HOW TO GET THE PERCENTAGE? EXAMPLES AND EXERCISES - CULTURE VOCABULARY - 2025

You can get a percentage with several methods. You can quickly calculate 10% of any number just by moving its decimal point one place to the left. For example, 10% of 100 is 10; 10% of 1000 is 100.

If you want to calculate more complex percentages such as 36% of 25 or 250% of 20, you need to use other methods. For cases where the 10% system is not applicable, the following methodologies can be taken into account.

Figure 1. Discounts with different percentages. How much do we save in each one? Source: Pixabay.

The term percentage means a certain part of every hundred and refers to the arithmetic operation carried out to find that part. For example, 20% (read "twenty percent") discount in pesos means that for every 100 pesos 20 pesos are discounted.

The percentage is used to calculate how much of the total a quantity represents. In this case, the total is taken to the scale of 100 and the percentage informs what quantity, based on those 100, is the part to be calculated.

Let's see how to do it with these examples. First of all we do it as a fraction:

• 20% = 20/100
• 5% = 5/100
• 0.7% = 0.7 / 100
• 100% = 100/100

Note that 100% is equal to 1. But percentages can also be written in decimal form:

• 20% = 0.20
• 5% = 0.05
• 0.7% = 0.007
• 100% = 1.0

When you express the percentage of a certain number in decimal form, you simply shift the comma of that number two places to the left. In the percentage, the proportionality rule also applies:

20% is 20 out of 100, therefore:

20% of 100 is 20, 20% of 200 is 40, 20% of 300 is 60, 20% of 50 is 10.

General rule for 20% of any amount

This rule can easily be extended to find any other desired percentage. Let's see how in the next section.

Exercise solved with formula to calculate n%

A formula to summarize the above and quickly calculate any percentage n is:

n% = (A * n) / 100

For example you want to calculate 25% of 400

So n = 25 and A = 400, which results in (400 * 25) / 100 = 100

Example

What percent of 60 is 24?

Solution

What is asked is equivalent to asking what is the n% of 60 that gives 24?

We propose the general formula:

We solve for n with this procedure:

-The 100 that is dividing in the left member of the equality, goes to the right member by multiplying.

-And the 60 that multiplies in the left member goes to the right member dividing.

It is concluded that 40% of 60 is 24.

Solved problems of calculation of percentage

Here are some simple exercises to start practicing the above.

Exercise 1

Find 50% of 90.

Solution

Here X = 90, n = 50% and we substitute:

90 * 50% = 90 * (50/100) = 4500/100 = 45

This one is pretty simple, because 50% of any amount is half that amount and half of 90 is 45.

Exercise 2

Find 30% of 90.

Solution

90 * 30% = 90 * (30/100) = 2700/100 = 27

Percentage increases

It is common in everyday life to hear about an increase in something, for example an increase in production, a salary increase or a rise in a product. It is almost always expressed as a percentage.

For example, a certain product cost € 300 but suffered a 30% increase. We ask ourselves: what is the new price of the product?

The first thing is to calculate the portion that corresponds to the increase. Since the increase is 30 parts of 100, then the increase portion, based on the original price of 300, is three times the 30 parts, that is, 3 * 30 = 90.

The product increased € 90, so the new final price will be what it cost before plus the increase:

We can build a formula for calculating the percentage increase. We use letters to symbolize prices, like this:

- f is the final value

-i is the initial value and

-n is the percentage of increase.

With these names, the final value would be calculated like this:

f = i + (i * n / 100)

But since i is repeated in both terms, it can be taken as a common factor to obtain this other expression, equally valid:

f = i * (1 + n / 100)

Let's verify with the case already solved, the product that cost € 300 and increased 30%. This is how we make sure that the formula works well:

Exercise 3

An employee earned € 1,500, but was promoted and his salary increased by 20%. What is your new salary?

Solution

Let's apply the formula:

The new salary of the employee is € 1800.

Percentage decreases

In the case of decreases, the formula for calculating the final value f of a certain initial quantity i that suffered a decrease of n% is:

f = i * (1 - n / 100)

It should be noted that the positive sign (+) of the formula in the previous section was replaced by a negative sign (-).

Figure 2. Notice of percentage discount. Source: Pixabay

Exercise 4

One product marked € 800, but received a 15% discount. What is the new price of the product?

Solution 4

The final price according to the formula is:

The final price with the 15% discount is € 680, which represents a saving of € 120.

