### Video Transcript

Uptown Pizzeria sells medium
pizzas for seven dollars each and charges a three-dollar delivery fee per
order. Is the cost of an order
proportional to the number of pizzas ordered?

In this question, we’ll go
through different scenarios for the cost of pizza and charges. And we’ll verify our answer
using a graph. So, let’s start by looking at
the cost of an order for different quantities of pizza. So, for one medium pizza, we’re
told that the cost is seven dollars and we have a three-dollar delivery
charge. So, the total cost of our order
would be 10 dollars. For two pizzas, we’d have two
times seven dollars. And adding on our three-dollar
delivery fee would give us 17 dollars in total. For three pizzas, we’d have
three times seven dollars, which is 21 dollars. And adding on our three-dollar
delivery would be 24 dollars in total.

In this question, we’re asked
if the cost of the order is proportional to the number of pizzas ordered. We can recall that if we have
two quantities 𝐴 and 𝐵, then they are in proportion when from one situation to
another both quantities have been multiplied by the same number. We could also consider this as
𝐴 sub one over 𝐵 sub one equals 𝐴 sub two over 𝐵 sub two, where 𝐴 sub one
and 𝐵 sub one are the quantities of 𝐴 and 𝐵 in one situation and 𝐴 sub two
and 𝐵 sub two are the quantities of 𝐴 and 𝐵 in another situation.

So, if we take our situation
with one pizza, we could say that the cost per pizza would be 10 over one, which
is equivalent to 10 dollars per pizza. In our second situation, the
cost per pizza would be 17 dollars over two, which is 8 dollars 50 per
pizza. In our third situation, we’d
have 24 dollars for the order divided by three pizzas, so that’s eight dollars
per pizza. So, in order to have a
proportional relationship, we would need to check if our fractions 10 over one,
17 over two, and 24 over three are equal. And no, they’re not equal. So, we can say that the cost of
an order and the number of pizzas are not proportional.

Let’s have a look at verifying
this using a graph. We can plot the number of
pizzas versus the cost of the order. Using the values we calculated
earlier that one pizza would have a total cost of the order of 10 dollars, two
pizzas would have an order cost of 17 dollars, and three pizzas would have an
order cost of 24 dollars, we can plot these and draw a line through them. Here, we have a straight line
which doesn’t pass through the origin. This would indicate a
nonproportional linear relationship.

In fact, if we look at the
point where it crosses the 𝑦-axis, we can see that this would be at the
coordinate zero, three, which is the slightly bizarre situation of ordering zero
pizzas and getting charged three dollars for delivery. If we have a graph of two
proportional quantities, then we would have a straight-line graph which passes
through the origin. As we don’t have this here,
then this confirms our original answer that the cost of an order and the number
of pizzas ordered are not proportional.