Part a) Isabella is going to drive to her cousin’s house. She has two options for her journey. The first option is 100 miles on single carriageways. The second option is 20 miles longer, but 70 miles of the journey is on the
motorway. She assumes that her average speed will be 55 miles per hour on single carriageways
and 70 miles per hour on the motorway. Using her assumptions, work out which option is faster.
There’s also a part b that we’ll come onto later. So in order to solve this problem, what I’m going to do is divide it into two parts
to start with. I’m gonna work out for the first option how long the time will be and then I’ll work
out for the second option how long the time will be for that.
So to enable us to calculate the time taken by each option, we’re gonna use the speed
distance time triangle. And we can use the speed distance time triangle to help us to find a formula for
Because we can see that if we have time, then the other two letters are 𝑑 and
𝑠. And the 𝑑 is above the 𝑠. So this tells us that time is equal to distance divided by speed.
The triangle would work if we, for instance, wanted to find distance because distance
will be equal to speed multiplied by time and that’s because it’s next to the
Okay, so now, we have our formula, let’s use this to find the time taken for option
one. So therefore, we can say that the time is going to be equal to 100. And that’s because 100 was the 100 miles travelled in the first option divided by the
Well, we’re told in the question that the average speed will be 55 miles per hour on
a single carriageway. And option one uses only single carriageways. So therefore, it’s gonna be 100 over 55 and that’s hours. I’m not gonna cancel it down at this point because I’m just gonna leave it in the way
that it is — that’s 100 over 55. And then, we’re gonna find out the time taken for option two and then compare
Well, for option two, we’ve got two sections. We’ve got the section that is driven on the single carriageway and we have the
section which is driven on the motorway.
Well, we’re told in the question that 70 miles of the journey is on the motorway. So therefore, I’m going to work out the time taken on the motorway first.
Well, the time taken on the motorway is going to be equal to 70 — and that’s because
70 is the distance, 70 miles — divided by 70 — and that’s because the average speed
on the motorway will be 70 miles per hour — which will be equal to one hour.
Okay, so that’s the time on the motorway. Now, let’s find the time on the single carriageway for option two.
Well, if we want to work out the time on the single carriageway, first of all, we’ll
need to work out what is the distance travelled on the single carriageway. So the distance travelled on a single carriageway can be worked out with 120 minus
And that’s because we’re told that the second option is 20 miles longer than the
first option. And the first option was 100 miles. So therefore, the second option is going to be 120 miles.
And then, we take away 70 because we’re told that 70 miles of the journey is on the
motorway. Then this is all divided by 55. And that’s because that’s the average speed on the single carriageway. And this is gonna give us 50 over 55 hours and that’s cause 120 minus 70 is 50.
Okay, great, so we’ve now worked out the time taken on the single carriageway. And we’ve got the time taken on the motorway. But what we need to do is combine them to find out the total time taken for option
Well, the total time is going to be equal to one plus 50 over 55. And that’s cause we’ve got one hour for the time of the motorway and 50 over 55 hours
for the time on the single carriageway. Well, we know that one is the same as fifty-five fifty-fifths or 55 over 55.
So therefore, if we add 55 over 55 and 50 over 55, we’re gonna get 105 over 55. So we can say that the total time for option two is 105 over 55 hours. Well, therefore, 105 over 55 is greater than 100 over 55. So therefore, option one is going to be faster.
Be careful here of a common mistake because we can see option two has the greater
amount because we know that 105 over 55 is greater. So a common mistake here for students is to go “Okay, in that case, option two must
be faster cause it’s a bigger amount.”
However, we’ve got to remember that we’re talking about time. So therefore, if we want something to be faster, the time taken will be less. And thus, that’s why option one is faster than option two.
Now, for part b, part b says that “Isabella’s assumptions about her average speeds
are too high. How does this affect her journey time?”
Well, if we think about Isabella’s assumptions, if the average speeds are too high in
her assumptions, therefore it means that in real life she’s going to be driving
slower. And if she’s driving slower, then therefore her journey time will be longer.