Frequency Trees

What you'll learn

How to recognise the total, groups, and subgroups in a frequency tree.
How to complete missing frequencies using adding and subtracting.
How to use fraction clues like “ $34\frac{3}{4}$ of the boys”.
How to answer probability questions from a completed tree.

1. Start with frequencies

A frequency is a count. For example, if 18 people prefer tea, the frequency for “prefer tea” is 18.

A category is a group something can go into, such as male/female, pass/fail, or Saturday/Sunday.

Most frequency tree questions use a simple idea: a total is split into parts.

Key Idea

Parts add to the total

If a total is split into two groups, then the two group frequencies add back to the total.

Example

Finding the missing part

A group has 90 students. 34 are in set A and the rest are in set B.

A simple split shows the total 90 students divided into set A and set B, with the missing part found by subtracting from the total.

Start with the total: 90 students.
One part is known: 34 students are in set A.
Subtract to find the other part: 90 - 34 = 56.
So 56 students are in set B.

2. What a frequency tree shows

A frequency tree is a branching diagram that sorts a total into smaller groups.

A frequency tree starts with a total and branches into categories and then subcategories, with numbers placed in nodes.

Definition

Frequency tree

A frequency tree shows counts as they split into categories. A node is an oval or box where a number goes. A branch is a labelled line showing which category you move into.

Read a frequency tree from left to right.

The first node usually contains the total.
The first split divides the total into two groups.
Each of those groups can then split again.
The final numbers at the ends are the combined groups, such as “girl and prefers drink A”.

At every split, use:

parent total = two children added together
missing child = parent total minus known child

Example

Completing a tree from totals

80 students choose either drink A or drink B. The tree splits first into boys and girls, then into drink A and drink B. There are 37 girls. 52 students choose drink A in total. 21 of the girls choose drink A.

The completed tree shows how 80 students split into boys and girls, then into drink A and drink B.

Put 80 at the start of the tree.
Find the number of boys: 80 - 37 = 43.
For the girls, 21 choose drink A, so girls choosing drink B is 37 - 21 = 16.
The total choosing drink A is 52. Since 21 of these are girls, boys choosing drink A is 52 - 21 = 31.
Boys total is 43, so boys choosing drink B is 43 - 31 = 12.
The completed end counts are: boys A 31, boys B 12, girls A 21, girls B 16.

Common Mistake

Putting an overall total on one branch

If you are told “52 students choose drink A”, that means all the drink A students together. It includes boys choosing A and girls choosing A.

The overall drink A total is made from both branches ending in drink A, not just one branch.

3. Using fraction clues

Sometimes the information includes a fraction, such as “ $34\frac{3}{4}$ of the football students go on Saturday”.

The words of the tell you which group to use.

For example:

“ $34\frac{3}{4}$ of the football students” means use the football total.
“ $25\frac{2}{5}$ of the girls” means use the girls total.
It does not always mean use the grand total.

The useful rule is:

total×fraction\text{part} = \text{group total} \times \text{fraction}

Tip

Underline the group

When you see a fraction, underline the words after “of the”. That is the total you should multiply by the fraction.

$The words after “of the” identify the group total to multiply by the fraction.$

Example

Completing a tree with a fraction

72 students go to either football club or rugby club. Each student goes on Saturday or Sunday. 48 students go to football club. $34\frac{3}{4}$ of the football students go on Saturday. 54 students go on Saturday in total.

$The fraction clue uses the football total, so the football branch is split into Saturday and Sunday first.$

The total is 72. Football is 48, so rugby is 72 - 48 = 24.
Find football on Saturday by using the football total:
$48 \times \frac{3}{4} = 36$
Football total is 48, so football on Sunday is 48 - 36 = 12.
Total Saturday is 54. Football Saturday is 36, so rugby Saturday is 54 - 36 = 18.
Rugby total is 24, so rugby Sunday is 24 - 18 = 6.

