In this section, we cover some of the most common applications where we use percentages.

How much is 60% of 30?[br][br][b]Solution:[/b] First, we convert 60% into a decimal number:[br][br][math] \large \frac{60}{100} = 0{,}6 [/math][br][br]Then we multiply the number 30 by 0,6:[br][br][math] \large 0{,}6 \cdot 30 = 18 . [/math][br][br]The answer is 18.

How many percents is 90 kg of 300 kg?[br][br][b]Solution:[/b] First, we calculate the ratio:[br][br][math] \large \frac{90 \, \mathrm{kg}}{300 \, \mathrm{kg}} = \frac{9}{30} = \frac{3}{10} = 0,3 . [/math][br][br]The ratio we obtained is a decimal number. Finally, we convert the decimal number into a percentage:[br][br][math] \large 0,3 = 0,3 \cdot 100\% = 30\% . [/math]

A product currently costs 420 euros, but this is only 80% of the original price. What is the original price?[br][br][b]Solution[/b]: Let us denote the original price [math]x[/math]. We can write 80% of [math]x[/math] as [math]\large \frac{80}{100} \cdot x = 0,8x[/math]. Since 420 euros is 80% of [math]x[/math], we can write[br][br][math] \large \begin{array}{rcll}[br]0,8x & = & 420 & | : 0,8 \\[br]x & = & \frac{420}{0,8} & | \ 0,8 = 4/5 \\[br]x & = & \frac{420}{4/5} & | \frac{a}{b} : \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c} \\[br]x & = & 420 \cdot \frac{5}{4} & | \ 420/4 = 105 \\[br]x & = & 105 \cdot 5 \\[br]x & = & 525[br]\end{array} [/math][br][br]So the original price is 525 euros.[br][br]We notice that the original value can be calculated by dividing the new value by the percentage value (after converting it to a decimal).

A rent of 490 euros is increased by 2%. What is the new rent?[br][br][b]Solution:[/b] There are two main methods for solving this problem.[br][br][b]Method 1:[/b] The new rent percentage is[br][br][math] \large 100\% + 2\% = 102\% , [/math][br][br]which converts to a coefficient of 1,02. The new rent is[br][br][math] \large 1,02 \cdot 490 \, \text{EUR} = 499,80 \, \text{EUR} . [/math][br][br]The answer can be obtained by, for example, long multiplication.[br][br][br][b]Method 2:[/b] Let us first calculate how much is 2% of 490 euros:[br][br][math]\large 0,02 \cdot 490 \, \text{EUR} = 9,80 \, \text{EUR} . [/math][br][br]This is straightforward to see, as 1% of 490 euros is 4,90 euros, so 2% is 9,80 euros. The new rent is[br][br][math]\large 490 \, \text{EUR} + 9,80 \, \text{EUR} = 499,80 \, \text{EUR} . [/math]

a) How much smaller is the number 4 compared to the number 5?[br]b) How much larger is the number 5 compared to the number 4?[br]Express your answer as percentages.[br][br][b]Solution:[/b] a) First, let us calculate the difference between the numbers 4 and 5:[br][math] \large 4-5 = -1 . [/math][br]Then we find the ratio between the difference and the number 5 ([b]note:[/b] since we are comparing to number 5, we must divide by 5, not by 4):[br][math] \large \frac{4-5}{5} = \frac{-1}{5} = -0,2 . [/math][br]Finally, we convert the decimal number into percentages:[br][math] \large -0,2 = -20\% . [/math][br][br]The negative sign expresses that the number 4 is smaller. Therefore, we conclude that the number 4 is 20% smaller than the number 5.[br][br]b) Note that while the number 4 is 20% smaller than the number 5, this does [b]not[/b] 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:[br][br][math] \large \frac{5-4}{4} = \frac{1}{4} = 0,25 . [/math][br][br]Finally, we convert the decimal 0,25 into a percentage:[br][br][math] \large 0,25 = 25\% . [/math][br][br]So the number 5 is 25% larger than the number 4.

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?[br][br][b]Solution:[/b] Let us denote the original price [math]x[/math]. After a 50% discount, the lowered price is[br][br][math] \large 100\% - 50\% = 50\% [/math][br][br]of the original price. We convert this 50% into a decimal number: [math]50\% = 0,5[/math]. So after the first discount, the lowered price can be written as[br][br][math] \large 0,5x . [/math][br][br]After the second 60% discount, the final price is[br][br][math] \large 100\% - 60\% = 40\% [/math][br][br]of the previous price. We convert this 40% into a decimal number: [math]40\% = 0,4[/math]. Next, we combine the discounts. The previous price was [math]0,5x[/math], so the final price is 40% of the previous price:[br][br][math] \large 0,4 \cdot 0,5 x = 0,2x . [/math][br][br]That is, the final price is 20% of the original price. To obtain the original price, we create the following equation:[br][br][math] \large \begin{array}{rcll} 0,2x & = & 104 \, \text{EUR} & | : 0,2 \\[br]x & = & \frac{104 \, \text{EUR}}{0,2} \\[br]x & = & \frac{520 \, \text{EUR}}{1} \\[br]x & = & 520 \, \text{EUR} . \end{array} [/math][br][br]So the original price was 520 euros.

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?[br][br][b]Solution:[/b] As both comparisons were made with Anna, let us denote the reward that Anna received [math]A[/math]. Since Bridgette received 20% more than Anna (that is, 120% of what Anna received), the reward received by Bridgette is [math]1,2 \cdot A[/math]. Similarly, as Cecilia received 10% less than Anna (that is, 90% of what Anna received), the reward received by Cecilia is [math]0,9 \cdot A[/math]. In total, the rewards received by the three people are[br][br][math] \large A + 1,2A + 0,9A . [/math][br][br]As the total reward is 620 euros, we create the following equation:[br][br][math] \large \begin{array}{rcll}[br]A + 1,2A + 0,9A & = & 620 \, \text{EUR} \\[br]3,1A & = & 620 \, \text{EUR} & | : 3,1 \\[br]A & = & \frac{620 \, \text{EUR}}{3,1} \\[br]A & = & 200 \, \text{EUR} .[br]\end{array} [/math][br][br]So Anna received 200 euros. Bridgette received[br][br][math] \large 1,2 \cdot 200 \, \text{EUR} = 240 \, \text{EUR} , [/math][br][br]and Cecilia received[br][br][math] \large 0,9 \cdot 200 \, \text{EUR} = 180 \, \text{EUR} . [/math][br][br]In total, they received [math]200+240+180 = 620[/math] euros.