).

Then 15x+20y = 16.50x+16.50y.

On simplifying we get x:y = 7:3.

=3.50/100= 0.035.
& 1.50/100 = 0.015.

How 3.50/100 = 35., 1.50/100 = 15.?

(20-16.50)=3.50.
(16.50-15)=1.50.

If values in points means should divide by 100 or 10 based on that because ratio should not be in points. So,

3.50/100 = 35.
1.50/100 = 15.

Respectively 35:15 = 7:3.

How 3.50 and 1.50 come?

There is a shortcut to solve such ratio questions.

1. Take the difference between the average value and the next quantity value. similarly, take the difference between the average value and the first told quantity.

So, here it is 20 -16.50 i.e. 3.50
Now, for the first quantity it is 16.5 - 15 i.e. 1.50

Now, the ratio is 3.5 /1.5 i.e. 7:3.

My question here - I understand from the equation formation that by taking such differences I will get the answer.

However, my question is to understand visually this process of solution. How come a difference gave us the right ratio? Is there any hidden fundamental behind it? Can someone decipher this difference technique via. Singapore Maths? Why difference gave me the answer is what I am puzzled. Please don't explain me from the equation point of view.

Is the below mentioned method correct.

15/2 = 7.5( consider it has 7).
20/2 = 10.
10-7 = 3.

So the answer is 7:3?

In the question Rs 15 and Rs 20 are mentioned the prices of 1st and 2nd varieties sequentially so we need to take Rs 20 -16.50 as the first variant in the solution.

Check the option first.

7:3 = 7*first variety/10 + 3*second variety/10 = 16.50.

7*15/10 = 10.5.

3*20/10 = 6.

By adding two you'll get 16.5.

So 7:3 is correct.

Let x kg first variety mixed with y kg of second variety.

So total cost of(x+y) kg is Rs. 15x+20y.

So cost of 1 kg mixture is Rs. (15x+20y)/x+y.

Now according to question,

(15x+20y)/(x+y)=16.50
=> (15x+20y)/(x+y)=33/2
=>33x+33y=30x+40y
=>3x=7y
=> x:y=7:3.

Answer is 7:3.