Convert 0,072 and [math]\frac{3}{4}[/math] into percentages.[br][br][b]Solution:[/b] To convert decimal numbers into percentages, we multiply by 100. This principally moves the decimal separator two steps to the right. Therefore,[br][br][math]\large 0,072=0,072\cdot100\%=7,2\%. [/math][br][br]To convert [math]\frac{3}{4}[/math] into a percentage, we first convert the fraction into a decimal number. That is, [math]\frac{3}{4} = 0.75[/math]. Finally, we multiply the decimal number by 100:[br][br][math]\large \frac{3}{4} = 0,75 = 0,75 \cdot 100 \% = 75 \% .[/math]

Convert 42,3% and 0,12% into decimal numbers.[br][br][b]Solution:[/b] To convert percentages into decimal numbers, we divide by 100. This principally moves the decimal separator two steps to the left, so[br][br][math] \large 42,3\% = \frac{42,3}{100} = 0,423 . [/math][br][br]Similarly,[br][br][math] \large 0,12\% = \frac{0,12}{100} = 0,0012 . [/math]

How many per cents is 90 kg of 300 kg?[br][br][b]Solution:[/b] We want to know the percentage of 90 kg. In this case, [i]b [/i] of the formula is 90 kg. The [i]a[/i] of the formula is 300 kg because it is compared to:[br][br][math] \large p\,=\dfrac b a =\frac{90 \, \mathrm{kg}}{300 \, \mathrm{kg}} = \frac{9}{30} = \frac{3}{10} =\large 0,3 = 0,3 \cdot 100\% = 30\% . [/math][br]

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.

A product currently costs 420 euros, but this is only 80% of the original price. What is the original price?[br][br][b]Solution 1: [/b]Mark the original price with variable x because we don't know it. This is also the formula a because it is compared to it. In addition, p = 80% = 80/100 and b = 420 euros.[br][br][math]\large \begin{array}{rcll}[br]p&=&\dfrac b a \\[br]\dfrac{80}{100} &=& \dfrac{ 420 \text{ euros}}{x} &|\text{cross multiplying} \\[br]80\cdot x &=& 100 \cdot 420 \text{ euros} &|: 80\\[br]x&=& \dfrac{42000 \text{ euros}}{80} = 525 \text{ euros}[br][br]\end{array}[br][/math][br][br]The original price is 525 euros. [br][b]Solution 2[/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. In the formula, p = 102 % = 1,2 and [i]a[/i] = 490 euroa. [br][br][math] \large \begin{array}{rcll}[br][br]p&=&\dfrac b a &|\cdot a\\[br]b&=&pa \\[br]b&=&102\,\%\cdot 490 \text{ euros}\\[br]b&=& 1,02\cdot 490 \text{ euros}\\[br]b&=& 499,80 \text{ euros}[br]\end{array}[/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] [b]a)[/b] First, let us calculate the difference between the numbers 4 and 5:[br][br][math] \large 4-5 = -1 . [/math][br][br]We are comparing to the number 5, so in the formula, [i]a[/i] = 5 and [i]b[/i] is difference -1. [br][br][math] \large p = \dfrac b a= \dfrac{4-5}{5} = \dfrac{-1}{5} = -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]b) [/b] In this case, we are comparing to the number 4. So, [i]a = 4 [/i] and [i]b[/i] = 5 - 4 in the formula. [br][br][math] \large p=\dfrac b a= \dfrac{5-4}{4} = \frac{1}{4} = 0,25=25\,\% . [/math][br][br][br]So the number 5 is 25% larger than the number 4.[br]

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 [math] \large 100\% - 50\% = 50\% [/math] 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. To obtain the original price, we create the following equation:[br][br][math] \large \begin{array}{rcl}[br]\text{paid price}&=& 104€\\[br]40\,\% \underbrace\text{discounted price}&=& 104€\\[br]40\,\%\cdot( 50\;\%\text{ original price})&=& 104€\\[br]0,4 \cdot 0,5 x &=& 104€\\[br]0,2x&=&104€ \\[br]x&=& 520€\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.