Examples
In this section, we cover some of the most common applications where we use percentages.
Example 1
How much is 60% of 30?
Solution: First, we convert 60% into a decimal number:
Then we multiply the number 30 by 0,6:
The answer is 18.
Example 2
How many percents is 90 kg of 300 kg?
Solution: First, we calculate the ratio:
The ratio we obtained is a decimal number. Finally, we convert the decimal number into a percentage:
Example 3
A product currently costs 420 euros, but this is only 80% of the original price. What is the original price?
Solution: Let us denote the original price . We can write 80% of as . Since 420 euros is 80% of , we can write
So the original price is 525 euros.
We notice that the original value can be calculated by dividing the new value by the percentage value (after converting it to a decimal).
Example 4
A rent of 490 euros is increased by 2%. What is the new rent?
Solution: There are two main methods for solving this problem.
Method 1: The new rent percentage is
which converts to a coefficient of 1,02. The new rent is
The answer can be obtained by, for example, long multiplication.
Method 2: Let us first calculate how much is 2% of 490 euros:
This is straightforward to see, as 1% of 490 euros is 4,90 euros, so 2% is 9,80 euros. The new rent is
Example 5
a) How much smaller is the number 4 compared to the number 5?
b) How much larger is the number 5 compared to the number 4?
Express your answer as percentages.
Solution: a) First, let us calculate the difference between the numbers 4 and 5:
Then we find the ratio between the difference and the number 5 (note: since we are comparing to number 5, we must divide by 5, not by 4):
Finally, we convert the decimal number into percentages:
The negative sign expresses that the number 4 is smaller. Therefore, we conclude that the number 4 is 20% smaller than the number 5.
b) Note that while the number 4 is 20% smaller than the number 5, this does not mean that the number 5 is 20% larger than the number 4. To verify this, you can calculate how much is 120% of 4. For the actual question, the solution method is the same as in part a): we calculate the difference between the numbers, and divide by the number that we are comparing to:
Finally, we convert the decimal 0,25 into a percentage:
So the number 5 is 25% larger than the number 4.
Example 6
It is time for a super sale. A washing machine was sold at a 50% discount. The customer had a special coupon that gave a further 60% discount, finally leading to a lowered price of 104 euros. What is the original price?
Solution: Let us denote the original price . After a 50% discount, the lowered price is
of the original price. We convert this 50% into a decimal number: . So after the first discount, the lowered price can be written as
After the second 60% discount, the final price is
of the previous price. We convert this 40% into a decimal number: . Next, we combine the discounts. The previous price was , so the final price is 40% of the previous price:
That is, the final price is 20% of the original price. To obtain the original price, we create the following equation:
So the original price was 520 euros.
Example 7
Anna, Bridgette and Cecilia share their reward of 620 euros so that Bridgette receives 20% more than Anna, and Cecilia receives 10% less than Anna. How many euros did each person receive?
Solution: As both comparisons were made with Anna, let us denote the reward that Anna received . Since Bridgette received 20% more than Anna (that is, 120% of what Anna received), the reward received by Bridgette is . Similarly, as Cecilia received 10% less than Anna (that is, 90% of what Anna received), the reward received by Cecilia is . In total, the rewards received by the three people are
As the total reward is 620 euros, we create the following equation:
So Anna received 200 euros. Bridgette received
and Cecilia received
In total, they received euros.