Frequency Trees: Edexcel GCSE Foundation maths revision notes

What a frequency tree shows

A frequency tree sorts a group of people or things in two steps. The first box holds the whole group. Its two branches split the group into two parts, and then each part is split into two again.

For example, 40 people were asked if they like football. The tree below sorts them into men and women first, and then into those who like football and those who do not.

Every number counts people. A box at the end of the branches counts the people who fit both labels on its path, so the 10 counts the men who like football.

In the exam you are usually asked to complete a frequency tree using information written in words, for 3 marks. A second part often asks for a probability, a fraction or a percentage from your tree, for 1 or 2 marks.

The branches add up

The two boxes at the end of a pair of branches always add up to the box they come from. In the football tree, 10 + 15 = 25 men, 6 + 9 = 15 women, and 25 + 15 = 40 people.

So a missing box is found by subtraction:

\text{missing box} = \text{the box before} - \text{the other box}

The four end boxes add up to the whole group too: 10 + 15 + 6 + 9 = 40. Use this to check a finished tree.

A frequency tree is not a probability tree. Its numbers are counts of people, so each pair of boxes adds up to the box before. In a probability tree, each pair of branches adds up to 1

Worked example 1: Reading a frequency tree

A farm park had 64 visitors one morning. The frequency tree shows how many were adults or children, and whether they fed the animals.

(a) How many of the children did not feed the animals?

(b) How many of the visitors fed the animals?

(a) Follow the children branch, then its “did not feed” branch. The box there says 7
(b) Visitors who fed the animals are on both branches: some are adults and some are children.
Your turn. How many of the visitors fed the animals?

Your answer

11 + 29 = 40 visitors fed the animals.
Answer: (a) 7 (b) 40

Worked example 2: Filling in missing boxes

85 people used a library one day. The frequency tree shows whether each person was under 18 or 18 or over, and whether they borrowed a book.

Work out the numbers in the boxes labelled P, Q and R.

P and 30 add up to 85, so P = 85 - 30 = 55
19 and Q add up to 30, so Q = 30 - 19 = 11
Your turn. R and 21 add up to P, which is 55. Work out R.

Your answer

R = 55 - 21 = 34
Check: 19 + 11 + 34 + 21 = 85
Answer: P = 55, Q = 11, R = 34

Completing a tree from facts in words

Put the total in the first box.
Read each fact and decide which box it belongs in. A fact such as “14 of the 40 men wear glasses” gives two numbers: 40 in the men box, and 14 in the box for men who wear glasses.
Tick off each fact in the question as you use it.
Find a pair of boxes with only one gap, and fill the gap by subtraction. Repeat until every box has a number.
Check that each pair adds up to the box before, and that the four end boxes add up to the total.

Write each subtraction down beside the tree. Examiners give marks for a correct method, even when a number in a box turns out to be wrong.

Worked example 3: Completing a tree from words

96 people visited a climbing wall on Saturday.

54 of the visitors had climbed there before. The rest were climbing there for the first time.

38 of the visitors who had been before reached the top.

15 of the first-timers did not reach the top.

Use this information to complete the frequency tree.

Put in the numbers you are told: 96 in the first box, 54 for been before, 38 for been before and reached the top, and 15 for first time and did not reach it.
First time: 96 - 54 = 42
Been before, did not reach the top: 54 - 38 = 16
Your turn. How many of the first-timers reached the top?

Your answer

42 - 15 = 27 first-timers reached the top.
Answer: the completed tree is below. Check: 38 + 16 + 27 + 15 = 96

Probabilities and fractions from a tree

When one person is chosen at random, each person in the group is equally likely to be picked. Then:

\text{probability} = \frac{\text{number in the box you want}}{\text{number in the group they are chosen from}}

Read carefully who can be chosen. “One of the people” means anyone in the tree, so divide by the number in the first box. “One of the women” means only the women, so divide by the number in the women box.

A question may ask what fraction or what percentage of a group fit a description. Work it out in the same way. If 12 of the 30 adults wear glasses, the fraction is \frac{12}{30} = \frac{2}{5}, and the percentage is \frac{12}{30} \times 100 = 40\%

Give a probability as a fraction, a decimal or a percentage, not as a ratio.

