Proportionality Between Variables:

Proportionality explains how two or more variables are related. We say that two variables are proportional if their product or their ratio is equal to a constant (proportionality constant). There are two types of proportionality: direct proportional and inverse proportional. For directly proportional variables, their ratio is equal to the proportionality constant. For inversely proportional variables, their product is equal to the proportionality constant.

Let the cost of {eq}x\; \rm lbs {/eq} of grapes be {eq}y {/eq}. Since the total cost increases as the weight of grapes increase, the cost {eq}y {/eq} is directly proportional to the weight {eq}x {/eq} and we can write this as:

• {eq}y\propto x {/eq}

Removing the proportionality sign and adding a proportionality sign, we get:

• {eq}y = kx {/eq}

If the cost of {eq}x = 3\; \rm lb {/eq} of grapes is {eq}y = \$6.57 {/eq}, then: • {eq}\$6.57 = 3k {/eq}

Solving for k, we get:

• {eq}k = \dfrac{\$6.57}{3} {/eq} • {eq}\boxed{\color{blue}{k = \$2.19}} {/eq}

The proportionality constant is \$2.19, which is the cost per lb of grapes.