Let us solve the motivational example we had, using the algorithm for solving proportion equations. For 2,5 cups of flour, add 2 tablespoons of oil. If we added 10 cups of flour, how much oil would we need?[br][br]First, let us denote the amount of oil by [math]x[/math]. We have the following table:[table][tr][td][b]flour[/b][/td][td][b]oil[/b][/td][/tr][tr][td]2,5 cups[/td][td]2 tbsp[/td][/tr][tr][td]10 cups[/td][td][math]x[/math][/td][/tr][/table][br][br][code][/code][table][tr][/tr][/table]We start by creating a proportion equation:[br][br][math]\frac{2{,}5 \, \mathrm{cups}}{10 \, \mathrm{cups}} = \frac{2 \, \mathrm{tbsp}}{x}[/math][br][br]Let us simplify the equation, as the units on the left-hand side are cancelled, and we can also simplify the numbers on the left-hand side by dividing both the numerator and the denominator by 2,5:[br][br][math]\frac{\textcolor{red} 1}{ \textcolor{blue} 4} = \frac{ \textcolor{blue}{2 \, \mathrm{tbsp}} }{ \textcolor{red} x}[/math][br][br]Now we perform the cross multiplication by multiplying the red terms, and the blue terms:[br][br][math]1 \cdot x = 4 \cdot 2 \, \mathrm{tbsp},[/math][br][br]that is,[br][br][math]x = 8 \, \mathrm{tbsp}.[/math][br][br]This is already the solution of the equation. Hence, we need 8 tablespoons of oil.

In February 2023, Euros and Japanese Yens had the following exchange rate: 1 EUR = 143 JPY. How many Euros is 2860 Japanese Yens?[br][br]We can solve this problem by using proportion equations. Let us create the following table first:[br][table] [tr][br] [td][b]Euros[/b][br][/td][br] [td][b]Yens[/b][br][/td][br][/tr][br] [tr][br] [td]1 EUR[/td][br] [td]143 JPY[/td][br][/tr][br] [tr][br] [td][math]x[/math][br][/td][br] [td]2860 JPY[/td][br][/tr][br][/table][br]We have a proportion equation:[br][br][math]\frac{1 \, \mathrm{EUR}}{x} = \frac{143 \, \mathrm{JPY}}{2860 \, \mathrm{JPY}}[/math][br][br]We can cancel the JPY units on the right-hand side:[br][br][math]\frac{ \textcolor{red}{1 \, \mathrm{EUR}} }{ \textcolor{blue} x} = \frac{ \textcolor{blue}{143} }{ \textcolor{red}{2860} }[/math][br][br]Now, let us perform the cross multiplication by multiplying the red terms, and the blue terms:[br][br][math]1 \, \mathrm{EUR} \cdot 2860 = x \cdot 143,[/math][br][br]that is,[br][br][math]2860 \, \mathrm{EUR} = 143x . [/math][br][br]We can swap the left-hand side and the right-hand side to obtain:[br][br][math]143x = 2860 \, \mathrm{EUR} . [/math][br][br]To solve this equation, we divide both sides by the coefficient of [math]x[/math], that is, we divide both sides by 143:[br][br][math]x = \frac{2860 \, \mathrm{EUR}}{143} = 20 \, \mathrm{EUR} . [/math][br][br]Therefore, 5330 Japanese Yens corresponds to 20 Euros.

We know that the average velocity ([math]v[/math]) can be calculated as the ratio of the distance travelled ([math]s[/math]) and the time spent ([math]t[/math]), that is, [math]v = \frac{s}{t}[/math]. If a car travels with a velocity of 80 km/h, how long does it take to travel 500 km?[br][br][b]Method 1:[/b] traditional equation solving.[br][br][math]\large \begin{array}{rcll}[br]v & = & \frac{s}{t} & | \cdot t \\[br]v\cdot t & = & s & | : v \\[br]t & = & \frac{s}{v} \\[br]t & = & \frac{500 \, \mathrm{km}}{80 \, \mathrm{km/h}} \\[br]t & = & 6{,}25 \, \mathrm{h} . [br]\end{array}[br][/math][br][br][b]Method 2:[/b] proportion equation.[br][br]Let us write [math]v = \frac{v}{1}[/math] to obtain a proportion equation:[br][br][math]\large \begin{array}{rcll}[br]\frac{ \textcolor{red} v}{ \textcolor{blue} 1} & = & \frac{\textcolor{blue} s}{ \textcolor{red} t} .[br]\end{array}[br][/math][br][br]Now, we can proceed by performing the cross multiplication:[br][br][math]\large \begin{array}{rcll}[br]v \cdot t & = & 1 \cdot s & | : v \\[br]t & = & \frac{s}{v} \\[br]t & = & \frac{500 \, \mathrm{km}}{80 \, \mathrm{km/h}} \\[br]t & = & 6{,}25 \, \mathrm{h} . [br]\end{array}[br][/math][br][br]It takes 6,25 hours to travel 500 kilometers.