Direct Proportion
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.
$y=kx$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.
Examples
Question 1
Consider the equation $P=90t$P=90t.
State the constant of proportionality.
Find the value of $P$P when $t=2$t=2.
Question 2
If $a$a varies directly with $x$x cubed, and $a=12$a=12 when $x=4$x=4:
find the constant of variation, $k$k
define $a$a in terms of $x$x
find the value of $a$a when $x=2$x=2
Question 3
Find the equation relating $a$a and $b$b given the table of values.
| $a$a | $0$0 | $1$1 | $2$2 | $3$3 |
|---|---|---|---|---|
| $b$b | $0$0 | $2$2 | $4$4 | $6$6 |