Problem Solving

Using Proportions

If you change a variable we can predict whether other variables in the same equation will increase or decrease. The first step is to decide if the relationship between the variables is directly proportional or inversely proportional.

$$ {\color{#09e}{\Uparrow} \atop y} {\atop = k } {\color{#09e}{\Uparrow} \atop x} \quad\quad\quad {\color{#f00}{\Downarrow} \atop y} {\atop = k } {\color{#f00}{\Downarrow} \atop x} \quad\quad\quad \frac{ y \color{#f00}{\Downarrow}}{x \color{#f00}{\Downarrow}} = k$$

directly proportional = a constant ratio, k, between two variables. Both variables increase and decrease together.


$$ {\color{#09e}{\Uparrow} \atop y} {\color{#f00}{\Downarrow} \atop x} {\atop = k } \quad\quad\quad {\color{#f00}{\Downarrow} \atop y} {\color{#09e}{\Uparrow} \atop x} {\atop = k } \quad\quad\quad y {\color{#f00}{\Downarrow}} = \frac{k}{x \color{#09e}{\Uparrow}} $$

inversely proportional = a constant ratio, k, between a variable and the inverse of a variable. Variables increase and decrease opposite to each other.

You can use proportions when everything else in an equation is constant except two variables. Then you predict how one will change when the other changes.

$$F = ma$$

Example: A force of 100 N produces an acceleration of 10 m/s² for a 10 kg mass. How should you change the force to reduce the acceleration?
solution $$ \text{m is constant}$$ $$F = ma$$ $$ \text{F and a are directly proportional}$$ $${\color{#f00}{\Downarrow} \atop F} {\atop =} { \atop m} {\color{#f00}{\Downarrow} \atop a} $$ $$ \text{Decreasing F will decrease a}$$

$$v = \frac{\Delta x}{\Delta t}$$

Example: You can run 100 m in about 15 s. If you want to decrease your time, should you increase or decrease your velocity?
solution $$ \text{Δx is constant}$$ $$v = \frac{\Delta x}{\Delta t}$$ $$v \Delta t = \Delta x$$ $$ \text{v and Δt are inversely proportional}$$ $${\color{#09e}{\Uparrow} \atop v} {\color{#f00}{\Downarrow} \atop \Delta t} {\atop =} { \atop \Delta x}$$ $$ \text{increasing v will decrease Δt}$$
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