Successive percentages

It appears when some quantity undergoes a percentage variation and then another is applied, also percentage. For example a product that has had two percentage discounts in a row. Another example is an employee who had two consecutive pay raises.

- Successive percentage increases

The solution basis for these cases is the same as for single increases, but it must be taken into account that the second percentage increase is made on the final value of the first increase.

Suppose a product that rose first 10% and then 5%. It is incorrect to say that it suffered an increase of 15%, it was actually more than this percentage.

The formulas for the final value would be applied like this:

-First the final value of the first increase of n1% is calculated

-And then, to find the final value of the second increase of n2%, the final value of f1 is taken as the initial value. Thus:

Exercise 5

A book originally cost € 55, but due to its success and high demand, it suffered two consecutive increases over the original price. The first increase was 10% and the second 20%. What is the final price of the book?

Solution

-First increase:

-Second increase

The final price is € 72.6.

Exercise 6

In reference to the previous exercise. The two consecutive increases: what percentage of a one-time increase over the original price of the book correspond to?

Solution

If we call the single percentage increase n%, the formula that relates this single percentage increase to the original value and the final value is:

That is to say:

Solving for the percentage increase n% = (n / 100), we have:

Thus:

A total percentage increase of 32% was applied to the price of the book. Note that this increase is greater than the sum of the two consecutive percentage increases.

- Successive percentage discounts

The idea is similar to that of successive percentage increases. The second percentage discount must always be applied to the final value of the first discount, let's see an example:

Exercise 7

A 10% discount followed by a second 20% discount on an item, what single percentage discount is equal to?

Solution

-First discount:

Substituting the first equation in the second it remains:

Developing this expression, we obtain:

Taking common factor i:

Finally, the percentages indicated in the question are replaced:

In other words, the successive discounts of 10% and 20% correspond to a single discount of 28%.

Advanced exercises

Let's try these exercises only when the ideas in the previous ones are clear enough.

Exercise 8

The base of a triangle measures 10 cm and the height 6 cm. If the length of the base decreases by 10%, by what percentage must the height be increased so that the area of ​​the triangle does not change?

Figure 3. Alternative solution to exercise 8. Prepared by F. Zapata.

Solution 8

The original area of ​​the triangle is:

Now if the base decreases by 10%, then its new value is:

The new value for the height will be X, and the original area should remain unchanged, so that:

Then the value of X is solved as:

Which means an increase of 0.666 compared to the original value. Let's see now what percentage of this represents:

0.666 = 6 * n / 100

The answer is: the height must be increased by 11.1% for the area of ​​the triangle to remain the same.

Exercise 9

If a worker's salary is increased by 20%, but then the tax deducts 5%, he asks himself: what is the real increase that the worker receives?

Solution

First we calculate the increase of n1%:

Then we apply the discount of n2%:

The first equation is replaced in the second:

The previous expression is developed:

Finally, i common factor is taken and the values ​​of n1 = 20 and n2 = 5 that appear in the statement are substituted:

The worker received a net raise of 14%.

Exercise 10

Decide what is more convenient between these two options:

i) Purchase t-shirts with a discount of 32% each.

ii) Buy 3 shirts for the price of 2.

Solution

We analyze each option separately and then choose the most economical:

i) Let X be the current price of a t-shirt, a 32% discount represents a final price of Xf:

Xf = X - (32/100) X = X - 0.32X = 0.68X

For example, buying 3 T-shirts means spending 3 x 0.68 X = 2.04X

ii) If X is the price of a t-shirt, for 3 t-shirts you will simply pay 2X.

Suppose that a T-shirt is worth 6 euros, with the 32% discount it would be worth 4.08 euros. Buying 1 shirt is not a valid option in the 3 × 2 offer. So if you only want to buy 1 shirt, the discount is preferable.

But if you want to buy by the dozen, the 3 × 2 offer is only slightly cheaper. For example, 6 t-shirts with the discount would cost 24.48 euros, while with the 3 × 2 offer they would cost 24 euros

References

1. Easy Classroom. The percentage. Recovered from: aulafacil.com
2. Baldor A. 2006. Theoretical practical arithmetic. Cultural Editions.
3. Educa Peques. How to learn to calculate percentages. Recovered from: educapeques.com
4. Gutiérrez, G. Notes on Financial Mathematics. Recovered from: csh.izt.uam.mx
5. Smart ticks. Percentage: what it is and how it is calculated. Recovered from: smartick.es