Video: GCSE Mathematics Foundation Tier Pack 1 • Paper 1 • Question 17

GCSE Mathematics Foundation Tier Pack 1 • Paper 1 • Question 17

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Video Transcript

A shop is selling orange juice. The juice comes in either one-liter or 1.5-liter cartoons. Each one-liter carton of orange juice costs one pound 23. Each 1.5-liter carton of orange juice costs one pound 52. Fatima estimates that she drinks on average two-fifths of a liter of orange juice per day. Fatima has no orange juice, but she wants to buy enough orange juice to last her for nine days. Part a) Using Fatima’s estimate, calculate the smallest amount of money Fatima needs to spend on orange juice to last her nine days. You must show how you reached your answer.

Let’s start by working out whether the one-liter carton or 1.5-liter carton of orange juice is cheaper per liter. We are told in the question that a one-liter carton of orange juice costs one pound 23. Therefore, the orange juice in this carton costs one pound 23 per liter.

Next, we can estimate the cost per liter of the orange juice in the 1.5-liter carton. In order to find the cost per liter of the orange juice in this carton, we will simply divide that cost, which is one pound 52, by the amount in the carton. So that’s 1.5 liters.

However, this is a difficult calculation to perform. And so we can estimate it. Since 1.52 is roughly equal to 1.5, we can say that 1.52 over 1.5 is roughly equal to 1.5 over 1.5. Then any number divided by itself is simply one. And so 1.5 divided by 1.5 gives us one. This gives us that our estimate for the cost per liter of the orange juice in the 1.5-liter carton is one pound. So from this, we can deduce that the orange juice in the 1.5-liter carton is cheaper per liter.

Next, let’s work out how much orange juice Fatima estimates that she drinks over nine days. The question tells us that she estimates that she drinks two-fifths of a liter of orange juice per day. And so to find the amount that she drinks in nine days, we simply need to multiply two-fifths by nine.

In order to multiply two-fifths by nine, let’s first write nine as a fraction. We can write nine as nine over one, since nine over one means nine divided by one and nine divided by one is equal to nine. So nine over one is equivalent to nine. And therefore, our calculation becomes two-fifths multiplied by nine over one.

In order to multiply two fractions, we simply multiply the two numerators to form the new numerator and then multiply the two denominators to form the new denominator. So the numerator of our new fraction will be two multiplied by nine and the denominator will be five multiplied by one. Then, two times nine is simply 18 and five times one gives us just five. So we are left with the fraction 18 over five.

Now, 18 over five is an improper or top heavy fraction, which we can turn into a mixed number. The first step of doing this is to see how many times five goes into 18. We can see that three lots of five is 15 and four lots of five is 20. Now, four lots of five is too many times five since 18 is less than 20. Therefore, five goes into 18 three times. Then since 18 minus 15 is three, that means that our remainder is three. Therefore, five goes into 18 three times with a remainder of three.

So we can write our improper fraction as three and three-fifths. So this number three and three-fifths tells us how many liters of orange juice Fatima needs to buy in order to last her nine days. Let’s consider the different ways in which Fatima can buy three and three-fifths liters of orange juice.

She could buy four of the one-liter cartons. The cost of buying four one-liter cartons of orange juice would be four multiplied by one pound 23. In order to multiply these two numbers, we can use long multiplication.

We start by multiplying the four and the three to give us 12. And so we write down the two and we carry the one over. Next, we multiply the four by the two to give us eight and then we add the one that we carried from the last step, leaving us with nine. Then, finally, we multiply the four by the one to give us four.

In order to work out where the decimal point goes, we count how many digits there are after the decimal point in the numbers which we’re multiplying together as we can see there are two digits after the decimal point in 1.23 and no digits after the decimal point in four. So in the number which we found, there will be two digits after the decimal point. Therefore, our decimal point goes before the nine. And we can say that four multiplied by one pound 23 is equal to four pounds and 92 pence.

Now, let’s consider another way in which Fatima could buy three and three-fifths liters of orange juice. We can see that she could buy three lots of the 1.5-liter cartons of orange juice. And this will cost her three multiplied by the cost of the 1.5-liter cartons, which is one pound 52.

