Video: AQA GCSE Mathematics Higher Tier Pack 4 • Paper 3 • Question 17

Gary is a van driver. He drives his van from Birmingham to Slough and then from Slough to Windsor. The table shows information about the fuel consumption of his van. On the journey from Birmingham to Slough, Gary has an average speed of 60 miles per hour and it takes him 1 hour and 48 minutes. The journey from Slough to Windsor is 4 miles and takes him 12 minutes. Using this information, show that Gary uses less than 3 gallons of fuel for his journey from Birmingham to Windsor via Slough.

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Video Transcript

Gary is a van driver. He drives his van from Birmingham to Slough and then from Slough to Windsor. The table shows information about the fuel consumption of his van. If he travels 50 miles per hour or more, the minimum miles travelled per gallon is 40. If he travels at less than 50 miles per hour, then the minimum miles travelled per gallon is 30. On the journey from Birmingham to Slough, Gary has an average speed of 60 miles per hour. And it takes him one hour and 48 minutes. The journey from Slough to Windsor is four miles and takes him 12 minutes. Using this information, show that Gary uses less than three gallons of fuel for his journey from Birmingham to Windsor via Slough.

So to free up some room so that we can show our calculations, I’m gonna write a synopsis about the information in the question. So first of all, we know that the journey from Birmingham to Slough, we’ve got an average speed of 60 miles per hour. And the time taken is one hour and 48 minutes. And the journey from Slough to Windsor is a distance of four miles. And it takes time of 12 minutes. And what we need to do is show that the fuel used is less than three gallons.

And in order to solve the problem, what we’re gonna use is the speed-distance-time triangle. And this helps us to find the speed, the distance, or the time. So we’re gonna start with the Birmingham–Slough journey. And for that journey, what we want to find is the distance, because we’ve got the average speed and the time. And using the distance-speed-time triangle, we can see that distance is equal to speed multiplied by time.

So we’re gonna get that distance is equal to 60, because that’s our speed, and then multiplied by 1.8, because that’s our time. Well, we can see how we got that 1.8 from one hour 48 minutes. Well, let’s deal with the 48 minutes. Well, 48 minutes out of an hour can be written as 48 over 60. That’s because there are 60 minutes in an hour. And we can divide both 48 and 60 by 12 to simplify our fraction.

So if we do that, we get 48 divided by 12, which is four, and that’s gonna be our numerator, and then 60 divided by 12, which is five. So that’ll be our denominator. So we can say that 48 minutes or 48 out of 60 is the same as four-fifths of an hour. And four-fifths is equal to 0.8. That’s because one-fifth is equal to 0.2. And we multiply that by two.

You could also work it out if you weren’t quite sure using the bus stop method. And you could do four divided by five. So therefore, one hour and 48 minutes is equal to 1.8 hours. And when we do that part of the calculation, we get 108 miles. So 60 multiplied by 1.8 gives us 108 miles. So that’s the distance from Birmingham to Slough.

So now we’re gonna look at Slough to Windsor. And for Slough to Windsor, well we’ve got the distance and we’ve got the time. So this time, we want to work out the speed. And we can use the triangle again to give us the formula for speed. And this is distance divided by time. So therefore, if we substitute in our values, we’re gonna have speed is gonna be equal to then we’re gonna have four because that’s our distance. And then 0.2 as we said is gonna be what four is divided by. And that’s because 12 minutes is equal to 0.2 hours. And we’re alluded to the fact that it would be 0.2 or one-fifth of an hour.

But I want to show again how we got there just so you’re sure. So 12 minutes is the same as 12 over 60, because it’s 12 over 60 as a part of an hour, because there are 60 minutes in an hour. So again, we can divide the numerator and denominator by 12. So if we divide 12 by 12, we get one. And that’s our numerator. And then 60 divided by 12 is five. So that’ll be our denominator. And a fifth is equal to 0.2.

So great, so we’ve got speed is equal to four divided by 0.2, which is gonna give us 20 miles an hour. And we can calculate that. But if we want to use a written method, we could think of it as four divided by a fifth because 0.2 is a fifth. Well, if you divide by a fraction, you then multiply by the reciprocal of that fraction. So you flip the fraction. So that will be the same as four multiplied by five over one, which is the same as four multiplied by five, which gives us our 20.

Now great, we found out both bits of information we’re looking for. So now what we need to do is show that the maximum fuel used for the journey is less than three gallons. So first, what we’re gonna do is work out the maximum fuel used for each stage of the journey. So the maximum fuel for Birmingham to Slough part of the journey is gonna be 108. And that’s because 108 was our distance. And that’s divided by 40. That’s because 40 is the minimum miles travelled per gallon, because when we’re travelling from Birmingham to Slough, Gary was traveling more than 50 miles an hour, because Gary was travelling 60 miles an hour. And that’s gonna give us a value of 2.7 gallons used. And that’s gonna be the maximum fuel used. And we know it’s the maximum because we’re told that 40 miles per gallon is the minimum miles travelled per gallon.

Well, we’re gonna use the same method to calculate the maximum fuel from Slough to Windsor. This time, the calculation is gonna be four divided by 30. That’s cause it’s the distance of four miles. And we know that it’s gonna be travelling at 30 minimum miles per gallon because, in the second part from Slough to Windsor, the speed which we calculated was 20 miles per hour. So therefore, it’s less than 50 miles per hour.

So that means, using the table, we can see that the minimum miles travelled per gallon is 30. So this gives us a total maximum fuel of 0.13 recurring gallons from the journey from Slough to Windsor. So therefore, the total fuel is gonna be equal to 2.7 plus 0.13 recurring, because that’s the maximum fuel used to each part of the journey, which is gonna be equal to 2.83 recurring gallons. And 2.83 recurring is less than three. So therefore, we’ve shown that the fuel use is less than three gallons and we’ve included all of our working.

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