# What is 3 5 6 8 3

### The division of fractions

You know what fractions are and you can add, subtract, multiply and divide a fraction by a natural number. Still missing? The division of two fractions!

As a reminder, here are the most important rules again! Then the rule for dividing will be easier for you!

RULE | example |
---|---|

A fraction consists of a numerator and a denominator: $$ (COUNTER) / (NEN NER) $$ | |

You multiply two fractions by multiplying the numerators and denominators, respectively. Or in short: and | $$1/2*3/4= (1*3)/(2*4)$$ $$=3/8$$ |

You divide a fraction by a natural number by adding the | $$4/5:3=4/((5*3))$$ $$=4/15$$ |

### What does it mean to divide two fractions?

The task: $$ 3/4: 3/8 $$

That means:

How often does the fraction $$ 3/8 $$ fit into the fraction $$ 3/4 $$?

As a picture:

Move the $$ 3/8 $$ pie piece and think about how it fits into the area of $$ 3/4 $$.

Exactly 2 of the $$ 3/8 $$ - cake fit into the $$ 3/4 $$ - cake:

So the calculation is: $$ 3/4: 3/8 = 2 $$

Do you remember? Even when dividing whole numbers, you wondered how often one number fits into another.

$$ 8: 2 = 4 $$ told you that the 2 fits into the 8 exactly 4 times

### An example when it doesn't fit that well

The task: $$ 6/9: 3/6 $$

That means: How often does the fraction $$ 3/6 $$ fit into the fraction $$ 6/9 $$?

Imagine it figuratively:

Move the $$ 3/6 $$ block: The block fits in a whole time and in addition to a fraction of $$ 1/3 $$. The $$ 3/6 $$ fit $$ 1 1/3 $$ times into $$ 6/9 $$.

The problem is: $$ 6/9: 3/6 = 1 1/3 = 4/3 $$

*kapiert.de*can do more:

- interactive exercises

and tests - individual classwork trainer
- Learning manager

### Do you already find the rule?

Try to derive a rule from the examples:

That of the result is derived from **multiplication** the one ruptures with that of the other.

That results from the **multiplication** the one ruptures with that of the other.

In brief the 3rd example: $$ 6/9: 3/6 = 6/9 * 6/3 = (6 * 6) / (9 * 3) = 36/27 $$

You turn the division task into a painting task! To do this, turn the second fraction over. In mathematical terms, that means: You build it **Reciprocal of the fraction**.

You divide two fractions by multiplying the first fraction by the reciprocal of the second fraction.

Example: $$ 5/3: 7/2 = 5/3 * 2/7 = (5 * 2) / (3 * 7) = 10/21 $$

**The reciprocal**:

For every break there is a valuable partner: the **Sweeping break** or **Reciprocal**. Swap the numerator and denominator and you get the reciprocal value.

The reciprocal of $$ 2/3 $$ is $$ 3/2 $$.

The reciprocal of $$ 5 = 5/1 $$ is $$ 1/5 $$.

### Examples, examples

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

$$5/6:2/7=5/6*7/2=35/12$$

### And with shortening

Skilful shortening is always good. :-)

$$11/7:22/35=11/7*35/22=(1*5)/(1*2)=5/2$$

$$24/15:16/25=24/15*25/16=(6*5)/(3*4)=(2*5)/(1*4)=(1*5)/(1*2)=5/2$$

Do not shorten it until you have converted the division problem into the painting problem.

### Division of mixed numbers

As with multiplying, mixed numbers are first converted into an improper fraction.

**Example:**

$$2 1/3:5 2/3=7/3:17/3=7/3*3/17=7/17$$

**Example 2:** with shortening

$$4 4/5:3 6/10=24/5 : 36/10=24/5*10/36=(2*2)/(1*3)=4/3=1 1/3$$

*kapiert.de*can do more:

- interactive exercises

and tests - individual classwork trainer
- Learning manager

### Double fractions

Do you remember: A fraction is nothing more than a division problem.

$$ 1/2 = 1: 2 = 0.5 $$ or $$ 3/4 = 3: 4 = 0.75 $$

You need that with **Double fractions**. Double fractions? They have a fraction in the numerator and denominator.

**Examples:**

$$(3/4)/(5/8)=3/4:5/8=3/4*8/5=6/5$$

$$(10/4)/(9/2)=10/4*2/9=10/18=5/9$$

### What do you need the division of fractions for?

You need the division when you divide a fraction evenly.

**Example:**

A bottle contains $$ 3/4 $$ liters of juice. How many glasses ever

Can you fill 150ml ($$ 3/20 $$ liters) with it?

Solution:

$$3/4:3/20=3/4*20/3=5$$

Answer: You can fill exactly 5 glasses.

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