4. Probability from a frequency tree

A probability is a chance, written as a fraction, decimal, or percentage. In these questions, it is often easiest to use a fraction.

Definition

Probability from frequencies

Probability is found by doing $possible\frac{\text{number wanted}}{\text{number possible}}$ . The denominator is the bottom number of the fraction and tells you the group you are choosing from.

The key exam skill is choosing the correct denominator.

If the question says:

“One person is chosen at random” — use the grand total.
“One of the left-handed people is chosen” — use the left-handed total.
“One of the people who do not like football is chosen” — use the total who do not like football.

This second type is called a conditional probability, because you are already told the person is in a particular group.

Example

Choosing the correct denominator

A completed frequency tree has 70 people in total. 30 are left-handed: 18 are male and 12 are female. The other 40 people are right-handed.

The denominator changes depending on whether the choice is from all 70 people or only the 30 left-handed people.

If one person is chosen from all 70 people, the denominator is 70.
The probability of choosing someone who is left-handed and female is $1270\frac{12}{70}$ , which simplifies to $635\frac{6}{35}$ .
If one of the left-handed people is chosen, the denominator changes to 30.
The probability that this left-handed person is female is $1230\frac{12}{30}$ , which simplifies to $25\frac{2}{5}$ .

Exam technique

In the exam

Fill in the total first, then work along the tree from left to right.
At each split, check that the two smaller numbers add to the number before the split.
For probability questions, circle the group being chosen from; that number is your denominator.

$Circling the group being chosen from helps identify the denominator for the probability fraction.$

Self review

Check yourself

If 62 people are surveyed and 27 are boys, how many are girls?
If 40 students play tennis and $35\frac{3}{5}$ go on Saturday, how many go on Saturday?
In the question “one of the girls is chosen”, what should your denominator be?

Frequency Trees

What you'll learn

How to recognise the total, groups, and subgroups in a frequency tree.
How to complete missing frequencies using adding and subtracting.
How to use fraction clues like “ $34\frac{3}{4}$ of the boys”.
How to answer probability questions from a completed tree.

1. Start with frequencies

A frequency is a count. For example, if 18 people prefer tea, the frequency for “prefer tea” is 18.

A category is a group something can go into, such as male/female, pass/fail, or Saturday/Sunday.

Most frequency tree questions use a simple idea: a total is split into parts.

Key Idea

Parts add to the total

If a total is split into two groups, then the two group frequencies add back to the total.

Example

Finding the missing part

A group has 90 students. 34 are in set A and the rest are in set B.

A simple split shows the total 90 students divided into set A and set B, with the missing part found by subtracting from the total.

Start with the total: 90 students.
One part is known: 34 students are in set A.
Subtract to find the other part: 90 - 34 = 56.
So 56 students are in set B.

2. What a frequency tree shows

A frequency tree is a branching diagram that sorts a total into smaller groups.

A frequency tree starts with a total and branches into categories and then subcategories, with numbers placed in nodes.

Definition

Frequency tree

A frequency tree shows counts as they split into categories. A node is an oval or box where a number goes. A branch is a labelled line showing which category you move into.

Read a frequency tree from left to right.

The first node usually contains the total.
The first split divides the total into two groups.
Each of those groups can then split again.
The final numbers at the ends are the combined groups, such as “girl and prefers drink A”.

At every split, use:

parent total = two children added together
missing child = parent total minus known child

Example

Completing a tree from totals

The completed tree shows how 80 students split into boys and girls, then into drink A and drink B.

Put 80 at the start of the tree.
Find the number of boys: 80 - 37 = 43.
For the girls, 21 choose drink A, so girls choosing drink B is 37 - 21 = 16.
The total choosing drink A is 52. Since 21 of these are girls, boys choosing drink A is 52 - 21 = 31.
Boys total is 43, so boys choosing drink B is 43 - 31 = 12.
The completed end counts are: boys A 31, boys B 12, girls A 21, girls B 16.

