Two-Way Tables

Read a two-way table, complete it from its totals, find fractions, percentages and ratios of a row, a column or the whole, and build one from a list or from words.

Before you start

A frequency table lists each value or category with its frequency, and the frequencies add up to the total (Tally and Frequency Tables).
A percentage of a group is the part out of that group's total, times 100%: 12 out of 48 is $\frac{12}{48} \times 100\% = 25\%$ .

What a Two-Way Table Shows

A frequency table records one thing about each person or thing: a colour, a number of siblings. A two-way table records two things about each person or thing at once, such as whether the student is a boy or a girl and which CCA group the student is in. One of the two things is listed down the side (the rows), the other along the top (the columns).

Each cell inside the table is the number of people (or things) that have both: the cell in the row "Sports" and the column "Girls" is the number of girls whose CCA is a sport.

The Total column gives the total of each row, and the Total row gives the total of each column. The cell in the bottom right-hand corner is the grand total: the number of people (or things) in the whole table.

Two-way tables

The cells of each row add up to that row's total. The cells of each column add up to that column's total.

The row totals add up to the grand total, and so do the column totals. Both ways must give the same number.

The grand total is the cell in the bottom right-hand corner.

Secondary 1 students by CCA group
CCA group	Boys	Girls	Total
Sports	32	24	56
Uniformed groups	20	26	46
Performing arts	12	30	42
Clubs and societies	16	20	36
Total	80	100	180

Example 1: Reading cells and totals

Every Secondary 1 student of a school is in one CCA group. The two-way table above shows the number of boys and of girls in each group.

(a) How many girls are in performing arts?

(b) How many Secondary 1 students are in uniformed groups?

(d) Which CCA group has the most girls?

(e) How many more girls than boys are in performing arts?

(f) Check that the table is consistent.

Solution

(a) Row "Performing arts", column "Girls": 30.

(b) Row "Uniformed groups", column "Total": 46.

(d) Down the column "Girls", the largest number is 30, in the row "Performing arts".

(e) In the row "Performing arts": $30 - 12 = 18$ .

(f) Each row: $32 + 24 = 56$ , $20 + 26 = 46$ , $12 + 30 = 42$ , $16 + 20 = 36$ .

Each column: boys $32 + 20 + 12 + 16 = 80$ ; girls $24 + 26 + 30 + 20 = 100$ .

Grand total both ways: $80 + 100 = 180$ and $56 + 46 + 42 + 36 = 180$ .

(a) 30 girls (b) 46 students (c) 80 boys (d) Performing arts (e) 18 more girls (f) Every row and column adds up, and both ways give 180.

Fractions, Percentages and Ratios

A fraction or percentage from a two-way table is always a part out of a whole, and the question says which whole. "What percentage of the boys" means out of the boys only, the total of the column "Boys". "What fraction of the students in performing arts" means out of that row's total. "What percentage of all the students" means out of the grand total.

The same cell can give different answers for different wholes, so find the whole first.

Choosing the whole

"Of the boys", "of the girls", "of the students in sports": the whole is that row's or that column's total.

"Of all the students": the whole is the grand total.

Fraction $= \frac{\text{part}}{\text{whole}}$ and percentage $= \frac{\text{part}}{\text{whole}} \times 100\%$ .

To compare two groups of different sizes, compare their percentages (or fractions), not their numbers.

Example 2: Fractions, percentages and ratios

Use the table of Secondary 1 students by CCA group in Example 1.

(a) What percentage of the boys are in sports?

(b) What percentage of the girls are in uniformed groups?

(d) What fraction of all the Secondary 1 students are boys?

(e) Find the ratio of boys to girls in clubs and societies.

Solution

(a) The whole is the boys, 80 in all. 32 of them are in sports: $\frac{32}{80} \times 100\% = 40\%$ .

(b) The whole is the girls, 100 in all. 26 of them are in uniformed groups: $\frac{26}{100} \times 100\% = 26\%$ .

(c) The whole is the students in performing arts, 42 in all. 30 of them are girls: $\frac{30}{42} = \frac{5}{7}$ (divide by the HCF, 6).

(d) The whole is all the students, the grand total 180. 80 of them are boys: $\frac{80}{180} = \frac{4}{9}$ (divide by 20).

(e) Boys : girls in clubs and societies $= 16 : 20 = 4 : 5$ (divide both terms by 4).

(a) 40% (b) 26% (c) $\frac{5}{7}$ (d) $\frac{4}{9}$ (e) 4 : 5

Example 3: "Of the girls" or "of all the students"?

All 200 Secondary 1 students of a school took a swimming test. The table shows who can swim 50 metres.

