# Deep arithmetic question. [duplicate]

here is my best attempt to explain my doubt: Look at the following case, when we have a fractions like 2/2 + 2/2 we don't solve them like 4/4 instead its 4/2 we don't increase the denominator.

If some alien saw this he would be why not add the denominator?, fortunately I know the reason, the denominator represents how many times an object has been divided in pure math the object are just numbers, since the amount of pieces something has been cut into does not add up by adding the pieces of some other object therefore the denominator does not add up.

As you can see from my above explanation the rules of not adding denominator makes sense but to someone who does no the explanation for not adding the denominators will just view them as mere rules.

That is the problem with my doubt, 2 x 2/2 to solve this we need to remove "2" from the numerator and the denominator but we don't divide both the "2's" only one, Now like for not adding the denominators there is some explanation, which I don't know.

My other question, we know that 4/2 is 2 and 2 x 2 = 4, so when we do 2 x 2 / 2 we don't divide both the numbers instead only one of them and arrive at the same answer, how?

These "reasons" I believe are never taught in school they are treated as rules, this is causing serious problems while solving questions because every concept in math depends on my understanding of these.

## marked as duplicate by Namaste, Don Thousand, Community♦Dec 4 '18 at 14:04

• In your first example you are adding halfs. Two and two. The result will again give halfs, they will be 4. Take two halfs of a pizza, add other two halfs of pizza :) – user376343 Dec 4 '18 at 13:38
• Have you read (and understood) the answers to your previous question math.stackexchange.com/q/3024246/42969? You are partially repeating the same “doubts.” – Martin R Dec 4 '18 at 13:39
• @MartinR I didn't realize OP posted that. I should not have answered this. – Don Thousand Dec 4 '18 at 13:42
• Simply put, to add two things you need to measure them in the same units. That's the purpose of common denominator. Adding one pound to one ton does not make sense. – Vasya Dec 4 '18 at 13:53
• I felt like no one understood my question. – MinigameZ more Dec 4 '18 at 13:58

Here's one way to build intuition.

Let's say that you have the expression $$\frac22+\frac22$$. Let's set that equal to a variable $$x$$. So we have $$x=\frac22+\frac22$$. Now, we can multiply every term in this equation by 2. So, we get $$2x=2+2$$.

On the other hand, with $$x=\frac{2\times2}2$$, we get $$2x=2\times2$$. Does this make more intuitive sense?

The point is, you shouldn't view this as "removing" or "adding" 2s to the fraction. Try to understand the underlying mechanism and it will make more sense.