In Keeping it in Proportion, we learnt about proportional relationships. The types of relationships we looked at in these previous chapters are actually called directly proportional relationships. If two amounts are directly proportional, it means that as one amount increases, the other amount increases at the same rate.

For example, if you earn $\$18$$18 per hour, your earnings are directly proportional to the number of hours worked because $\text{earnings }=18\times\text{hours worked }$earnings =18×hours worked .

The mathematical symbol for "is directly proportional to..." looks like a stretched alpha symbol:

So, if we use the example above, we could write $e$e$\propto$∝$h$h. In other words, "Your earnings ($e$e) are directly proportional to the number of hours you work ($h$h)."

The constant of proportionality

The constant of proportionality is the value that relates the two amounts. In the example above, the constant would be $18$18.

We can write a general equation for amounts that are directly proportional.

General Equation for Amounts that are Directly Proportional

$y=kx$y=kx

where $k$k is the constant of proportionality

Once we solve the constant of proportionality, we can use it to answer other questions in this relationship.

Worked Examples

Question 1

Dave can wash$2$2cars in $4$4 minutes.

How long would it take for him to wash 1 car?

How long would it take him to wash$20$20cars?

Question 2

When a rock sample from a rock bed was examined, it was found that $4.1$4.1 kilograms of the total mass of the sample was copper.

If this sample contains $10%$10% of the total copper in the rock bed, how many kilograms of copper were there in the rock bed?

Question 3

Uther earns $\$423.13$$423.13 in $17$17 hours. How many hours does he work in a week in which he earns $\$273.79$$273.79?