# What is a fifth of 100

### Fractions and percentages

Lisa and Jannis train for a sports badge.

Lisa has already met 80 percent of the requirements for a gold badge. Jannis has not yet achieved a fifth of the required performance.

Confusing? Which of the two is now the bigger sports fan? This is so difficult to say because the proportions are given once as a fraction and once as a percentage.

You can state proportions not only as fractions, but also as a percentage. How are these two statements related?

### What does percent $$% $$ actually mean?

In order to be able to compare proportions more easily, there is this trick with $$% $$: You divide the whole thing into $$ 100 $$ equal parts, no matter how big the whole thing is. One part is then a hundredth.

One hundredth is one percent.

In short: $$ 1/100 = 1 $$ $$% $$

As a picture: You color 1 box out of 100 boxes.

What if you ink more boxes?

Here 43 boxes out of 100 boxes are colored. These are $$ 43/100 $$ or $$ 43 $$ $$% $$.

You can specify proportions as a fraction or with percent $$% $$.

You can easily convert hundredths of a fraction into percent.

The following applies: $$ 1/100 = 1 $$ $$% $$

**percent** (lat.):

**Per:** of**centus:** hundred

Percentages always relate to the whole. 43% of 100 students are different from 43% of 1000 students. You will learn how this is all connected later. :)

### Which fraction is the same as 80%?

Back to the task:

$ 80% $ means nothing else than $ 80 $ from $ 100 $ or $ 80/100 $.

Actually, you don't need to convert anything here. You just write the percentage on the fraction line (in the numerator) and a $$ 100 $$ below it (in the denominator). If possible, trim the fraction.

So:

$$80/100 = 8/10 = 4/5$$

So if Lisa has met $$ 80% $$ of the requirements,

then there are always $$ 4 $$ of each $$ 5 $$ athletic achievements.

So she was pretty good there, wasn't she?

How to convert a percentage to a fraction:

- Write the percentage in the numerator and 100 in the denominator.
- Brevity.

Example: $$ 10% = 10/100 = 1/10 $$

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### What percentage is $$ 1/5 $$?

The reverse is not much more difficult either. You only need to expand or reduce the fraction until the denominator is $$ 100 $$. Then the numerator is your percentage.

At $$ 1/5 $$ you expand with $$ 20 $$ and get $$ 20/100 $$.

So:

$$ 1/5 stackrel (20) = (1 * 20) / (5 * 20) = 20/100 = 20% $$

So you can read the percentage directly in the meter.

So Jannis has not yet provided $$ 20% $$ of the required services.

### Can you think of what?

Lisa did $$ 80% $$,

Jannis are still missing $$ 20% $$.

$$ 100% $$ always means "everything".

In this case "all the services to get the sports badge".

If Lisa has achieved $$ 80% $$, then she automatically lacks $$ 20% $$ of the achievements.

Lisa and Jannis are both equally well prepared for the sports badge.

It didn't sound like that at first.

How to convert a fraction to a percentage:

- Extend the fraction to a fraction of a hundred.
- The numerator is the percentage you are looking for.

Example: $$ 3/5 stackrel (20) = 60/100 = 60% $$

### Is it always that easy?

Actually already. However, there are denominators that cannot easily be expanded or reduced to $$ 100 $$. In this case, you take a few more steps to get the result.

**Example 1:**

Enter the fraction $$ 42/60 $$ as a percentage.

Because $$ 100 $$ is not a multiple of $$ 60 $$, you cannot simply expand to $$ 100 $$ here. But you can reduce the fraction with $$ 6 $$. That gives $ 7/10 $.

$$42/60 = (42 : 6)/(60 : 6) = 7/10$$

You can expand this fraction with $$ 10 $$ and get $$ 70/100 $$, i.e. $$ 70% $$.

$$7/10 = (7 * 10)/(10 * 10) = 70/100 = 70 %$$

**Example 2:**

What percentage is $$ 27/45 $$?

It's best to shorten it with $$ 9 $$. Then you have $$ 3/5 $$. Now you only need to expand with $$ 20 $$ and you get $$ 60/100 $$ or $$ 60% $$ as the result.

$$27/45 = (27 : 9)/(45 : 9) = 3/5$$

$$ 3/5 = (3 * 20)/(5*20) = 60/100 = 60 %$$

Unfortunately, it doesn't work so well with all fractions ... For example, you cannot expand $$ 1/3 $$ to a fraction of 100. But you don't have to be interested in that at first, you'll learn that later.

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