Graphed coordinates (x,y){\displaystyle (x,y)} can also be written as a ratio y:x{\displaystyle y:x} or a fraction yx{\displaystyle {\frac {y}{x}}}. You can also think of x{\displaystyle x} as the independent variable and y{\displaystyle y} as the dependent variable, since y{\displaystyle y} changes in relation to the change in x{\displaystyle x}. The constant of proportionality can never be 0{\displaystyle 0}. [2] X Research source
The same equation can also be written k=yx{\displaystyle k={\frac {y}{x}}}. Use this version if you’re trying to find the constant of proportionality and already have the values for the two variables. The formula d=rt{\displaystyle d=rt} (distance = rate x time), which you might recognize from science class, is another version of the equation for the constant of proportionality. [4] X Research source
The same equation can also be written k=yx{\displaystyle k=yx}. Use this version to find the constant of proportionality when you already have the values for the other two variables.
For example, the ordered pairs (4,20){\displaystyle (4,20)}, (8,40){\displaystyle (8,40)}, and (12,60){\displaystyle (12,60)} are the same as the ratios 20:4{\displaystyle 20:4}, 40:8{\displaystyle 40:8}, and 60:12{\displaystyle 60:12}. As fractions, they are 204{\displaystyle {\frac {20}{4}}}, 408{\displaystyle {\frac {40}{8}}}, and 6012{\displaystyle {\frac {60}{12}}}.
For example, look at the set 204{\displaystyle {\frac {20}{4}}}, 408{\displaystyle {\frac {40}{8}}}, and 6012{\displaystyle {\frac {60}{12}}}. In each ratio, the values of the numerator and the denominator both increase, so you’re looking at direct proportionality. What if you had 112{\displaystyle {\frac {1}{12}}}, 26{\displaystyle {\frac {2}{6}}}, and 34{\displaystyle {\frac {3}{4}}}? The values of the denominator decrease as the values of the numerator increase, so you’re looking at inverse proportionality.
To continue from the previous example, take the first ratio of 204{\displaystyle {\frac {20}{4}}}. Since the set is directly proportional, you’d use k=204=5{\displaystyle k={\frac {20}{4}}=5}. Your constant of proportionality is 5{\displaystyle 5}. What about the inverse set? Take the ratio 112{\displaystyle {\frac {1}{12}}} and plug the numbers into the equation k=yx{\displaystyle k=yx} to get k=(1)(12)=12{\displaystyle k={(1)}{(12)}=12}. Your constant of proportionality is 12{\displaystyle 12}.
Try this on your own with the ratios in the set 204{\displaystyle {\frac {20}{4}}}, 408{\displaystyle {\frac {40}{8}}}, and 6012{\displaystyle {\frac {60}{12}}}. You’ll see that you get 5{\displaystyle 5} for each ratio, which tells you this ratios are indeed directly proportional! This also applies to ratios that are inversely proportional. For example, in the set 112{\displaystyle {\frac {1}{12}}}, 26{\displaystyle {\frac {2}{6}}}, and 34{\displaystyle {\frac {3}{4}}}, plugging each ratio into the equation k=yx{\displaystyle k=yx} gets you a constant of 12{\displaystyle 12} for each ratio, so they are inversely proportional.
Use the equation y=kx{\displaystyle y=kx} if the set of ratios is directly proportional. If x{\displaystyle x} is 7{\displaystyle 7} and k{\displaystyle k} is 5{\displaystyle 5}, y{\displaystyle y} would be 35{\displaystyle 35} (y=(7)(5)=35{\displaystyle y=(7)(5)=35}). Use the equation y=kx{\displaystyle y={\frac {k}{x}}} if the set of ratios is inversely proportional. If x{\displaystyle x} is 4{\displaystyle 4} and the constant of proportionality is 12{\displaystyle 12}, y{\displaystyle y} would be 3{\displaystyle 3} (y=124=3{\displaystyle y={\frac {12}{4}}=3}).
Hint: All of the values increase, so use the equation for direct proportionality: k=yx{\displaystyle k={\frac {y}{x}}}.
Hint: The y{\displaystyle y} values are decreasing, so use the equation for inverse proportionality: k=yx{\displaystyle k=yx}.
Hint: 100 yards is 300 feet.
Hint: Gas mileage is the number of miles your car can drive on one gallon of gas, so you’re looking for the constant of proportionality here.
Since all the ratios in the set have the same constant and the values are all increasing, you also know that this series is directly proportional.
For 105−2{\displaystyle {\frac {105}{-2}}}, k=(105)(−2)=−210{\displaystyle k=(105)(-2)=-210}. For −1052{\displaystyle {\frac {-105}{2}}}, k=(−105)(2)=−210{\displaystyle k=(-105)(2)=-210}. For −356{\displaystyle {\frac {-35}{6}}}, k=(−35)(6)=−210{\displaystyle k=(-35)(6)=-210}. For −2110{\displaystyle {\frac {-21}{10}}}, k=(−21)(10)=−210{\displaystyle k=(-21)(10)=-210}. For −1514{\displaystyle {\frac {-15}{14}}}, k=(−15)(14)=−210{\displaystyle k=(-15)(14)=-210}. Since all of the ratios have the same constant of proportionality, they are inversely proportional.
To find how far Flash will go in 5{\displaystyle 5} minutes, use the equation for direct proportionality, y=kx{\displaystyle y=kx}. Here, x{\displaystyle x} is 5{\displaystyle 5}, so your answer is 30{\displaystyle 30} (y=(6)(5)=30{\displaystyle y=(6)(5)=30}). To find out how long it will take Flash to complete the 100-yard (300 feet) dash, use the same equation, but solve for x{\displaystyle x} instead. Start with 300=6x{\displaystyle 300=6x}, then divide each side by 6{\displaystyle 6} to get your answer, 50{\displaystyle 50}.
k=2248=28{\displaystyle k={\frac {224}{8}}=28} k=28010=28{\displaystyle k={\frac {280}{10}}=28} k=1124=28{\displaystyle k={\frac {112}{4}}=28}
When graphing ratios, the first number (or numerator of a fraction) is the y{\displaystyle y} coordinate and the second number (or denominator of a fraction) is the x{\displaystyle x} coordinate. [19] X Research source
Remember to use k=yx{\displaystyle k={\frac {y}{x}}} if the values of both variables go up and k=yx{\displaystyle k=yx} if the value of one variable goes up and the other goes down.