Common Mistake

Putting an overall total on one branch

If you are told “52 students choose drink A”, that means all the drink A students together. It includes boys choosing A and girls choosing A.

The overall drink A total is made from both branches ending in drink A, not just one branch.

3. Using fraction clues

Sometimes the information includes a fraction, such as “ $34\frac{3}{4}$ of the football students go on Saturday”.

The words of the tell you which group to use.

For example:

“ $34\frac{3}{4}$ of the football students” means use the football total.
“ $25\frac{2}{5}$ of the girls” means use the girls total.
It does not always mean use the grand total.

The useful rule is:

total×fraction\text{part} = \text{group total} \times \text{fraction}

Tip

Underline the group

When you see a fraction, underline the words after “of the”. That is the total you should multiply by the fraction.

$The words after “of the” identify the group total to multiply by the fraction.$

Example

Completing a tree with a fraction

$The fraction clue uses the football total, so the football branch is split into Saturday and Sunday first.$

The total is 72. Football is 48, so rugby is 72 - 48 = 24.
Find football on Saturday by using the football total:
$48 \times \frac{3}{4} = 36$
Football total is 48, so football on Sunday is 48 - 36 = 12.
Total Saturday is 54. Football Saturday is 36, so rugby Saturday is 54 - 36 = 18.
Rugby total is 24, so rugby Sunday is 24 - 18 = 6.

4. Probability from a frequency tree

A probability is a chance, written as a fraction, decimal, or percentage. In these questions, it is often easiest to use a fraction.

Definition

Probability from frequencies

Probability is found by doing $possible\frac{\text{number wanted}}{\text{number possible}}$ . The denominator is the bottom number of the fraction and tells you the group you are choosing from.

The key exam skill is choosing the correct denominator.

If the question says:

“One person is chosen at random” — use the grand total.
“One of the left-handed people is chosen” — use the left-handed total.
“One of the people who do not like football is chosen” — use the total who do not like football.

This second type is called a conditional probability, because you are already told the person is in a particular group.

Example

Choosing the correct denominator

A completed frequency tree has 70 people in total. 30 are left-handed: 18 are male and 12 are female. The other 40 people are right-handed.

The denominator changes depending on whether the choice is from all 70 people or only the 30 left-handed people.

If one person is chosen from all 70 people, the denominator is 70.
The probability of choosing someone who is left-handed and female is $1270\frac{12}{70}$ , which simplifies to $635\frac{6}{35}$ .
If one of the left-handed people is chosen, the denominator changes to 30.
The probability that this left-handed person is female is $1230\frac{12}{30}$ , which simplifies to $25\frac{2}{5}$ .

Exam technique

In the exam

Fill in the total first, then work along the tree from left to right.
At each split, check that the two smaller numbers add to the number before the split.
For probability questions, circle the group being chosen from; that number is your denominator.

$Circling the group being chosen from helps identify the denominator for the probability fraction.$

Self review

Check yourself

If 62 people are surveyed and 27 are boys, how many are girls?
If 40 students play tennis and $35\frac{3}{5}$ go on Saturday, how many go on Saturday?
In the question “one of the girls is chosen”, what should your denominator be?

Frequency Trees

What you'll learn

1. Start with frequencies

2. What a frequency tree shows

3. Using fraction clues

4. Probability from a frequency tree

Recap questions

You've reached the end

Frequency Trees

What you'll learn

1. Start with frequencies

2. What a frequency tree shows

3. Using fraction clues

4. Probability from a frequency tree

Recap questions

You've reached the end

Frequency Trees

What you'll learn

1. Start with frequencies

2. What a frequency tree shows

3. Using fraction clues

4. Probability from a frequency tree

Recap questions

You've reached the end

Frequency Trees Revision Guide

Frequency Trees

What you'll learn

1. Start with frequencies

2. What a frequency tree shows

3. Using fraction clues

4. Probability from a frequency tree

Recap questions

You've reached the end

Frequency Trees Revision Guide