The Answers

Fry’s DelightMiss Fry was given a box containing a variety of chocolates. Although she likes chocolates, she is not greedy, so she decided to share her chocolates and make them last. Her method of consumption was to eat one on the first day, and give away 10% of the remainder, to eat 2 on the second day and give 10% of the remainder away, eat three on the third day and give away 10% of the remainder, and continue in this way until no chocolates were left.

How many chocolates were in the box and how many days did they last?

Fry’s Delight

We begin with the assumption that she doesn’t eat or give away parts of chocolates.

This means that the minimum she can give away is 1.

1 is 10% of 10, which means she would have 9 left to eat, which would make it the 9th day.

Working backwards gives:

Fry’s DelightDay No at

startEat Leaving Give

AwayEnd of

Day9 9 9 0 0 08 18 8 10 1 97 27 7 20 2 186 36 6 30 3 275 45 5 40 4 364 54 4 50 5 453 63 3 60 6 542 72 2 70 7 631 81 1 80 8 72

Bournville Dreams

A chocolate factory makes two different types of chocolate: dark and light. Each day, it receives 1200L of milk and 1200kg of cocoa. A batch of dark chocolate requires 20L milk and 50kg of cocoa. A batch of light chocolate requires 40L of milk and 30kg cocoa. The management insists that the workers produce at least 25 batches in total each day and that only full batches can be produced. The company makes a profit of £15 on a batch of dark chocolate and £10 on a batch of light chocolate. Work out how many batches of each type need to be produced to maximise profits.

Bournville Dreams

A chocolate factory makes two different types of chocolate: dark and light. Each day, it receives 1200L of milk and 1200kg of cocoa. A batch of dark chocolate requires 20L milk and 50kg of cocoa. A batch of light chocolate requires 40L of milk and 30kg cocoa. The management insists that the workers produce at least 25 batches in total each day and that only full batches can be produced. The company makes a profit of £15 on a batch of dark chocolate and £10 on a batch of light chocolate. Work out how many batches of each type need to be produced to maximise profits.

First we identify constraints.

Cocoa

50d + 30l ≤ 1200

Milk

20d + 40l ≤ 1200

Min Batches

d + l ≥ 25

d ≥ 0, l ≥ 0

Bournville Dreams

Bournville DreamsMaximise: profit = 15d + 10l

The optimum solution will always be at a corner of the feasible region. These are:• (0,30) = £300• (0,25) = £250• (22,3) = £360• (9,25) = £385• (8,26) = £380• (8,25) = £370

8 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 925

25.1

25.2

25.3

25.4

25.5

25.6

25.7

25.8

25.9

26

Number of Dark Batches

Nu

mb

er o

f L

igh

t B

atch

es

Milk = 9 × 20 + 25 × 40 = 1180L Cocoa = 9 × 50 + 25 × 30 = 1200kg

Christmas Celebrations

At Christmas, Ken likes to buy chocolates for family, friends and carol singers. He decides to spend £200 and has 100 people to buy for. He divides his spending between family-sized tins for £10, small selection packs for friends at £2, and fun-sized chocolate bars for carol singers at 20p each. Given that Ken has more friends than he anticipates carol singers, how many of each item should he buy?

Christmas Celebrations

T + S + B = 100

10T + 2S + 0.2B = 200

B must be a multiple of 5 and S > B

2T + 2S + 2B = 200 Þ 8T - 1.8B = 0Þ 8T = 1.8BÞ 40T = 9B

Christmas Celebrations

40T = 9B Þ B = 0, 40 or 80Þ S > B, so B ≠ 80

B = 0 => T = 0 => S = 100

B = 40 => T = 9 => S = 51

So 40 carol singers, 9 family and 51 friends.

Box Clever

The gift box for some Fairtrade hot chocolate powder is to be made from a single square sheet of card, with side length of 24cm, by cutting smaller squares (of integer side length) from the corners of the original square. It will then be folded up to make an open box (The lid is made separately).

What is the maximum volume that can be obtained in this way? Show, using a graph, that this is the maximum volume.

Box Clevercut out base volume

0 24 0

1 22 484

2 20 800

3 18 972

4 16 1024

5 14 980

6 12 864

7 10 700

8 8 512

9 6 324

10 4 160

11 2 44

12 0 0

Box Clever

cut out base volume0 24 01 22 4842 20 8003 18 9724 16 10245 14 9806 12 8647 10 7008 8 5129 6 324

10 4 16011 2 4412 0 0

0 1 2 3 4 5 6 7 8 9 10 11 120

200

400

600

800

1000

1200

Size of cut out (cm)

Vo

lum

e o

f b

ox

(cm

³)

Maximum volume of

1024cm³ when cut out is 4cm

Six After Eight

A small selection bag of posh Fairtrade chocolates contains 6 chocolates. These are a random selection from 8 different chocolates. A quality-control system is put in place to ensure that no selection contains more than 2 of the same chocolate or fewer than 4 different chocolates. How many different bags of chocolates are possible?

Six After Eight

There are 3 possible scenarios:

6 unique chocolates

1 pair and 4 other unique chocolates

2 pairs and 2 other unique chocolates

Six After Eight

6 unique chocolates

282

78

!2!6

!8

Six After Eight

1 pair and 4 other unique chocolates

28057823

5678

!3!4

!78

Six After Eight

2 pairs and 2 other unique chocolates

42057124

5678

!4!2

!6

2

78

Six After Eight

28 + 280 + 420 = 728

Seg-sational!

There are 20 segments in a chocolate orange. Modelling a chocolate orange as a sphere of diameter 5cm, with a cylindrical hollow of diameter 5mm running down the core, what is the volume of a segment?

You may discount the small dome of chocolate at each of the ‘poles’ of the Chocolate Orange.

Seg-sational!

Volume of sphere of chocolate

333 45.655.23

4

3

4cmr

Seg-sational!

Volume of hollow cylinder

322 98.0525.0 cmhr

Seg-sational!

Volume of segment

322.32047.64

47.6498.045.65

cm

Breaking the Mould

Anna and Billy play a game with a rectangular chocolate bar that is 5 squares by 10 squares. Anna starts. They take turns breaking a piece of the bar (only one piece can be broken in one turn, always along the lines between the squares). The first player to break off a 1 by 1 square piece wins. Who wins the game?!

Breaking the Mould

Anna Billy

A Year’s Supply of Chocolate

Assume that all of the Cadbury’s Dairy Milk bars bought in England were the standard length of 11.4cm. When laid end-to-end, they would form a line around the Earth along the line of latitude passing through Liverpool (53½°N). Using the Earth’s circumference at 40,075km at the equator, how many bars are sold in England each year?

A Year’s Supply of Chocolate

Radius of Earth =

40075 ÷ 2π = 6378.134km

Radius at Liverpool

6378.134 × cos (53.5) = 3793.860km

Circumference at Liverpool = 3793.860 × 2π

= 23837.523km

A Year’s Supply of Chocolate

Circumference at Liverpool = 3793.860 × 2π

= 23837.523km

= 2383752317.9cm

Number of bars

= 2383752317.9 ÷ 11.4

= 209101080.5118

≈ 209,101,081 bars