	Can swim 50 m	Cannot swim 50 m	Total
Boys	96	24	120
Girls	72	8	80
Total	168	32	200

(a) What percentage of the boys can swim 50 m?

(b) What percentage of the girls can swim 50 m?

(d) What fraction of the students who can swim 50 m are girls?

(e) Ravi says, "More boys than girls can swim 50 m, so the boys are better swimmers." Is he right?

Solution

(a) Out of the 120 boys: $\frac{96}{120} \times 100\% = 80\%$ .

(b) Out of the 80 girls: $\frac{72}{80} \times 100\% = 90\%$ .

(d) Out of the 168 who can swim 50 m: $\frac{72}{168} = \frac{3}{7}$ (divide by the HCF, 24).

(e) 96 boys and 72 girls can swim 50 m, but there are more boys (120) than girls (80). Out of their own totals, 80% of the boys and 90% of the girls can swim 50 m.

(a) 80% (b) 90% (c) 84% (d) $\frac{3}{7}$ (e) No: a greater share of the girls (90%) than of the boys (80%) can swim 50 m.

Completing a Two-Way Table

When some cells are missing, the totals give them back. A row or column with only one unknown cell can be finished at once: the unknown is the total minus the other cells. Each cell found may leave another row or column with only one unknown, so work round the table until it is full.

When two cells are linked (one is twice the other, say), write them in one unknown and solve an equation.

Completing a two-way table

Find a row or a column with only one unknown. The unknown $=$ the total $-$ the other cells of that row or column.

Repeat until every cell is found.

Check at the end: every row and every column adds up, and both sets of totals give the grand total.

Example 4: Missing cells found from the totals

The 40 students of Class 1A were asked how they travel to school. The two-way table shows some of the results. Find the numbers $a$ , $b$ , $c$ , $d$ , $e$ and $f$ .

How the students of Class 1A travel to school
	MRT	Bus	Walk	Total
Boys	8	$a$	5	22
Girls	$b$	7	$c$	$d$
Total	15	$e$	$f$	40

Solution

Row "Boys" has only one unknown: $8 + a + 5 = 22$ , so $a = 22 - 13 = 9$ .

Column "Total": $22 + d = 40$ , so $d = 18$ (there are 18 girls).

Column "MRT": $8 + b = 15$ , so $b = 7$ .

Row "Girls" now has only one unknown: $7 + 7 + c = 18$ , so $c = 4$ .

Column "Bus": $e = 9 + 7 = 16$ .

Column "Walk": $f = 5 + 4 = 9$ .

Check the Total row: $15 + 16 + 9 = 40$ , the grand total.

The completed table
	MRT	Bus	Walk	Total
Boys	8	9	5	22
Girls	7	7	4	18
Total	15	16	9	40

$a = 9$ , $b = 7$ , $c = 4$ , $d = 18$ , $e = 16$ and $f = 9$ .

Example 5: Two linked cells

The Library Club has 45 members from Secondary 1 and Secondary 2. The number of Secondary 1 girls is twice the number of Secondary 1 boys.

	Secondary 1	Secondary 2	Total
Boys	$x$	11
Girls	$2x$
Total	24		45

(a) Find the value of $x$ .

(b) Complete the table.

Solution

(a) Column "Secondary 1": $x + 2x = 24$ , so $3x = 24$ and $x = 8$ (divide both sides by 3).

So there are 8 Secondary 1 boys and $2 \times 8 = 16$ Secondary 1 girls.

(b) Row "Total": Secondary 2 has $45 - 24 = 21$ members.

Column "Secondary 2": girls $= 21 - 11 = 10$ .

Row "Boys": $8 + 11 = 19$ . Row "Girls": $16 + 10 = 26$ .

Check: $19 + 26 = 45$ , the grand total.

The completed table
	Secondary 1	Secondary 2	Total
Boys	8	11	19
Girls	16	10	26
Total	24	21	45

(a) $x = 8$ (c) $\frac{5}{13}$ of the girls are in Secondary 2.

Building a Two-Way Table

A two-way table can be built from a list of raw data in which each item records two things, such as "BY" for a boy who answered Yes. Draw the table with a row for each value of one thing and a column for each value of the other, then go through the list once, counting each item into its cell.

It can also be built from a description in words. Draw the empty table first, write each number given into its cell, and then complete the table from its totals.

Example 6: A two-way table from a list

Mrs Goh asked 20 students, "Did you have breakfast before school today?" She wrote B for a boy or G for a girl, then Y for Yes or N for No:

BY, GN, GY, BY, BN, GY, BY, GY, GN, BY, BN, GY, BY, GN, GY, BN, BY, GY, GN, BY

(a) Make a two-way table of the answers.

