A percentage is a way to define a part, or fraction of a total.

When we talk about percentages, we should always ask ourselves the following: "the percentage of what?". Saying 50% without specifying 50% of what, makes no sense. However, if we say "the $50\%$ of $80$" is $40$. In summary, the percentage represents what part of $100$ is the number in question.

The percentage symbol is $\%$: when we want to express that $a\%$ We will write it as: $a \over 100$

To solve percentage problems, we will use the following formula

If we want to know what percentage $(Y)$ of a certain number $(X)$ is, the formula to use is:

$\frac{X}{100}×Y$

If we want to know what percentage $A$ is of $B$, the formula to use is:

$\frac{A}{B}×100$

How to calculate the percentage?

To better assimilate the topic of percentages, you must first understand the concept behind it. Imagine you have a board of squares, as in the following drawing:

This board has $10$ columns and $10$ rows, so there are a total of $100$ squares. One percent of the board is just one square. The $5\%$ will be five squares. And the $100\%$ will be the entire board.

In other words, one percent is a hundredth, or one hundredth part. When we are given percentages, we can always represent them as fractions, where the numerator is the percentage, and the denominator is $100$. This is represented with the percentage sign $\%$. The line between the two small circles represents the fraction line, and the two circles indicate the two zeros in the number $100$.

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So, $100\%$, are 100 within $100$. $100 \over 100$That is, the whole total.

How do you calculate the percentage of something?

Another topic we must understand clearly is the concept of "total". When we talk about percentages, we always have to ask ourselves, "the percentage of what?". Saying $50\%$ without specifying $50\%$ of what, makes no sense.

Let's look at a very simple example: $50\%$ of $100$ is $50$; $50\%$ of a million is $500,000; 50\%$ of $8$ is $4$, and so on.

In other words, to solve percentage problems, first, you have to understand what the total is, and what $100\%$ is before proceeding to calculate the percentage itself.

First, you must understand the function of each piece of data: the total and the percentage. For example: a shirt that costs $200$ dollars, will be the total, while the percentage will be, let's say as an example, a $25\%$ discount. Without understanding the function of each of these pieces of data, it will be very difficult for you to solve this type of exercises.

Let's suppose we are asked how much we should pay for a shirt that costs 200, when we receive a discount of $25\%$. In this case, we must do the following calculation: $25$ times $200 = 5000$. This is the initial step to know what discount you will receive. In summary, you must perform a multiplication operation between the percentage (discount) and the total (price).

Now you must divide $5000$ by $100$. The result obtained is $50$. So in this case, a discount of $25\%$, actually means that the discount received is $50$ dollars. How much will the shirt cost you after the discount? $150$ pesos. This is a classic example of an exercise where you must demonstrate your knowledge on how percentages are calculated.

How to Calculate the Discount Percentage, Exercise with Solution

Imagine you have won $120$ dollars in a bet, and that you have promised to give your little brother $30\%$ of what you have won. How much do you owe your brother? The number we want to find will be represented by the letter $X$.

We know that $X$ is a part of the money we have won in the bet. Therefore, we can write it as follows: $X \over 100$

What part does $X$ represent? the $30\%$ So that means: $\frac{X}{120}=30\%$ As we have already learned, $30\%$ is: $30 \over 100$ Therefore, we can say that:

$\frac{X}{120}=\frac{30}{100}$ We can easily solve this equation. We multiply crosswise and this is the result: $100\times X=30\times 120$

$100X=3600$ $X=36$

So, the $30\%$ of $120$, is $36$. Keep in mind that the value of the percentage represents the real value that the percentage represents. In this case, the value of the percentage represents the money we will give to our brother, for having promised him the $30\%$. In summary, the percentage is $30\%$, and the value of the percentage is $36$.

Let's look at another example

How to calculate the percentage of a quantity

Alejandra and Natalia bought $30$ hair ribbons. Alejandra took $10\%$ of the ribbons, and the rest were given to Natalia as a gift. How many did Alejandra keep? We will perform the calculation in exactly the same way:

Since the total is $30$, and the percentage is $10$. First, we multiply $10$ by $30$, which gives us $300$. Then we divide this figure by $100$, resulting in $3$. Therefore, Alejandra kept $3$ hair ribbons, and the rest ($27$), were given to Natalia as a gift.

