Ratios and Rates

Lesson

We have already learnt about proportional relationships, where two variables vary in such a way that one is a constant positive multiple of the other. In other words, they always vary by the same constant. We call this constant the constant of proportionality.

Proportional relationships are always in the form $y=kx$`y`=`k``x`. We know that $k$`k` represents the multiplicative factor. However, it also represents the constant of proportionality. When we graph these relationships, they produce straight lines with positive gradients that always pass through the origin $\left(0,0\right)$(0,0).

For example, let's say a shop is selling apples for $\$3$$3.

We know that five apples will cost $\$3$$3, ten apples will cost $\$6$$6, fifteen apples will cost $\$9$$9 and so on.

We know that the price will increase at a constant rate.

We could graph these two variables in a table.

Number of apples ($x$x) |
$0$0 | $5$5 | $10$10 | $15$15 |

Cost ($y$y) |
$0$0 | $3$3 | $6$6 | $9$9 |

Since we know $5$5 apples cost $\$3$$3, we can work out how much one apple costs:

$3\div5=0.6$3÷5=0.6

This means that each apple costs $60$60 cents and we can say that this is the constant of proportionality.

Further, we can write this as an equation: $y=0.6x$`y`=0.6`x`.

Remember!

The constant of propotionality is always positive.

Since proportional relationships are in the form $y=kx$`y`=`k``x`, we can also calculate the constant of proportionality ($k$`k`) by rearranging this equation and we find:

$k=\frac{y}{x}$`k`=`y``x`

Consider the equation:$y=8x$`y`=8`x`.

a) What is the constant of proportionality for the given equation?

b) How do you know that this equation is directly proportional? There may be more than one correct option.

In the following proportionality table, the second row is obtained by multiplying the top row by the constant of proportionality. Complete the table and find that constant.

$7$7 | $8$8 | $10$10 | $\editable{}$ |

$\editable{}$ | $72$72 | $\editable{}$ | $117$117 |

a) Complete the table:

b) What is the constant of proportionality?

Fred is making batches of bread rolls. He knows he can make $60$60 bread rolls in $10$10 hours, and $120$120 bread rolls in $20$20 hours. What is the constant of proportionality?

Apply direct and inverse relationships with linear proportions

Apply numeric reasoning in solving problems