(b) What percentage of the girls did not have breakfast?

Solution

(a) Rows for Boys and Girls, columns for Yes and No. Count each code: BY 7, BN 3, GY 6, GN 4.

	Yes	No	Total
Boys	7	3	10
Girls	6	4	10
Total	13	7	20

Check: the grand total is 20, the number of students asked.

(b) The whole is the 10 girls, and 4 answered No: $\frac{4}{10} \times 100\% = 40\%$ .

(b) 40% of the girls (c) 65% of all the students

Example 7: A two-way table from words

On Sports Day, each of 120 Secondary 1 students took part in exactly one of two events, the sprint or the long jump. 70 of the students were boys. 45 of the boys ran the sprint, and 30 of the girls did the long jump.

(a) Draw and complete a two-way table.

(b) What fraction of the students who did the long jump were girls?

Solution

(a) Rows for Boys and Girls, columns for Sprint and Long jump. Write in the numbers given: boys 70, boys in the sprint 45, girls in the long jump 30, grand total 120.

Girls: $120 - 70 = 50$ .

Boys in the long jump: $70 - 45 = 25$ .

Girls in the sprint: $50 - 30 = 20$ .

Sprint: $45 + 20 = 65$ . Long jump: $25 + 30 = 55$ . Check: $65 + 55 = 120$ .

	Sprint	Long jump	Total
Boys	45	25	70
Girls	20	30	50
Total	65	55	120

(b) The whole is the 55 long jumpers, and 30 of them were girls: $\frac{30}{55} = \frac{6}{11}$ (divide by 5).

(b) $\frac{6}{11}$ (c) 40%

Common Mistakes

Mixing up rows and columns: "boys in sports" is the cell where the row "Sports" meets the column "Boys".
Taking the wrong whole: a percentage "of the boys" is out of the total for boys, not out of the grand total; "of the students who can swim" is out of that column's total.
Comparing groups of different sizes by their numbers. More boys than girls may be able to swim simply because there are more boys; compare percentages.
Adding a Total cell in with the cells of its row or column, which counts the same people twice.
Not checking: every row and column must add up, and the row totals and the column totals must give the same grand total.

Practice

Try each question before you open its solution.

Practice 1: Books borrowed by level

The school library recorded the books borrowed by Secondary 1 and Secondary 2 students in one week.

	Fiction	Non-fiction	Total
Secondary 1	64	36	100
Secondary 2	45	35	80
Total	109	71	180

(a) How many non-fiction books did Secondary 2 students borrow?

(b) How many books did Secondary 1 students borrow altogether?

(d) What percentage of the books borrowed by Secondary 1 students were fiction?

ShowHide solution

(a) Row "Secondary 2", column "Non-fiction": 35.

(b) Row "Secondary 1", column "Total": 100.

(d) Out of the 100 books of Secondary 1: $\frac{64}{100} \times 100\% = 64\%$ .

(a) 35 (b) 100 (c) 109 (d) 64%

Practice 2: Hot and iced drinks

A drinks stall at a hawker centre sold 120 drinks one afternoon, each hot or iced. Find the numbers $a$ , $b$ , $c$ , $d$ and $e$ , and then the percentage of the iced drinks that were teh.

	Kopi	Teh	Barley	Total
Hot	30	$a$	15	70
Iced	$b$	10	$c$	$d$
Total	50	35	$e$	120

ShowHide solution

Row "Hot": $30 + a + 15 = 70$ , so $a = 70 - 45 = 25$ .

Column "Kopi": $30 + b = 50$ , so $b = 20$ .

Column "Total": $70 + d = 120$ , so $d = 50$ .

Row "Iced": $20 + 10 + c = 50$ , so $c = 20$ .

Column "Barley": $e = 15 + 20 = 35$ .

Check: column "Teh" $25 + 10 = 35$ , and the Total row $50 + 35 + 35 = 120$ .

Iced drinks that were teh: $\frac{10}{50} \times 100\% = 20\%$ .

$a = 25$ , $b = 20$ , $c = 20$ , $d = 50$ , $e = 35$ ; 20% of the iced drinks were teh.

Practice 3: Who owns a pet?

150 students were asked whether they own a pet.

	Own a pet	Do not own a pet	Total
Boys	15	45	60
Girls	27	63	90
Total	42	108	150

(a) What percentage of the boys own a pet?

(b) What percentage of the girls own a pet?

(d) What fraction of the pet owners are girls?

ShowHide solution

(a) Out of the 60 boys: $\frac{15}{60} \times 100\% = 25\%$ .

