Multiplying Fractions



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You’re probably wondering how to multiply fractions? Well, multiplying fractions is very easy, much easier than adding and subtracting fractions.

No need for adding or subtracting, and no need for finding a LCD, or worrying whether we have like denominators or not. Whew, that’s a relief.

But before we begin you have to be familiar with things like numerators and denominators, what is expanding or simplifying fractions etc.

Anyway this is the basic formula:


a × c = a×c
b d b×d

We multiply the numerators in the "attic", and denominators in the "cellar".

The letters "a", "b", "c" and "d" represent any number (except "b" and "d" mustn’t be zero).

Then if applicable:

  • we break to factors,
  • look for common factors and
  • reduce (simplify) as much as we can
  • and what’s left is our solution

I guess the easiest way to get it is by showing examples.

Let’s look at the following cases:

If we disregard the usual pizza pie images, and someone asks us how much is 2 times 1 half – we would spontaneously say 2 halves, i.e. 1 whole.

So I have an idea how to explore this first with you. As usual we’ll start with our second numerator being 1, and then we’ll build up as we go.

But first, let’s look at plain old multiplication, I mean whole numbers multiplication.




Multiplying fractions by whole numbers


First case of
Multiplying Fractions by whole numbers

  • Numerator > 1
  • Denominator = 1


× =
  4   ×   2   =   4×2 = 8 = 8
  1     1     1×1 1

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
Multiplying Fractions by whole numbers

  • Numerator = 1
  • Denominator > 1

But what happens when we put a less-than-one (proper) fraction as a second factor, say 1/2? We can interpret the case below as: what is the 1/2 of a 4.


× =
  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:

  • number 2 as a numerator has made our number 4 two times bigger
  • number 2 as a denominator has made our number 4 two times 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).










Third case of
Multiplying Fractions by whole numbers

  • Numerator > 1 and
  • Denominator > 1

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:

  • 3 times 2 is 6
  • 6 divided by 5 is 6/5

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:
simplifying fractions makes our job much easier. So we should factor our numerator and denominator and eliminate the same factors.





Multiplying fractions (by fractions)



First case of
Multiplying Fractions (by fractions)

  • Numerators = 1
  • Denominators > 1

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.






Second case of
Multiplying Fractions (by fractions)

  • Numerator > 1
  • Denominator > 1


× =
  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.






Multiplying Mixed Fractions

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:

  • 10 as 5 × 2
  • 9 as 3 × 3
  • 6 as 3 × 2
  • We left a 4
    (no need to factor it because it hasn’t any common factors with 5 and 3)

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
for for multiplying fractions page:

  • If necessary turn mixed fractions into improper ones
  • We factor the numerators and the denominators
  • We reduce the common factors
  • we multiply the numerators
  • we multiply the denominators
  • turn back our solution into mixed form if applicable






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