Two Way Tables

What you'll learn

How to read a two-way table using rows and columns.
How to fill in missing numbers using totals.
How to answer simple probability questions from a completed table.
How to build a table from worded information and percentages.

1. Reading a two-way table

A two-way table organises information in two different ways. For example, it might sort students by gender and by how they travel to school.

Definition

Rows, columns and cells

A row goes across the table.
A column goes down the table.
A cell is one box inside the table.
A row total adds across a row. A column total adds down a column.
The grand total is the total for the whole table, usually in the bottom-right corner.

A labelled two-way table showing what rows, columns, cells, row totals, column totals and the grand total mean.

Example

Reading one box in a table

The table shows 60 students’ favourite drink.

The Girls row and Water column meet at the cell containing 18.

	Water	Juice	Total
Boys	14	16	30
Girls	18	12	30
Total	32	28	60

To find how many girls chose water, look along the Girls row.
Look down the Water column.
The row and column meet at 18, so 18 girls chose water.
The Water column total is 32, so 32 students chose water altogether.

2. Completing missing values

Most two-way table questions are about filling in blanks. A blank does not mean zero — it means the number is missing.

Key Idea

Totals must match

Every row must add across to its row total, and every column must add down to its column total. Use subtraction to work backwards from totals.

Example

Completing a travel table

100 students were asked how they travel to school.

A partially completed travel table where blanks are found by working backwards from row, column and grand totals.

	Walk	Bus	Other	Total
Boys	18			54
Girls		25	9
Total			20	100

The grand total is 100 and there are 54 boys, so the girls’ total is 100 - 54 = 46.
In the Girls row, the missing Walk number is 46 - 25 - 9 = 12.
In the Other column, the missing Boys number is 20 - 9 = 11.
In the Boys row, the missing Bus number is 54 - 18 - 11 = 25.
Now add down the columns: Walk is 18 + 12 = 30, and Bus is 25 + 25 = 50.

The completed table is:

The completed travel table can be checked by adding every row across and every column down.

	Walk	Bus	Other	Total
Boys	18	25	11	54
Girls	12	25	9	46
Total	30	50	20	100

Tip

Quick check

When you finish, check both ways: rows across and columns down. If one total does not work, one of your entries is wrong.

3. Probability from a two-way table

A probability is a measure of chance. If one person is chosen at random, every person is equally likely to be picked.

For GCSE, you often use:

Probability = favourable outcomes divided by total outcomes.

The favourable outcomes are the ones the question asks for.

Example

Finding a probability from the table

Use the completed travel table above. One student is chosen at random. Find the probability that the student walks to school.

For a whole-table probability, use the Walk column total over the grand total.

The question says one student is chosen from the whole group, so use the grand total: 100.
The number who walk is the Walk column total: 30.
Write the probability as a fraction: $30100\frac{30}{100}$ .
Simplify the fraction by dividing top and bottom by 10, so the probability is $310\frac{3}{10}$ .

Common Mistake

Using the wrong total

If the question says one student from the whole table, use the grand total. Only use a row total if the question says something like “one of the boys is chosen”.

4. Making a table from words

Sometimes the table is empty and the information is written in sentences. Start by putting in the totals you know, then fill the inside cells.

Example

Building a table from a worded question

160 Year 8 students either walk, get the bus, or cycle. There are 76 boys. 42 students walk. 18 of the walkers are boys. 50 girls get the bus. 38 students cycle.

The worded information is organised into a two-way table with gender as rows and travel method as columns.

Put in the grand total 160 and the boys’ total 76.
Find the girls’ total: 160 - 76 = 84.
Put in the Walk total 42 and Cycle total 38, so the Bus total is 160 - 42 - 38 = 80.
“18 of the walkers are boys” means Boys Walk is 18, so Girls Walk is 42 - 18 = 24.
Girls Bus is 50, so Girls Cycle is 84 - 24 - 50 = 10.
Boys Bus is 80 - 50 = 30, and Boys Cycle is 38 - 10 = 28.

	Walk	Bus	Cycle	Total
Boys	18	30	28	76
Girls	24	50	10	84
Total	42	80	38	160

5. Percentages in two-way tables

A percentage means “out of 100”. If the question gives a total number, change the percentages into numbers first.

Common Mistake

Percent of all or percent of a group?

A phrase like “18% are boys who get the bus” usually means 18% of everyone, not 18% of the boys, unless the question clearly says “of the boys”.