In fact, if we formulate it as a formula, we can find the missing data. The formula will be:

$\frac{X}{30}=\frac{10}{100}$

This formula will be used in any percentage exercise. You must understand the data well, and introduce it very carefully into the formula. To show you various cases where you will use this formula (each time in a slightly different way), we will show you the following examples:

Suppose that in a clothing store, you find a shirt you want to buy. The shirt costs $200$ dollars, but there is a $20\%$ discount.

How much will the shirt cost after the discount?

Solution: Let's look at the formula we wrote earlier, and place in it the data we already have. It must be taken into account that if the shirt is sold with a $20\%$ discount, it currently costs $80\%$, so our percentage will be $80$. The percentage $=80$. The initial amount, that is, the original price is $200$ The percentage value is what we are missing, since we do not know how much we will have to pay after the discount. Therefore, the percentage value will be $X$. Which we will formulate in the following way: $\frac{80}{100}=\frac{X}{200}$ We multiply crosswise and this is the result: $100X=16000$ $X=160$

After the discount, the price of the shirt is $160$ dollars.

Another way to reach the same result is to calculate what the discount is

So, the first thing would be to reveal what the amount of the discount is. Then, we will have to subtract this discount from the original price, and only then will we arrive at the price of the shirt after the discount:

Percentage $=20$ Initial amount $=200$ What we do not know is what the discount amount ($20\%$) will be. We will call this the Percentage Value. We call this missing data $X$. Which can be represented as follows: $\frac{20}{100}=\frac{X}{200}$ We multiply crosswise and this is the result: $100X=4000$ $X=40$

Whose result is: $X=40$ $40$ is not the amount we will have to pay after the discount. $40$ represents the true value of $20\%$ of$200$. So $40$ dollars is the discount when buying the shirt.

Look again at what we have been asked in this exercise: the price of the shirt after the discount. Now that we know that the discount is $40$ dollars. We can calculate:

Tricky question that might confuse you

The shorts are sold with a $40\%$ discount. After the discount, they cost $300$ dollars (they seem to be luxury pants). How much did the pants originally cost, before the discount? Keep in mind, that we have the price after the discount and the discount percentage. If the pants were sold with a $40\%$ discount, now it costs $60\%$ of the price. We can deduce that: the percentage is equal to $60$. We know that the value of $60\%$ of the pants is equal to $300$ dollars, because $300$ dollars is the price of the pants after the discount. Therefore, the value of the percentage is equal to $300$.

What remains for us to find is the initial amount. We will call it $X$. We will formulate it as follows: $\frac{60}{100}=\frac{300}{X}$ We multiply crosswise and this is the result: $60X=30000$ $X=500$

Therefore, $500$ is the original price of the pants before the discount. (We have already told you that these are luxury pants).

So far, we have known the value of the percentage. Now, we want to find the percentage itself.

Let's look at the following example

A bracelet originally costs $50$ dollars. Now, the selling price is $58$ dollars. What? Has the price gone up? Yes! By what percentage did the price of the bracelet increase? Solution: Keep in mind that in this case, the initial amount is given and we can also find the percentage value, but the percentage itself is what is missing. The initial amount is $50$. Its current value after the price increase is $58$. The percentage is what is missing ($X$). Which we will formulate as follows: $\frac{X}{100}=\frac{58}{50}$ We multiply crosswise and this is the result: $50X=5800$ $X=116$

In fact, the percentage we received is higher than $100$.

What does this mean? It means that the bracelet is sold at $116\%$ of its original price. The bracelet has become more expensive, and therefore now costs more than it did before. How much more? A $16\%$. $116-100=16$

Sometimes, you will find complex exercises that have several stages to be solved. For example, a product that costs a certain price, and the value has gone up by $20\%$. After that, it was reduced by $10\%$ of the new price. How much does the product cost now? First, you must calculate the new price after the price increase. Only then can you calculate the new price, that is, the reduction of $10\%$ of the new price.