(b) Out of the 90 girls: $\frac{27}{90} \times 100\% = 30\%$ .

(d) Out of the 42 pet owners: $\frac{27}{42} = \frac{9}{14}$ (divide by 3).

(a) 25% (b) 30% (c) 28% (d) $\frac{9}{14}$

Practice 4: Who wears spectacles?

The two-way table shows the number of students in three Secondary 1 classes who wear spectacles and who do not.

Class	Wear spectacles	Do not wear spectacles	Total
1A	18	22	40
1B	17	17	34
1C	14	26	40
Total	49	65	114

(a) How many students in 1B wear spectacles?

(b) How many of the 114 students do not wear spectacles?

(c) Ali says, "1A has the greatest share of students who wear spectacles, because 18 is the largest of the three class numbers in that column." Is he right? Explain with percentages.

ShowHide solution

(a) Row "1B", column "Wear spectacles": 17.

(b) Row "Total", column "Do not wear spectacles": 65. (Check: $22 + 17 + 26 = 65$ .)

1A: $\frac{18}{40} \times 100\% = 45\%$

1B: $\frac{17}{34} \times 100\% = 50\%$

1C: $\frac{14}{40} \times 100\% = 35\%$

1B has the greatest share, 50%, although 1A has the most students who wear spectacles.

(a) 17 (b) 65 (c) No: 1B has the greatest share (50%, against 45% for 1A and 35% for 1C).

Practice 5: The school choir

The school choir has 50 members from Secondary 1 and Secondary 2, and 35 of them are girls. The numbers of Secondary 1 girls and Secondary 2 girls are in the ratio 2 : 3, so they are written $2x$ and $3x$ .

	Secondary 1	Secondary 2	Total
Boys	6	9	15
Girls	$2x$	$3x$	35
Total			50

(a) Find the value of $x$ .

(b) Complete the table.

ShowHide solution

(a) Row "Girls": $2x + 3x = 35$ , so $5x = 35$ and $x = 7$ (divide both sides by 5).

Secondary 1 girls: $2 \times 7 = 14$ . Secondary 2 girls: $3 \times 7 = 21$ . (Check: $14 + 21 = 35$ .)

(b) Secondary 1: $6 + 14 = 20$ . Secondary 2: $9 + 21 = 30$ . Check: $20 + 30 = 50$ .

	Secondary 1	Secondary 2	Total
Boys	6	9	15
Girls	14	21	35
Total	20	30	50

(a) $x = 7$ (c) 70% of the Secondary 1 members are girls.

Practice 6: Homework in the library or at home

Farah asked 24 students in Secondary 1 and Secondary 2 whether they did their homework yesterday in the school library or at home. She wrote L for the library or H for home, then 1 for Secondary 1 or 2 for Secondary 2:

L1, H2, L2, L1, H1, L2, L1, L2, H1, L2, L1, H2, H1, L2, L1, L2, H1, L1, L2, H2, L1, H1, L2, L2

(a) Make a two-way table.

(b) What percentage of the Secondary 2 students did their homework in the library?

ShowHide solution

(a) Count each code: L1 7, H1 5, L2 9, H2 3.

	Library	Home	Total
Secondary 1	7	5	12
Secondary 2	9	3	12
Total	16	8	24

Check: the grand total is 24, the number of students asked.

(b) Out of the 12 Secondary 2 students: $\frac{9}{12} \times 100\% = 75\%$ .

(b) 75% (c) $\frac{5}{8}$

Practice 7: Bicycles in the Science Club

There are 48 students in the Science Club, and 20 of them are girls. 21 of the boys and 12 of the girls own a bicycle.

(a) Copy and complete the two-way table.

	Own a bicycle	Do not own a bicycle	Total
Boys
Girls
Total

(b) How many students in the club do not own a bicycle?

(d) What fraction of the students who own a bicycle are boys?

ShowHide solution

(a) Number of boys: $48 - 20 = 28$ .

Boys who do not own a bicycle: $28 - 21 = 7$ . Girls who do not: $20 - 12 = 8$ .

Own a bicycle: $21 + 12 = 33$ . Do not: $7 + 8 = 15$ . Check: $33 + 15 = 48$ .

	Own a bicycle	Do not own a bicycle	Total
Boys	21	7	28
Girls	12	8	20
Total	33	15	48

(b) Row "Total", column "Do not own a bicycle": 15.

(d) The whole is the 33 students who own a bicycle, and 21 of them are boys: $\frac{21}{33} = \frac{7}{11}$ (divide by 3).

