

No wisdom here, we all know this: 4 × 2 = 8. But if we look at this as a multiplication of fractions – we see that bigger numerators make our product bigger. So numerators "help" a product "grow". What happened with our number 4? Our 2/1 has made our 4/1 two times bigger, i.e. 8.
Second case of

×  =  
4  ×  1  =  4×1  =  4  = 2  
1  2  1×2  2 
Aha, now instead of making it bigger, it made our product smaller.
We’re noticing:
So we proudly conclude – if we’re multiplying by a fraction, i.e. a fraction is our multiplier, its numerator increases the product by multiplying the multiplicand, and a denominator decreases the product by dividing the multiplicand. What a mouthful!
Or we could simply put: multiplying by a fraction is like doing two operations at once – we’re multiplying by a numerator and dividing by a denominator (I like this one better).
Let’s try with second numerator > 1: We can interpret the case below as: what are the 2/5 of a 3
×  =  =  
3  ×  2  =  3×2  =  6  = 
1 
1  
1  5  1×5  5  5 
So, we have 3 (wholes) and we’re multiplying it by 2 and dividing it by 5 at the same time (i.e. by 2/5). It’s a combined operation we could say. :)
If we separate it in steps:
We can divide 6 by 5 but we would get a decimal number, and we’re not dealing now with decimals, so we’re leaving it as is.
So you see, this multiplying fractions with whole numbers is easy.
×  =  
4  ×  3  =  4×3  =  3  
1  4  1×4  1 
But we should notice something, there’s no point in multiplying something by a number and then right away dividing it by that same number. So we reduce our product by eliminating the common factor 4 found above and below.
So that’s why I didn’t, and you never should, multiply 4×3 = 12, if you’re going to divide with that same 4 right afterwards.
Remember: 
Finally, we’re kickin’ out the whole numbers.
×  =  
1  ×  1  =  1×1  =  1  
2  2  2×2  4 
We see that whatever’s down is making our product smaller. The proper question for the case above would be: what is the half of a half.
×  =  
1  ×  1  =  1×1  =  1  
4  3  4×3  12 
In this case we’re asking: what is the third of one quarter. And if we divide 1/4 to 3 parts – we’ll get 1/12.
×  =  
2  ×  4  =  2×4  =  8  
3  5  3×5  15 
In this case above we’re wondering what are the four fifths of two thirds.
In this next case we’ll throw in some reducing
×  =  
3  ×  4  =  3×4  =  1  
4  6  4×3×2  2 
We factored 6 to 3×2. We do not necessarily have to factor every number to primes, the moment we spot common factors above and bellow the line – we reduce.
Well, the first thing we should do when want to multiply mixed fractions is – not to do it. :)
We should turn that mixed fractions multiplication into an improper fraction multiplication. I.e. we turn our mixed fractions into improper fractions.
Say we have this problem:
×  = ?  
1 
4  × 
2 
1  = ?  
6  4 
As we said, we’re going to turn both of them into the improper form using the following formula:
Improper Numerator = = (Denominator × Number of Wholes) + + Mixed Numerator 
And when we apply that to our first mixed fraction we get this:
=  
1 
4  =  1×6 + 4  =  10  
6  6  6 
We could have reduced but we're not going to, for the sake of practicing we're saving it for multiplication.
And here it is for the second mixed fraction.
=  
2 
1  =  2×4 + 1  =  9  
4  4  4 
Phew! Finally we’re in business! Now, as usual, we factor to find as many common factors for reducing.
×  =  =  
10  ×  9  =  5×2×3×3  =  15  = 
3 
3  
6  4  3×2×4  4  4 
We factored:
Please, DO NOT ever multiply if you have a chance to reduce!
One must admit that 15/4 is a much more elegant solution than 90/24. :)
Conclusion
