Two-Way Tables

Build a two-way table from sentences, read the right cell or total, and find a fraction or percentage of everyone or of one group.

In the exam for Edexcel IGCSE Maths 4MA1, Cambridge O Level Maths 4024 · 4 real past-paper questions below

اقرأ هذا الدرس مشروحاً بالعربي ←

What a Two-Way Table Shows

A two-way table sorts the same people in two ways at once. Here 35 students are sorted by club, down the side, and by boys or girls, across the top.

Club	Boys	Girls
Chess	9	6
Drama	7	13
Total	16	19

Rule

Each inner cell fits both its row and its column. The 6 means girls and chess.

Key idea

Rows add across, columns add down. Chess is $9 + 6 = 15$ , drama is $7 + 13 = 20$ , and $15 + 20 = 16 + 19 = 35$ .

Common mistake

Adding the Total row in again. The totals are not more people: every student is already in one inner cell.

Building the Table From Sentences

The 3 or 4-mark question gives the facts as sentences. 80 students each chose swimming or running. 35 are boys. 18 boys chose swimming. 30 students chose running.

Rule

Draw the table first, with a Total row and a Total column. Put each number in its cell before you work anything out. A fact about everyone, like "30 students chose running", goes in a Total.

Worked example

1
$girls = 80 - 35 = 45$ — everyone is a boy or a girl
2
$boys, run = 35 - 18 = 17$ — the boys add to 35
3
$girls, run = 30 - 17 = 13$ — 30 run in total
4
$girls, swim = 45 - 13 = 32$ — the girls add to 45
5
$swim = 18 + 32 = 50$ — add across the swim row
6
$50 + 30 = 80 ✓$ — the rows give the grand total

Sport	Boys	Girls
Swim	18	32
Run	17	13
Total	35	45

Common mistake

Putting "30 students chose running" in the boys cell or the girls cell. It is about everyone, so it is the running total.

Key idea

Some facts come as a ratio. "60 cards, squares to circles $5 : 1$ " means 6 parts of 10 cards each: 50 squares and 10 circles. Write those in the Total cells.

A Fraction or Percentage of Everyone

The 2-mark question: "What percentage of the students are girls who swim?" Use the table above.

Worked example

1
$girls who swim = 32$ — one cell: a girl and a swimmer
2
$\frac{32}{80}$ — out of all the students
3
$\frac{32}{80} \times 100 = 40%$ — times 100 for a percentage
4
$run: \frac{30}{80} = \frac{3}{8}$ — anyone who runs: the run total

Rule

Top: the cell or total the question describes. Bottom: the group you count out of. Here that is everyone, 80.

Common mistake

Using the swim total, 50, for "girls who swim". That counts the boys too. Girls who swim is one cell, 32.

“Of the Girls” or “Girls Who…”?

The 2-mark question that catches most students. The same cell can sit over a different total.

Girls who run

\frac{13}{80}

out of everyone

Of the girls, run

\frac{13}{45}

out of the girls only

Worked example

1
$girls who run = 13$ — the same cell both times
2
$of the girls: \frac{13}{45}$ — only the 45 girls count
3
$of the boys: \frac{18}{35}$ — what percentage of the boys swim?
4
$\frac{18}{35} \times 100 = 51.4%$ — to 3 significant figures

Common mistake

Dividing by 80 when the question says "of the girls". Those words make the group smaller: use the girls total.

Key idea

Say the question as "out of …". Whatever comes after "of the" is the bottom of the fraction.

Which Method? Read the Question

Find the cell first, then ask: out of whom?

You see	You do
Facts as sentences	draw the table, place each number, then fill
“… girls who swim”	one cell, over the grand total
“of the girls …”	the cell over the girls total
“What percentage …”	the fraction × 100
A table with a few gaps	See Tally Charts and Frequency Tables
“picked at random”	See Combined Events with Two-Way Tables
“given that …”	See Conditional Probability

Key idea

Check the table before you answer: the rows and the columns must both add up to the grand total.

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Real past-paper questions on two-way tables

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