Again, we can use long hand multiplication in order to complete this calculation. Two multiplied by three gives us six. Three multiplied by five gives us 15. So we write down the five and carry the one. Then, three times one gives us three. And we simply add the one, which we carried from before, to give us four.

Then, we can see in the two numbers we multiplied that there were two digits after decimal points. So our answer will also have two digits after the decimal point, giving us an answer of 4.56. Therefore, three lots of the 1.5-liter cartons of orange juice will cost Fatima four pounds 56. And this is cheaper than buying four one-liter cartons of orange juice since four pounds 56 is less than four pounds 92.

Therefore, we know that Fatima should not be buying four one-liter cartons of orange juice. Now, we found it would be cheaper for Fatima to buy three 1.5-liter cartons of orange juice than four one-liter cartons of orange juice. And this agrees with the calculation we performed earlier, where we found that the 1.5-liter cartons of orange juice are cheaper per liter of orange juice.

There is one more case we should consider here since three lots of 1.5-liter cartons will give her three times 1.5 liters of orange juice. And we can find three times 1.5 by doing long multiplication.

Three times five gives us 15. So we write five and carry the one. Then three times one is equal to three. So we add the one which we carried earlier, giving us four. Now, in the numbers we multiplied together, there is one digit to the right of the decimal point. So therefore, in our answer, there will be one digit to the right of the decimal point also. So we found that in three 1.5-liter cartons of orange juice, there is in fact 4.5 liters of orange juice.

So the other case we should consider is if she buys two 1.5-liter cartoons and one one-liter carton since this will give her two timesed by 1.5 liters plus one timesed by one liter of orange juice. And two timesed by 1.5 is simply three and one multiplied by one gives us one. So this is equal to three plus one or four liters of orange juice, which is more than she needs to last her for nine days since four is greater than three and three-fifths.

The cost of buying two 1.5-liter cartons and one one-liter carton will be two multiplied by one pound 52 plus one multiplied by one pound 23. Let’s start by finding one pound 52 multiplied by two. We first multiply the two by the two to give us four. Then, we multiply the two by the five to give us 10. So we write down the zero and carry the one.

Then, we multiply two by one to give us two and then we add the one which we carried earlier to give us three. Then, we count the number of digits after the decimal point in the two numbers we multiplied together. So that’s two digits. And so therefore, our answer will also have two digits after the decimal point. So our answer will be 3.04.

And we can say that two timesed by one pound 52 is equal to three pounds and four pence. Then, we have that one multiplied by one pound 23 is simply equal to one pound 23 pence. So the calculation which we have left to perform is three pounds and four pence plus one pound and 23 pence.

We can add these two numbers using column addition. We have 3.04 plus 1.23. When adding numbers using the column method, the decimal point stays in the same place. And so we can write this in here. Now, we start by adding the two digits on the right. So that’s four plus three, which gives us seven. Then, we had the next two digits, so that’s zero plus two, giving us two. And finally, we have the two digits on the left, so that’s three plus one, to give us four. This gives us that 3.04 plus 1.23 is equal to 4.27.

And so this gives us that the total cost of two 1.5-liter cartons and one one-liter carton is four pounds and 27 pence. And we can clearly see that this is cheaper than buying the three 1.5-liter cartons since four pounds and 27 is less than four pounds 56.

And so we found that using Fatima’s estimate the smallest amount of money Fatima needs to spend on orange juice to last her nine days is four pounds and 27 pence, which happens when she buys two 1.5-liter cartoons and one one-liter carton. Fatima actually drinks more than two-fifths of a liter of orange juice per day.

Part b) Explain how this might affect the amount of money spent on orange juice.

We’re told that Fatima actually drinks more than she estimated per day. And so over the nine-day period, she’d drink more than the three and three-fifths liters of orange juice, which we predicted that she would drink in nine days using her estimate. And so therefore, we can conclude that she may need to spend more than four pounds 27 on orange juice in order to last her nine days.

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