Example

Using percentages with a known total

200 students either walk or get the bus. 60% walk. 45% are boys. 18% are boys who get the bus. Work out how many girls walk.

Percentages are first converted into numbers out of the total of 200, then placed in the table.

60% of 200 is 120, so 120 students walk.
The other students get the bus, so 200 - 120 = 80 get the bus.
45% of 200 is 90, so there are 90 boys and 110 girls.
18% of 200 is 36, so 36 boys get the bus.
Boys who walk is 90 - 36 = 54.
Girls who walk is 120 - 54 = 66.

If no total number is given, you can pretend the total is 100, because percentages are already out of 100.

Example

Using percentages when no total is given

At a conference, 14% of people are staff and the rest are students. 46% are from Centre A. 50% are students from Centre B. Find the probability that a person chosen at random is staff from Centre A.

When no total is given, pretending there are 100 people turns the percentages into table entries.

Pretend there are 100 people.
There are 14 staff and 86 students.
Centre A has 46 people, so Centre B has 54 people.
Centre B has 50 students, so Centre B has 54 - 50 = 4 staff.
There are 14 staff in total, so Centre A staff is 14 - 4 = 10.
The probability is $10100\frac{10}{100}$ , which simplifies to $110\frac{1}{10}$ .

Exam technique

In the exam

Copy the table neatly and include the Total row and Total column.
Fill in the numbers you are directly given before doing calculations.
Use subtraction from totals to find missing boxes, then check rows and columns.
For probability, use the correct denominator: whole table, row total, or column total.

Self review

Check yourself

Can you explain the difference between a row total and a column total?
If 24 out of 120 students walk to school, what fraction would represent that probability?
In a percentage question with no total, why is it useful to pretend there are 100 people?

Two Way Tables

What you'll learn

How to read a two-way table using rows and columns.
How to fill in missing numbers using totals.
How to answer simple probability questions from a completed table.
How to build a table from worded information and percentages.

1. Reading a two-way table

A two-way table organises information in two different ways. For example, it might sort students by gender and by how they travel to school.

Definition

Rows, columns and cells

A row goes across the table.
A column goes down the table.
A cell is one box inside the table.
A row total adds across a row. A column total adds down a column.
The grand total is the total for the whole table, usually in the bottom-right corner.

A labelled two-way table showing what rows, columns, cells, row totals, column totals and the grand total mean.

Example

Reading one box in a table

The table shows 60 students’ favourite drink.

The Girls row and Water column meet at the cell containing 18.

	Water	Juice	Total
Boys	14	16	30
Girls	18	12	30
Total	32	28	60

To find how many girls chose water, look along the Girls row.
Look down the Water column.
The row and column meet at 18, so 18 girls chose water.
The Water column total is 32, so 32 students chose water altogether.

2. Completing missing values

Most two-way table questions are about filling in blanks. A blank does not mean zero — it means the number is missing.

Key Idea

Totals must match

Every row must add across to its row total, and every column must add down to its column total. Use subtraction to work backwards from totals.

Example

Completing a travel table

100 students were asked how they travel to school.

A partially completed travel table where blanks are found by working backwards from row, column and grand totals.

	Walk	Bus	Other	Total
Boys	18			54
Girls		25	9
Total			20	100

The grand total is 100 and there are 54 boys, so the girls’ total is 100 - 54 = 46.
In the Girls row, the missing Walk number is 46 - 25 - 9 = 12.
In the Other column, the missing Boys number is 20 - 9 = 11.
In the Boys row, the missing Bus number is 54 - 18 - 11 = 25.
Now add down the columns: Walk is 18 + 12 = 30, and Bus is 25 + 25 = 50.

The completed table is:

The completed travel table can be checked by adding every row across and every column down.

	Walk	Bus	Other	Total
Boys	18	25	11	54
Girls	12	25	9	46
Total	30	50	20	100

Tip

Quick check

When you finish, check both ways: rows across and columns down. If one total does not work, one of your entries is wrong.

3. Probability from a two-way table

A probability is a measure of chance. If one person is chosen at random, every person is equally likely to be picked.

For GCSE, you often use:

Probability = favourable outcomes divided by total outcomes.

The favourable outcomes are the ones the question asks for.

Example

Finding a probability from the table

Use the completed travel table above. One student is chosen at random. Find the probability that the student walks to school.

For a whole-table probability, use the Walk column total over the grand total.

