Hannah's Sweets - The Hardest GCSE Maths Exam Question Ever?
What does it mean to complete the square?
The Summer 2015 Higher Tier GCSE maths paper from Edexcel featured a particularly infamous question - Hannah's Sweets!
This question went viral with students complaining about how difficult it was and how they were going to lose out on top grades because of it. It even made the news, featuring on the BBC News website, the Guardian and others.
Was it actually as difficult as people made out? Was it perhaps the most difficult maths exam question ever? Let's have a look at the question and how to solve it.
This question went viral with students complaining about how difficult it was and how they were going to lose out on top grades because of it. It even made the news, featuring on the BBC News website, the Guardian and others.
Was it actually as difficult as people made out? Was it perhaps the most difficult maths exam question ever? Let's have a look at the question and how to solve it.
The question - Hannah's sweets
There are n sweets in a bag.
6 of the sweets are orange.
The rest of the sweets are yellow.
Hannah takes at random a sweet from the bag.
She eats the sweet.
Hannah then takes at random another sweet from the bag.
She eats the sweet.
The probability that Hannah eats two orange sweets is 1/3.
a. Show that n^2 - n - 90 = 0
6 of the sweets are orange.
The rest of the sweets are yellow.
Hannah takes at random a sweet from the bag.
She eats the sweet.
Hannah then takes at random another sweet from the bag.
She eats the sweet.
The probability that Hannah eats two orange sweets is 1/3.
a. Show that n^2 - n - 90 = 0
Why is this question so tricky?
At first glance, the Hannah's sweets question doesn't seem too difficult. The first four paragraphs set up what looks like a fairly standard probability question. We have a number of sweets in a bag, of which 6 are orange. We're told that Hannah takes two sweets and eats them (in other words, she is not returning the sweets to the bag after each choice). We're even told the probability of getting two orange sweets.
It's only right at the end that we're thrown a curveball by being given a seemingly random quadratic equation and being asked to show that it is correct.
It's this linking together of two parts of mathematics; probability and algebra, that seems to have thrown people. One student was even quoted as saying that 'Hannah's sweets in particular made me want to cry'.
It's only right at the end that we're thrown a curveball by being given a seemingly random quadratic equation and being asked to show that it is correct.
It's this linking together of two parts of mathematics; probability and algebra, that seems to have thrown people. One student was even quoted as saying that 'Hannah's sweets in particular made me want to cry'.
How to solve the Hannah's sweets question
If you break the question down into its separate parts, it isn't anywhere near as difficult as it first appears.
The first thing is to realise that, as well as being given a value for the probability of Hannah eating two orange sweets, we can also express this algebraically using the information given at the beginning.
To calculate the probability of Hannah eating two orange sweets, we need to find the probability that the first sweet is orange and then multiply this by the probability that the second sweet is orange.
The probability that the first sweet is orange is nice and simple. There are 6 orange sweets out of n sweets in total, hence p(first sweet is orange) = 6/n.
For the second sweet, we need to remember that Hannah has already removed and eaten an orange sweet. Therefore both the number of orange sweets and the total number of sweets have reduced by 1, to 5 and n−1 respectively. We then get that p(second sweet is orange) = 5/(n−1).
The first thing is to realise that, as well as being given a value for the probability of Hannah eating two orange sweets, we can also express this algebraically using the information given at the beginning.
To calculate the probability of Hannah eating two orange sweets, we need to find the probability that the first sweet is orange and then multiply this by the probability that the second sweet is orange.
The probability that the first sweet is orange is nice and simple. There are 6 orange sweets out of n sweets in total, hence p(first sweet is orange) = 6/n.
For the second sweet, we need to remember that Hannah has already removed and eaten an orange sweet. Therefore both the number of orange sweets and the total number of sweets have reduced by 1, to 5 and n−1 respectively. We then get that p(second sweet is orange) = 5/(n−1).
Setting up the quadratic equation
As mentioned earlier p(both sweets are orange) = p(first sweet is orange) × p(second sweet is orange). It therefore equals 6/n × 5/(n−1) = 30/n(n−1).
As we are also told in the question that this probability equals 1/3, we can set these equal to each other, giving us:
1/3 = 30/n(n−1)
Multiplying each side by n(n−1) gives us:
1/3 n(n−1) = 30
Multiplying both sides by 3 gives us:
n(n−1) = 90
We can then expand the brackets on the left:
n^2 − n = 90
Finally we subtract 90 from both sides:
n^2 − n − 90 = 0
There we have it. The question has been solved without any particularly high-level maths skills. The difficulty is all in how the question has been set-up. Mixing the probability and algebra, without giving an obvious starting point for the answer method, made this question one of the trickier and most memorable GCSE maths exam questions.
As we are also told in the question that this probability equals 1/3, we can set these equal to each other, giving us:
1/3 = 30/n(n−1)
Multiplying each side by n(n−1) gives us:
1/3 n(n−1) = 30
Multiplying both sides by 3 gives us:
n(n−1) = 90
We can then expand the brackets on the left:
n^2 − n = 90
Finally we subtract 90 from both sides:
n^2 − n − 90 = 0
There we have it. The question has been solved without any particularly high-level maths skills. The difficulty is all in how the question has been set-up. Mixing the probability and algebra, without giving an obvious starting point for the answer method, made this question one of the trickier and most memorable GCSE maths exam questions.
Part b - Calculate n
Part b of this question was a lot simpler. We are asked to solve the equation n^2 − n − 90 = 0 to calculate how many sweets there were in the bag to start with.
To do this we can factorise the equation by finding two numbers that multiply to give −90 and add to give −1. These are 9 and −10. We then get:
(n + 9)(n − 10) = 0
This gives us answers of −9 and +10. Obviously there wasn't a negative number of sweets in the bag and so the answer must be 10.
To do this we can factorise the equation by finding two numbers that multiply to give −90 and add to give −1. These are 9 and −10. We then get:
(n + 9)(n − 10) = 0
This gives us answers of −9 and +10. Obviously there wasn't a negative number of sweets in the bag and so the answer must be 10.
Comments
So what do you think? Was this the trickiest GCSE maths exam question ever? Was it all a fuss about nothing?
Don't forget to leave your comments below.
Don't forget to leave your comments below.