(b) 15 students (c) 60% (d) $\frac{7}{11}$

Practice 8: By bus or not

There are 200 Secondary 1 students in a school, and 45% of them are girls. 60% of the girls and 50 of the boys take a bus to school.

(a) Draw and complete a two-way table with columns "By bus" and "Not by bus".

(b) What percentage of all the students take a bus to school?

ShowHide solution

(a) Girls: $\frac{45}{100} \times 200 = 90$ . Boys: $200 - 90 = 110$ .

Girls who take a bus: $\frac{60}{100} \times 90 = 54$ . Girls who do not: $90 - 54 = 36$ .

Boys who do not take a bus: $110 - 50 = 60$ .

By bus: $50 + 54 = 104$ . Not by bus: $60 + 36 = 96$ . Check: $104 + 96 = 200$ .

	By bus	Not by bus	Total
Boys	50	60	110
Girls	54	36	90
Total	104	96	200

(b) $\frac{104}{200} \times 100\% = 52\%$

(b) 52% (c) 5 : 3

Practice 9: Who did better?

The table shows the results of a Mathematics test for a group of students.

	Passed	Failed	Total
Boys	30	10	40
Girls	35	15	50
Total	65	25	90

Siti says, "The girls did better than the boys, because more girls passed." Is she right? Explain with percentages.

ShowHide solution

35 girls and 30 boys passed, but there are 50 girls and only 40 boys, so compare out of each group's own total.

Boys: $\frac{30}{40} \times 100\% = 75\%$ passed.

Girls: $\frac{35}{50} \times 100\% = 70\%$ passed.

No: a greater share of the boys passed (75%) than of the girls (70%).

Practice 10: Find the mistake

Mr Tan made a two-way table of the sport chosen by Secondary 1 and Secondary 2 students for a games afternoon. When he copied it out, he wrote one cell (not a total) wrongly.

	Football	Badminton	Swimming	Total
Secondary 1	14	9	7	30
Secondary 2	11	13	5	28
Total	25	21	12	58

Which cell is wrong, and what should it be?

ShowHide solution

Check each row: Secondary 1, $14 + 9 + 7 = 30$ , right. Secondary 2, $11 + 13 + 5 = 29$ , not 28.

Check each column: Football $14 + 11 = 25$ , right. Badminton $9 + 13 = 22$ , not 21. Swimming $7 + 5 = 12$ , right.

The totals agree with each other: $30 + 28 = 58$ and $25 + 21 + 12 = 58$ .

The only cell in both the wrong row and the wrong column is Secondary 2, Badminton. It should be $28 - 11 - 5 = 12$ (and also $21 - 9 = 12$ ).

The cell for Secondary 2, Badminton should be 12, not 13.

Practice 11: Sports and other CCAs

There are 160 Secondary 1 students, and the ratio of boys to girls is 9 : 7. Each student is in a sports CCA or in another CCA. $\frac{2}{3}$ of the boys and $\frac{2}{5}$ of the girls are in a sports CCA.

(a) Draw and complete a two-way table.

(b) What fraction of the students in other CCAs are girls?

ShowHide solution

(a) Boys: $\frac{9}{16} \times 160 = 90$ . Girls: $\frac{7}{16} \times 160 = 70$ .

Boys in a sports CCA: $\frac{2}{3} \times 90 = 60$ , so $90 - 60 = 30$ boys are in other CCAs.

Girls in a sports CCA: $\frac{2}{5} \times 70 = 28$ , so $70 - 28 = 42$ girls are in other CCAs.

Sports: $60 + 28 = 88$ . Other: $30 + 42 = 72$ . Check: $88 + 72 = 160$ .

	Sports CCA	Other CCA	Total
Boys	60	30	90
Girls	28	42	70
Total	88	72	160

(b) Out of the 72 students in other CCAs: $\frac{42}{72} = \frac{7}{12}$ (divide by 6).

(b) $\frac{7}{12}$ (c) 15 : 7

Key Points

A two-way table records two things about each person or thing: one down the side (rows), one along the top (columns). Each cell counts the people or things that have both.
The cells of each row add up to the row total, and the cells of each column to the column total.
The row totals and the column totals both add up to the grand total, in the bottom right-hand corner.
A fraction or percentage is a part out of a whole: "of the boys" is out of the boys' total, "of all the students" out of the grand total.
Compare groups of different sizes by percentages, not by numbers.
A missing cell is the total minus the other cells of a row or column with only one unknown; linked cells are found from an equation.
To build a two-way table from a list, count each item into its cell; from words, write in the numbers given and complete the table from its totals.
Check every completed table: rows, columns and the grand total both ways.