This kind of addition of two fractions is almost working as the "usual" way of adding fractions. But before I start to explain how it works, we should create an example. In this case it would be 3/7 + 5/8 (= 59/56).
First you need to multiply the numerator of the first fraction with the denominator of the second fraction. So in this example you need the 3 of the first fraction and the 8 of the second fraction.
After that you do the same with the remaining numbers: Multiply the denominator of the first fraction with the numerator of the second fraction. Put both products into the numerator and add them together ⇒ (3⋅8 + 7⋅5)/x = (24+35)/x = 59/x.
The new denominator consists of the product of the two denominators which you've got already. So multiply 7 and 8. Put this result into the denominator ⇒ 59/ (7⋅8) = 59/56. As you can see we've got the same result as in the second paragraph. Cool, isn't it?
But if you take a closer look you'll find out that this is almost the same way as the "regular" adding of two fractions. Let's have a closer look …
… We multiply both numerators with both numbers of the other fraction.

7⋅5 and 7⋅8

8⋅3 and 8⋅7

In my opinion the Vedic Mathematics is easier than the 'normal' way of adding fractions. Let's assume two big fractions like 56/65 + 34/43. I think it will be more difficult to multiply 56 and 43 😉.
Click on the "+" to try it yourself. Enter new values if you like:

Subtracting Fractions
Maybe you have already understood, how the subtraction works. The only difference to the addition of fractions is that you are not adding the two numbers in the denominators, instead you subtract them.

5/8 − 3/7 = 5⋅7 − (8⋅3)/(8⋅7) = 35 − 24/56 = 11/56
Click on the "−"−button and change the values of the fractions if you like:

Multiplying with two two digit Numbers
In my opinion this kind of the Vedic Mathematics is the most interesting part of these three types of calculation. Let's take our "unsolvable" problem −56⋅43−.
At first we change the way we write this example: So write both numbers under each other and leave much space between the digits. Do it like this:

 5 6 4 3

Now multiply the left row and the right row:

 5 6 ↓ ↓ 4 3 20 18

After doing that multiply the top left number with the number on the bottom right and so do it with the number on the bottom left with the number on the top right. Now add both results:

 5 6 24 ↓ ✖ ↓ + 4 3 15 20 18 39

Now place your last result between the products of your first two multiplications. If the numbers in the row are two digit numbers the first digits are going to change into carries. (The red numbers are the carries):

 5 6 ↓ ↓ 4 3 20 39 18
20|39|18
2|0+3|9+1|8
2|3|10|8
2|3+1|0|8
2|4|0|8
2.408
Now key in the calculating task into a calculator. You'll see, it's the same result as we calculated before. Cool, isn't it?