### 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.