Video Transcript
Hannah works as a baby
sitter. The proportional relationship
between the number of hours she works and the total amount of money she earns is
shown in the graph. Which of the following
statements is not true? Option A) the point 𝑄 shows
that Hannah would earn 60 dollars if she worked four hours. Option B) the unit rate of this
proportional relationship is 15 dollars per hour. Option C) any point of
coordinates 𝑥, 𝑦 on this graph shows that Hannah would earn 𝑦 dollars if she
worked 𝑥 hours. Option D) if Hannah worked for
10 hours, she would earn 150 dollars. Option E) if Hannah worked for
four hours, she would earn 15 dollars.
So, here, we have the graph of
a proportional relationship between the number of hours that Hannah works and
the total amount of money she earns. We can confirm that it is a
proportional relationship because it’s a straight-line graph and it passes
through the origin. The point zero, zero would be
the situation where Hannah works zero hours and gets paid zero money. So, let’s have a look at the
statements and decide if they are true or false.
Let’s start with statement A,
the point 𝑄 shows that Hannah would earn 60 dollars if she worked four
hours. So, if we look at four on our
𝑥-axis, we can see that point 𝑄 also has the same 𝑥-coordinate. And in fact, the 𝑦-coordinate
would be 60, indicating 60 dollars. So, here, we would have Hannah
works for four hours and earns 60 dollars. So, this is point 𝑄 and
indicates that statement A is true.
Looking at statement B, we have
the term unit rate. We can recall that a unit rate
is a proportion with different quantities which has a denominator of one. In our case, our proportion
will be the dollars, or the money earned, over the number of hours. To find the unit rate, we need
to have how many dollars for one hour. Using our graph to help us
then, if we look at one hour on the 𝑥-axis, this will be the point one, 15,
which is equivalent to Hannah earning 15 dollars in one hour. And so, our statement B is
true. She earns 15 dollars per
hour.
Let’s have a look then at
statement C, any point of coordinates 𝑥, 𝑦 on this graph shows that Hannah
would earn 𝑦 dollars if she worked 𝑥 hours. Well, let’s look at the
coordinate zero, zero. This would be when she works
zero hours and gets zero dollars. At point one, 15, she works one
hour and gets paid 15 dollars. Equally, the coordinate two, 30
means she works two hours getting paid 30 dollars. And the coordinate three, 45
means she works three hours and gets paid 45 dollars. So, for any coordinate 𝑥, 𝑦,
this means she works 𝑥 hours and gets paid 𝑦 dollars, which is equivalent to
statement C. So, it must be true as
well.
Looking at statement D, if
Hannah worked for 10 hours, she would earn 150 dollars. So, let’s see if we can use the
graph. If we look on our 𝑥-axis where
the number of hours equals 10, we can see that the line does not go through this
point. But we can use another piece of
information to work out the value. And that is that Hannah earns
15 dollars per hour. We can do this by writing the
statement that 15 over one is equal to what over 10.
We can write this because we
know that if quantities 𝐴 and 𝐵 are in proportion, then 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 a situation and 𝐴 sub two and 𝐵 sub two are the
quantities of 𝐴 and 𝐵 in a different situation. So, returning to our
calculation then, we can see that if we take the denominator in our fraction 15
over one and multiply it by 10, this would give us 10. So, we can also multiply our
numerator by 10, which gives 150 over 10. So, in 10 hours, Hannah would
earn 150 dollars. Therefore, our statement D is
true.
Looking at our final statement
E then, if Hannah worked for four hours, she would earn 15 dollars. So, if we look at four hours on
our 𝑥-axis, we can see that this would be at the point four, 60 on the
line. This means that in four hours,
Hannah earns 60 dollars. We could also consider that
when Hannah earns 15 dollars, she will have done one hour of work. Using either of these
approaches, we could say that this statement that if Hannah worked for four
hours, she would earn 15 dollars is definitely not true. And so, option E is the
statement which is no true.