If we take any number, increase it by a certain percentage, and then subtract that same percentage from the resulting number, we get a number that is lower than the initial number! And also, keep the following in mind: $X\%$ of $Y$ is exactly equal to $Y\%$ of $X$.

The secret to success in this type of exercise is practice! Practice the entire topic of percentages using the indicated formulas, and try to solve all kinds of exercises. This way, you will know how to use the formulas effectively, and you will get the correct answers.

Examples of exercises

Exercise 1

How much is $¼$ of $20$?

Solution:

When we are asked to find a part ($¼$) of a whole number ($20$), we must multiply the part (in this case $¼$), by the whole number.

The price of a shirt was $40$ dollars, and now that there are discounts; its price has dropped by $20\%$.

We are asked to calculate the price of the shirt, after the $20\%$ discount.

Solution:

After the price discount, the shirt costs $80\%$ (of the original price).

The $80\%$ are actually $\frac{80}{100}=\frac{8}{10}=0.8$

To calculate $80\%$ of $40$ we will perform a simple multiplication: $40\times 0.8=32$

Therefore, the price of the shirt after the price discount, is $32$ dollars.

Exercise 3

Luis bought Juana a gift for the end-of-year holidays. When Juana asked him how much the gift cost, Luis replied that its real value is $400$ dollars, but that he got a $30\%$ discount. How much did Luis pay for the gift?

Here is the calculation:

$30\times 400=12000$

$\frac{12000}{100}=120$

The discount Luis received on the purchase of the gift was $20\%$ dollars. Here is the complete calculation: $400-120=280$, so he paid $280$ dollars for the gift.

Ana and María bought $50$ cookies. Ana ate $20\%$ of the cookies and María ate the rest. How many cookies did María eat?

This is the calculation:

The total number of cookies is $50$, and Ana ate $20\%$ of them. Therefore, we will perform the following multiplication: $20\times 50=1000$

The number we obtained was $1000$, divided by $100=10$. So, Ana ate $10$ cookies.

The complete calculation is:$50-10=40$. Therefore, María ate $40$ cookies.

Exercise 5

How much is $40\%$ of $500$?

This is a question that is formulated differently compared to the previous questions. Therefore, we will have to use a different calculation formula. In this case, you should divide the whole number by $100$ and then multiply by the percentage.

They filled a fish pond over two days. The first day they filled $180$ cubic meters of water, and this amount constitutes $40\%$ of the amount filled over two days.

Task:

How much water was filled over two days?

Solution:

Given that $180$ cubic meters were filled in a pond and it is $40\%$ of the amount filled over two days and it is $100\%$.

The question asks us to calculate how many cubic meters were filled in two days.

We calculate:

$40\%\text{ = 180}$

100=?

$\frac{180\times 100}{40}=\frac{1800}{4}=450$

The amount after two days is: $450$m³

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Here we can also reduce the 2 in the numerator and the denominator:

$\frac{3\times21}{10}=\frac{63}{10}=6.3$

Answer

6.3

Exercise #2

A 20% discount is applied to a toy ordinarily priced at $40.

What is the new price after the discount?

Video Solution

Step-by-Step Solution

$40-8=32$To find percentages, we must use two pieces of information we have: the total amount ($40) and the discount (20%).</p><p>This information can be placed in the following formula:</p><p><span class="katex">\( \frac{Price\times\text{Percentage}}{100}$Which allows us to find the percentage of something.

We replace:

$40\times\frac{20}{100}=$ $\frac{800}{100}=$

$8$

We discover that the value of the discount is 8.

But we are not finished yet!

We need to subtract the discount from the original amount to find the sale price: \( 40-8=32

Answer

32

Exercise #3

The price of to movie ticket htos risen from 40 to 45 pesos By whtot percenttoge did the price incretose?

Step-by-Step Solution

To answer the question, first we have to understand how much the ticket costs:

45-40=5

That is, the price of the ticket went up by 5$.

Now we would like to know what is the percentage value of 5 and, for that, we will divide the increase by the original price and multiply it by 100 to convert it into a percentage.

5/40 * 100

First we can start by converting the 100 into fraction form.

5/40 * 100/1

When there is multiplication of fractions, we can multiply numerator by numerator and denominator by denominator.