Warning: Layman question. Treat me as a 10 years old child

I could write this on the physics channel, but despite the context, my problem is intrinsically related to math.

First, let's point out the context

• There are two kinds of charge, positive and negative
• Like charges repel, unlike charges attract

To calculate the charge of one object we must consider that, the more charged the more the attraction or repulsion between this object and a second referential object will exist.

Considering the two objects at a distance $d$. Both attracting each other. One force, $F_1$, will be noticed.

If we want to know and create an unity to calculate the physical quantity related to the charge of the first object, we'll need a third auxiliary object.

What we will do is compare the force $F_1$ between the object 1 and 3. Then, compare the force $F_2$ between the object 2 and 3.

So we will put the object 1 at a distance $d$ from the object 3 and calculate the exerted force.

Then we will put the object 2 at a distance $d$ from the object 3 and calculate the exerted force.

If both forces are equal, then both charges (object 1 and 2) are also equal.

But if the object 2 have a charge grater then the object 1, we need to find out the difference.

Now the math problem

So far I understand that we must consider the multiplicity, in other words, the object 2 will have

$$F_2 = n.F_1$$

Reading a physics book they come out with the following:

If both charges are equal then:

$$\frac{a}{b} = c$$

If the charge 2 are grater then the 1, then:

$$\frac{a_1}{b_1} = \frac{a_2}{b_2} = ... = \frac{a_n}{b_n} = K$$

Now the math layman question

I can't understand why they represent the multiplicity ($a = n.b$) in that way:

$$\frac{a_1}{b_1} = \frac{a_2}{b_2} = ... = \frac{a_n}{b_n} = K$$

My questions is: What is the $a_1$, $a_2$, ..., $a_n$?

Is $a$ related to the object 1 and $b$ related to the object 2 (with grater charge)?

In this case, we divided the lower value by the larger one?

Why each part are equal and finally equal to $K$ ? $\frac{a_1}{b_1} = \frac{a_2}{b_2} = ... = \frac{a_n}{b_n} = K$

Where is the third object represented?

What is $K$ ?

Looking for an intuitive concept

I'm looking for purely intuitive concept about that. To understand my question type, please read that one:

The logic behind the rule of three on this calculation

Try to use simple concepts like take as base only on multiplicity and division. At least I think it is sufficient to point out this operation. (Sorry about my leak of understanding)

If one value is $x$ times grater the $y$ and we divided $\frac{x}{y}$ we will get that common value between the two variables and then will be possible determine the multiplicity.
We have $$\frac{a_1}{b_1} = \frac{a_2}{b_2} = ... = \frac{a_n}{b_n} = K$$ because we are considering more then the two mentioned objects(??) ...