What is 5 out of 1 2

Multiply fractions together

You already know how to multiply a fraction by a natural number. You multiply the fraction:



And how do you multiply two fractions ??

For example $$ 3/4 * 2/3 $$ or $$ 2/6 * 4/5 $$?

In these cases you are looking for the fraction of a fraction. Or more precisely: the Fraction of a fraction.

Sounds complicated? Pictures help! Here we go:

As a reminder, you multiply a fraction by a whole number by multiplying the numerator of the fraction by the number and keeping the denominator.

pictures say more than words

The task: $$ 2/3 * 3/8 $$

That means:

You are looking for the fraction $$ 2/3 $$ of $$ 3/8 $$.
Imagine the $$ 3/8 $$. You divide this fraction into 3 parts, because you are looking for $$ 2/3 $$ of it. In relation to a whole, you will then receive $$ 1/4 $$.

You can see it much better with a picture:



(Take your time to understand the picture! Nobody straps that at first glance. :-))

One more picture

You can also imagine the picture for the tasks differently:

$$2/6*4/5$$

Again you are looking for the fraction $$ 2/6 $$ of $$ 4/5 $$.

Divide the whole and mark the one break. Here are 4 of 5 lines.



Now you divide the whole thing in the other direction and mark the other fraction. Here are 2 of 6 columns.



The fraction you are looking for is the double-colored area. That's 8 out of 30 fields.

The result is called $$ 2/6 * 4/5 = 8/30 $$.

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Do you find the rule

And now you should paint these pictures for every task?!?!?

No, but you can use it to find the calculation rule!

$$2/6*4/5=8/30$$

How do you come up with 8 in the numerator and 30 in the denominator?

Exactly: $$ 2 * 4 = 8 $$ and

$$5*6=30$$

In short, the rule for multiplying fractions is: and.

Imagine that many students find it much easier to multiply fractions than to add them. "Calculating times" easier than "plus calculating" !! ?? Yes, because one rule is easy to remember!

Testing the rule

Check with the first example whether the rule fits.



According to the picture: $$ 2/3 * 3/8 = 1/4 $$.

Apply the rule (numerator times numerator and denominator times denominator):

$$2/3*3/8=(2*3)/(3*8)=6/24$$

Oops, that's not the same at all?

Don't forget to shorten : $$ 6/24 $$ abbreviated with 6 is $$ 1/4 $$.

You multiply two fractions by multiplying the numerators and denominators, respectively.
Or in short: and.

Examples

$$1/3*2/5=(1*2)/(3*5)=2/15$$

$$20/3*4/13=(20*4)/(3*13)=80/39$$

With mixed numbers:

Convert mixed numbers to fractions first:

$$4 2/3*3 1/5=14/3*16/5=(14*16)/(3*5)=224/15=14 14/15$$

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Skilful shortening simplifies the calculation

$$4/2*6/3=(4*6)/(2*3)=24/6=4$$

That pays off well. But the task can get easier if you cut before multiplying.

$$4/2*6/3=(4*6)/(2*3)=(2*2)/(1*1)=4/1=4$$

Sometimes you can shorten before you start:

$$4/2*6/3=2/1*2/1=2*2=4$$

Clever shortening can make life a lot easier, huh?

It can be worthwhile to shorten several times.

Multiply multiple fractions

Sure, you can multiply more than 2 fractions. Before you do the math, see if you can cut back.

Example 1:

$$2/3*4/5*5/2=(2*4*5)/(3*5*2)=4/3$$

Example 2:

Here you can shorten several times. You can reduce the numerator and denominator of different fractions to the same number. All numerators and all denominators are connected by a multiplication sign.

$$21/3*5/14*6/10=(21*5*6)/(3*14*10)=(7*1*6)/(1*14*2)=42/28=3/2$$

Example 3:


Finally an example for "short-artist":

$$15/12*4/10*9/20*16/6=(15*4*9*16)/(12*10*20*6)=(5*2*3*4)/(4*5*5*2)=3/5$$