## Percentage calculation

The percentage calculation is always applied when a Shares of a Whole should be determined. This is the case, for example, with winter sales. There the percentage calculation appears disguised as a discount: "25% on everything". How you can calculate the final price with this statement you will learn, among other things, in this article.

• Final sample exercise Percentage calculation

• ### Percent calculator

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### Percentage calculation formulas

Most students are taught the percentage calculation using three different formulas. In the context of these formulas, the following three terms, including their abbreviations, play a central role in the percentage calculation:

\ begin {align *}
& \ textrm {Basic value} (G) = \ frac {\ textrm {Percentage} (W) \ \ cdot \ 100} {\ textrm {Percentage} (p)} \ \
& \ textrm {percentage value} (W) = \ frac {\ textrm {basic value} (G) \ \ cdot \ \ textrm {percentage} (p)} {100} \ \
& \ textrm {Percentage} (p) = \ frac {\ textrm {Percentage} (W) \ \ cdot \ 100} {\ textrm {Basic value} (G)}
\ end {align *}

The following tasks are intended to clarify the above formulas and briefly show how they are used. Always think of the units in your answer sentence!

### Calculate percentage

Task 1) Calculate 10 percent of 500 kg

In this task, the percentage value \$ W \$ is sought. So we use our formula for the percentage and get:

\ [\ textrm {Percentage} (W) = \ frac {\ textrm {Basic value} (G) \ cdot \ textrm {Percentage} (p)} {100} = \ frac {500kg \ cdot 10} {100} = \ frac {5000kg} {100} = 50 \ kg \]
At this point it may be easier and in any case faster to calculate 10% of 500 kg in a different way. To do this, let's make it clear that the following relationship applies:
\ [10 \% = \ frac {10} {100} = 0.1. \]
With the help of this knowledge we now calculate: \$ 0.1 \ cdot 500 \ kg = 50 \ kg \$.
You can of course decide for yourself which calculation method suits you better. Which of the two paths you ultimately use does not matter in the test.
2. What percentage is 60 cm of 300 cm? We find the percentage and calculate with the appropriate formula:
\ [p = \ frac {W \ cdot 100} {G} = \ frac {60 \ cdot 100} {300} = \ frac {6000} {300} = 20 \ \% \]
Answer: 60cm is 20 percent of 300cm.

To repeat the topic of percentage calculation, watch the following explanatory video.

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### Calculate the basic value

As a reminder, the formula to calculate the base value is:

\ begin {align *}
\ textrm {Basic value} (G) = \ frac {\ textrm {Percentage} (W) \ \ cdot \ 100} {\ textrm {Percentage} (p)}
\ end {align *}

The tasks on the increased and decreased basic value play an equally important role in the percentage calculation. We also want to look at one task at a time.

The price of a pair of pants has increased by 25 percent and is now € 200. What was the original price of the pants?

Here we have to take into account that the basic value has already been increased by 25 percent and our percentage value is therefore 25 percent more. This means that our percentage is 125%. We are looking for the original price of our pants, i.e. the basic value. We put our corresponding values ​​in the formula and get:

\ begin {align *}
G = \ frac {W \ cdot 100} {p} = \ frac {200 € \ cdot 100} {125} = \ frac {20,000 €} {125} = 160
\ end {align *}

Answer: So the original price of our pants was € 160.

Percentage calculation, increased, decreased basic value with the rule of three | Math by Daniel Jung

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### Decreased basic value

The price of a pair of pants has been reduced by a percentage of 20% and is now € 120. What was the original price of the pants?
Our basic value has been reduced by 20 percent. The percentage that remains now corresponds to \$ 100 \% - 20 \% = 80 \% \$. So we are looking for our original basic value again. We put the values ​​we know into the formula and get:

\ [G = \ frac {W \ cdot 100} {p} = \ frac {120 € \ cdot 100} {80} = \ frac {12000 €} {80} = 150 \ \]

Answer: Originally the pants cost 150 €.

There are already 20 m of a path paved. That is 40% of the total length. What is the total length of the path? In this case, the basic value is sought. We use the formula we know and get:

\ [G = \ frac {W \ \ cdot \ 100} {p} = \ frac {20m \ \ cdot \ 100} {40} = \ frac {2000m} {40} = 50m \]

Answer: The path has a total length of 50m

### Calculate percentage

To calculate the percentage, we use the following formula:

\ begin {align *}
\ textrm {Percentage} (p) = \ frac {\ textrm {Percentage} (W) \ \ cdot \ 100} {\ textrm {Basic value} (G)}
\ end {align *}

What percentage is 60 cm of 300 cm? We find the percentage and calculate with the appropriate formula:

\ [p = \ frac {W \ cdot 100} {G} = \ frac {60cm \ cdot 100} {300} = \ frac {6000cm} {300} = 20 \ \% \]

Answer: The percentage is 20 percent.

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### Final sample exercise Percentage calculation

A can of 125g fruit gum costs € 1.50. A discounter advertises with the following poster:

Offer! 125g + 30% more content for only 1.99 €

1. Calculate how many grams of fruit gums are being sold on offer.

Solution:

Exercise part 1:

According to the offer, we receive an additional amount of 30%. At this point, we can use a simple invoice to calculate the quantity we will receive in the offer:

\ begin {align *}
125g \ cdot 1.3 = 162.5g
\ end {align *}

So we are getting \$ 162.5g fruit gums in the offer.

Exercise part 2:

We should now find out whether the offer is really cheaper compared to the original price. Therefore we now calculate the respective price per \$ 100g \$:

\ begin {align *}
1.50: 125 \ cdot 100 = 1.20
\ end {align *}

If we take the original price as a basis, fruit gums cost exactly \$ 1.20.

Then we calculate the price per \$ 100? \$ Based on the offer price:

\ begin {align *}
1.99: 162.5 \ cdot 100 \ approx 1.22
\ end {align *}

Based on the offer price, fruit gums cost approximately \$ 1.22 €. The offer described is therefore not an offer at all, just a sham package.

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