Video: Understanding the Meaning of a Point on the Graph of a Proportional Relationship

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? [A] The point 𝑄 shows that Hannah would earn $60 if she worked 4 hours. [B] The unit rate of this proportional relationship is $15/hour. [C] Any point of coordinates (π‘₯, 𝑦) on this graph shows that Hannah would earn $𝑦 if she worked π‘₯ hours. [D] If Hannah worked for 10 hours, she would earn $150. [E] If Hannah worked for 4 hours, she would earn $15.

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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 π‘₯, 𝑦 on this graph shows that Hannah would earn 𝑦 dollars if she worked π‘₯ 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 π‘₯-axis, we can see that point 𝑄 also has the same π‘₯-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 π‘₯-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 π‘₯, 𝑦 on this graph shows that Hannah would earn 𝑦 dollars if she worked π‘₯ 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 π‘₯, 𝑦, this means she works π‘₯ 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 π‘₯-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 𝐡 are in proportion, then 𝐴 sub one over 𝐡 sub one equals 𝐴 sub two over 𝐡 sub two, where 𝐴 sub one and 𝐡 sub one are the quantities of 𝐴 and 𝐡 in a situation and 𝐴 sub two and 𝐡 sub two are the quantities of 𝐴 and 𝐡 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 π‘₯-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.

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