confusion related to elementary operation on numbers Let's take for example an fraction:
$\dfrac{1+4}{2-4}$
and another fraction  $\dfrac{1*4}{2*4}$. In the second fraction we can cancel 4 from both numenator and denominator but on the first we cannot do so.I have learnt how to do this but I do not have exact idea why it happens. Will anybody explain to me as clearly as possible as why that happens?
 A: One way to think about it: when we cancel a common factor, we are dividing the top and the bottom by the same number. Now, division is the opposite of multiplication. Thus:
$$\frac{(1*4)/4}{(2*4)/4}=\frac{1}{4}$$However, if we do this with your other example, we get:$$\frac{(1+4)/4}{(2-4)/4}$$It's incorrect to  cancel, because $(1+4)/4\neq1$ and $((2-4)/4\neq2$; the division by 4 does not cancel out the addition or subtraction of 4.
We can also look at it pictorially. I'll change your minus to a plus, to make it easier to draw one of the figures.
First let me illustrate the equation $\dfrac{1*4}{2*4}=\dfrac{1}{2}$:

On the left, we have a rectangle, half of which is shaded. On the right, we have a big rectangle made up of 4 copies of the rectangle on the left. The fraction that is shaded is the same on both the left and the right.
Now let's look at the inequality $\dfrac{1+4}{2+4}\neq\dfrac{1}{2}$.

On the left we have the rectangle, half shaded. On the right we've added 4 boxes, so the shaded fraction is $\dfrac{1+4}{2+4}$. (That is, $1+4$ shaded boxes out of $2+4$ boxes total.) As you can see, the shaded fraction is quite different.
A: Because $\dfrac{a*4}{b*4} = \dfrac{a}{b}.\dfrac{4}{4} = \dfrac{a}{b}.\dfrac{1}{1} = \dfrac{a}{b}$, this is because only the operations of multiplication and division are involved, and you can swap the order around.
But, $\dfrac{a+4}{b-4} \ne \dfrac{a}{b} +/- \dfrac{4}{4}$, this is because the operations of addition and subtraction are mixed with division and you can't swap them round.
But $a + 4 - b - 4$ only involves addition and subtraction and you can swap them around (as with multiplication and division), so that $a + 4 - b - 4$ = $a + 4 - 4 - b$ = $a + 0 - b $ = $a - b$
So remember that if a calculation only involves multiplication and division (or only addition and subtraction) you can perform the individual operations in any order, but if you mix the two types then you can't.
A: Look at the meaning of equality of fractions. You have
$${a\over b} = {c\over d}$$
if and only if $$ad = bc.$$
Suppose we have $a/b = c/d$ and that $z$ is a number
Then 
$${a + z\over b+z} = {a\over b}$$
if and only if 
$$b(a + z) = a(b + z)$$
Multiply this out and you have
$$ab + bz = ab + az$$
or if $az = bz$.  Unless $z = 0$ or $a = b$, you are sunk.
You should do the same test with 
$${az\over bz} = {a\over b}$$ and see what happens.
