How do I solve Hannah’s sweet question?

This refers to a question in the Edexcel GCSE paper this year which took students to Twitter venting their frustration. The question is as follows:

‘There are n sweets in a bag. 6 of the sweets are orange. The rest of the sweets are yellow.

Hannah takes a random 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.

Show that n² – n – 90 = 0’

Seemingly out of nowhere you’re asked to prove that a certain quadratic equation holds using the information provided. The first three lines set up the situation whilst the fourth line provides you with some extra information to use to obtain the answer. Intuition should tell you that you need to calculate the probability that Hannah eats two orange sweet using the first three lines and then apply what you’re given in the fourth line.

So let’s do that. What’s the probability that the first sweet she eats from the bag is orange? There are n sweets in the bag, 6 of which are orange. So the probability is 6/n.

What’s the probability that the second sweet she eats from the bag is orange? Now there are n-1 sweets in the bag, 5 of which are orange (since she has eaten an orange sweet!). So the probability is 5/(n-1).

These two events are separate from one another, so the probability that both happen (i.e. both the sweets are orange) are the two probabilities multiplied together: 6/n × 5/(n-1) = 30/n(n-1)

But you’re told that this probability is 1/3! So all you need to do is set the expression equal to 1/3, rearrange and (hopefully!) obtain the required quadratic equation.

30/n(n-1) = 1/3

⇒ 90/n(n-1) = 1           (multiplying both sides by 3)

⇒ 90 = n(n-1)               (multiplying both sides by n(n-1))

⇒ n(n-1) – 90 = 0         (subtracting 90 from both sides)

⇒ n² – n – 90 = 0         (expanding the brackets)

Tah-dah. We’ve found the required equation and we’re done. This question was only worth three marks; a bit stingy in my opinion!

George B. GCSE Maths tutor, A Level Maths tutor, A Level Further Math...

2 years ago

Answered by George, a GCSE Maths tutor with MyTutor

Still stuck? Get one-to-one help from a personally interviewed subject specialist


£20 /hr

Hannah B.

Degree: MMath(Hons) in Mathematics (Other) - Manchester University

Subjects offered: Maths, Further Mathematics + 1 more

Further Mathematics
-Personal Statements-

“Top tutor from the renowned Russell Group university, ready to help you improve your grades.”

MyTutor guarantee

£20 /hr

Emily L.

Degree: Law (Bachelors) - Durham University

Subjects offered: Maths, English Literature+ 1 more

English Literature

“I am a first year undergraduate reading Law at Durham University. Prior to this, I attended the Tiffin Girls' School in Kingston upon Thames where I studied English Literature, Chemistry and Fine Art at A2 Level (with additional Mathe...”

£24 /hr

Joel G.

Degree: Economics with a year in industry (Bachelors) - Liverpool University

Subjects offered: Maths, Geography+ 1 more


“I am an economics student at the University of Liverpool. I am really interested in economics and mathematics and I would love to share this intererest with yourselves. I I have taught children from year 7 to year 11 how to debate thr...”

About the author

George B.

Currently unavailable: for new students

Degree: Mathematics (Masters) - Warwick University

Subjects offered: Maths, Further Mathematics + 1 more

Further Mathematics

“Third year undergraduate at one of the top universities for Maths. Eager to tutor and help improve your grades.”

You may also like...

Posts by George

Given that a and b are distinct positive numbers, find a polynomial P(x) such that the derivative of f(x) = P(x)e^(−x²) is zero for x = 0, x = ±a and x = ±b, but for no other values of x.

How do I solve Hannah’s sweet question?

Let P(z) = z⁴ + az³ + bz² + cz + d be a quartic polynomial with real coefficients. Let two of the roots of P(z) = 0 be 2 – i and -1 + 2i. Find a, b, c and d.

What values of θ between 0 and 2π satisfy the equation cosec(θ) + 5cot(θ) = 3sin(θ)?

Other GCSE Maths questions

Solve the simultaneous equations 2x + y = 7 and 3x - y = 8.

Solve these equations simultaneously: (1) 5x - 10z = -45 and (2) 9x = -5z + 80

Expand and simplify 3(2x + 5) – 2(x – 4)

How can I find x and y?

View GCSE Maths tutors


We use cookies to improve our service. By continuing to use this website, we'll assume that you're OK with this. Dismiss