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
Aha, now instead of making it bigger, it made our product smaller.
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
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.
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.
Finally, we’re kickin’ out the whole numbers.
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.
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.
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
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:
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:
|4||=||1×6 + 4||=||10|
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.
|1||=||2×4 + 1||=||9|
Phew! Finally we’re in business! Now, as usual, we factor to find as many common factors for reducing.
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. :-)