The question says one student is chosen from the whole group, so use the grand total: 100.
The number who walk is the Walk column total: 30.
Write the probability as a fraction: $30100\frac{30}{100}$ .
Simplify the fraction by dividing top and bottom by 10, so the probability is $310\frac{3}{10}$ .

Common Mistake

Using the wrong total

If the question says one student from the whole table, use the grand total. Only use a row total if the question says something like “one of the boys is chosen”.

4. Making a table from words

Sometimes the table is empty and the information is written in sentences. Start by putting in the totals you know, then fill the inside cells.

Example

Building a table from a worded question

160 Year 8 students either walk, get the bus, or cycle. There are 76 boys. 42 students walk. 18 of the walkers are boys. 50 girls get the bus. 38 students cycle.

The worded information is organised into a two-way table with gender as rows and travel method as columns.

Put in the grand total 160 and the boys’ total 76.
Find the girls’ total: 160 - 76 = 84.
Put in the Walk total 42 and Cycle total 38, so the Bus total is 160 - 42 - 38 = 80.
“18 of the walkers are boys” means Boys Walk is 18, so Girls Walk is 42 - 18 = 24.
Girls Bus is 50, so Girls Cycle is 84 - 24 - 50 = 10.
Boys Bus is 80 - 50 = 30, and Boys Cycle is 38 - 10 = 28.

	Walk	Bus	Cycle	Total
Boys	18	30	28	76
Girls	24	50	10	84
Total	42	80	38	160

5. Percentages in two-way tables

A percentage means “out of 100”. If the question gives a total number, change the percentages into numbers first.

Common Mistake

Percent of all or percent of a group?

A phrase like “18% are boys who get the bus” usually means 18% of everyone, not 18% of the boys, unless the question clearly says “of the boys”.

Example

Using percentages with a known total

200 students either walk or get the bus. 60% walk. 45% are boys. 18% are boys who get the bus. Work out how many girls walk.

Percentages are first converted into numbers out of the total of 200, then placed in the table.

60% of 200 is 120, so 120 students walk.
The other students get the bus, so 200 - 120 = 80 get the bus.
45% of 200 is 90, so there are 90 boys and 110 girls.
18% of 200 is 36, so 36 boys get the bus.
Boys who walk is 90 - 36 = 54.
Girls who walk is 120 - 54 = 66.

If no total number is given, you can pretend the total is 100, because percentages are already out of 100.

Example

Using percentages when no total is given

At a conference, 14% of people are staff and the rest are students. 46% are from Centre A. 50% are students from Centre B. Find the probability that a person chosen at random is staff from Centre A.

When no total is given, pretending there are 100 people turns the percentages into table entries.

Pretend there are 100 people.
There are 14 staff and 86 students.
Centre A has 46 people, so Centre B has 54 people.
Centre B has 50 students, so Centre B has 54 - 50 = 4 staff.
There are 14 staff in total, so Centre A staff is 14 - 4 = 10.
The probability is $10100\frac{10}{100}$ , which simplifies to $110\frac{1}{10}$ .

Exam technique

In the exam

Copy the table neatly and include the Total row and Total column.
Fill in the numbers you are directly given before doing calculations.
Use subtraction from totals to find missing boxes, then check rows and columns.
For probability, use the correct denominator: whole table, row total, or column total.

Self review

Check yourself

Can you explain the difference between a row total and a column total?
If 24 out of 120 students walk to school, what fraction would represent that probability?
In a percentage question with no total, why is it useful to pretend there are 100 people?

Two Way Tables

What you'll learn

1. Reading a two-way table

2. Completing missing values

3. Probability from a two-way table

4. Making a table from words

5. Percentages in two-way tables

Recap questions

You've reached the end

Two Way Tables

What you'll learn

1. Reading a two-way table

2. Completing missing values

3. Probability from a two-way table

4. Making a table from words

5. Percentages in two-way tables

Recap questions

You've reached the end

Two Way Tables

What you'll learn

1. Reading a two-way table

2. Completing missing values

3. Probability from a two-way table

4. Making a table from words

5. Percentages in two-way tables

Recap questions

You've reached the end

Two Way Tables Revision Guide

Two Way Tables

What you'll learn

1. Reading a two-way table

2. Completing missing values

3. Probability from a two-way table

4. Making a table from words

5. Percentages in two-way tables

Recap questions

You've reached the end

Two Way Tables Revision Guide