Worked example 4: A probability and a percentage

60 students were asked if they walk to school. The frequency tree shows the results.

(a) One of the 60 students is chosen at random. Find the probability that this student is a girl who walks to school. Give your answer as a fraction in its simplest form.

(b) What percentage of the boys walk to school?

(a) Any of the 60 students can be chosen, so the bottom of the fraction is 60
Your turn. Follow the girls branch, then its walk branch. Write the probability as a fraction in its simplest form.

Your answer

12 girls walk: \frac{12}{60} = \frac{1}{5}
(b) Only the boys count now, so divide by the number of boys, 28, not by 60
7 of the 28 boys walk: \frac{7}{28} \times 100 = 25\%
Answer: (a) \frac{1}{5} (b) 25\%

A total that covers both branches

Some facts count people on both branches. Suppose a tree splits people into men and women first, then into those who own a bike and those who do not. “45 of the people own a bike” counts men and women, so 45 does not go in any one box.

It is the total of two end boxes, one on each branch. If 30 of the women own a bike, then 45 - 30 = 15 of the men own a bike.

Before you write a number in a box, ask: does this fact count everyone, or only the people on one branch?

Worked example 5: A total across both branches

160 people at a music festival were asked if they camped and if they bought food at the festival.

95 of the people camped.

112 of the people bought food.

20 of the people who camped did not buy food.

Use this information to complete the frequency tree.

Put in 160 in the first box, 95 for camped, and 20 for camped but did not buy food.
Did not camp: 160 - 95 = 65. Camped and bought food: 95 - 20 = 75
The 112 who bought food are on both branches: 75 of them camped, and the rest did not camp.
Your turn. How many of the people who did not camp bought food?

Your answer

112 - 75 = 37 people who did not camp bought food.
Did not camp, did not buy food: 65 - 37 = 28
Answer: the completed tree is below. Check: 75 + 37 = 112 bought food, and 75 + 20 + 37 + 28 = 160

Drawing your own tree

Some questions sort a group in two ways but give no tree or table. Draw your own frequency tree, or a two-way table, so that every number has its place. Examiners report that students who did this did better than those who wrote numbers all over the page.

Draw the tree big enough to write in, and label every branch. Put the total in the first box, then use the facts to fill the other boxes, as before.

A fact may be a fraction or a percentage of a group. Work out the number first: if \frac{2}{3} of 45 people are adults, there are 45 \div 3 \times 2 = 30 adults.

Worked example 6: Drawing a tree to find a probability

A bakery sold 150 pies on Saturday. Each pie was large or small, and had a meat filling or a vegetable filling.

\frac{3}{5} of the pies were large.

54 of the large pies had a meat filling.

24 of the small pies had a vegetable filling.

One of the 150 pies is chosen at random. Work out the probability that it had a vegetable filling.

No tree is given, so draw one: 150 in the first box, branches for large and small, then meat and vegetable on each.
Large: \frac{3}{5} of 150 = 150 \div 5 \times 3 = 90. Small: 150 - 90 = 60
Large with a vegetable filling: 90 - 54 = 36. Small with a meat filling: 60 - 24 = 36
Your turn. Vegetable pies are on both branches. How many of the pies had a vegetable filling?

Your answer

36 + 24 = 60 pies had a vegetable filling.
Any of the 150 pies can be chosen: \frac{60}{150} = \frac{2}{5}
Answer: the probability is \frac{2}{5}, and the completed tree is below.

Common mistakes

Putting a total in the wrong box. A fact such as “112 bought food” can count people on both branches, so split it between two end boxes.
Filling in only the numbers you are given. Work out every empty box, so that each pair adds up to the box before.
Not checking the tree. Check that the four end boxes add up to the total, and write each calculation down.
Making a pair of branches add up to 1. That is a probability tree; in a frequency tree each pair adds up to the box before.
Dividing by the wrong total. For a probability, fraction or percentage, divide by the group that can be chosen: everyone, or one branch only.
Writing a probability as a ratio. Give it as a fraction, a decimal or a percentage.