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Either paper 3 marks
Every member of a sports club plays either tennis or squash. The two-way table shows information about the members.
Tennis Squash Total Men 14 15 29 Women 18 35 Total 32 32 64 (a)Find the number of women who play squash.
(b)One of the members is chosen at random.
Find the probability that the member is a man who plays tennis.
Show answer
(a) \(17\)
(b) \(\frac{7}{32}\)Along the Women row: \(35 - 18 = 17\).
Down the Squash column: \(32 - 15 = 17\).
Both routes give 17, which is the check that the table adds up.
14 of the 64 members are men who play tennis.
\(P = \frac{ 14 }{ 64 }\)
As a fraction in its simplest form this is \(\frac{7}{32}\).
- (b) M114 men who play tennis out of 64 members
- (b) A1probability \(\frac{7}{32}\)
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Either paper 5 marks
Every student in Year 9 and Year 10 at a school chose either French or Spanish. The two-way table shows some information about their choices.
French Spanish Total Year 9 7 22 Year 10 7 Total 18 40 (a)Complete the two-way table.
(b)One of these students is chosen at random.
Find the probability that the student is in Year 10 and chose French.
Show answer
(a) Year 9 Spanish 15, Year 10 French 11, Year 10 total 18, Spanish total 22
(b) \(\frac{11}{40}\)Work along each row and down each column, using the total that is already there.
Year 9 row: \(22 - 7 = 15\) chose Spanish.
French column: \(18 - 7 = 11\) Year 10 students chose French.
Total column: \(40 - 22 = 18\) students are in Year 10.
Spanish column: \(40 - 18 = 22\) students chose Spanish.
Check the Year 10 row: \(11 + 7 = 18\), which agrees.
From the completed table, 11 of the 40 students are in Year 10 and chose French.
\(P = \frac{ 11 }{ 40 }\)
As a fraction in its simplest form this is \(\frac{11}{40}\).
- (a) M1subtracts a known entry from its total, e.g. \(22 - 7\)
- (a) A1Year 9 Spanish \(15\) and Year 10 French \(11\)
- (a) A1Year 10 total \(18\) and Spanish total \(22\)
- (b) M1Year 10 French \(11\) as the numerator over total students \(40\)
- (b) A1probability \(\frac{11}{40}\)
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Either paper 4 marks
Everyone at a leisure centre one morning did exactly one activity. The two-way table shows what they did.
Swimming Gym Tennis Total Adults 11 12 11 34 Children 10 8 13 31 Total 21 20 24 65 (a)One of these people is chosen at random.
Find the probability that the person went swimming.
(b)Find the probability that the person is a child who played tennis.
Show answer
(a) \(\frac{21}{65}\)
(b) \(\frac{1}{5}\)The swimming column totals \(11 + 10 = 21\), out of 65 people altogether.
\(P(\text{swimming}) = \frac{ 21 }{ 65 }\)
As a fraction in its simplest form this is \(\frac{21}{65}\).
One cell of the table answers this: 13 children played tennis.
\(P = \frac{ 13 }{ 65 }\)
As a fraction in its simplest form this is \(\frac{1}{5}\).
- (a) M1swimming total \(21\) over everyone \(65\)
- (a) A1probability \(\frac{21}{65}\)
- (b) M1child tennis cell \(13\) over everyone \(65\)
- (b) A1probability \(\frac{1}{5}\)
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Either paper 3 marksTomas sells bicycles. The table shows the number of bicycles Tomas sold in each quarter of three years.
Quarter 1 Quarter 2 Quarter 3 Quarter 4 2015 14 59 24 54 2016 13 22 59 54 2017 80 22 26 20 (a)In which year did Tomas sell the most bicycles?
You must show your working.
(b)Tomas adds up the sales for each quarter over all three years, and uses these totals to decide how many bicycles to stock.
At the start of which quarter should Tomas stock the most bicycles?
Circle your answer.
Quarter 1 Quarter 2 Quarter 3 Quarter 4
Show answer
(a) 2015: 2015 sold 151, 2016 sold 148, 2017 sold 148
(b) Quarter 4: its three years total 128, more than the other quartersAdd up the four quarters in each year.
2015: \(14 + 59 + 24 + 54 = 151\)
2016: \(13 + 22 + 59 + 54 = 148\)
2017: \(80 + 22 + 26 + 20 = 148\)
The largest total is \(151\), so the year is 2015.
The largest single number in the table, \(80\), is only one quarter of 2017, so it does not decide the year.
Add the three years in each quarter.
Quarter 1: \(14 + 13 + 80 = 107\)
Quarter 2: \(59 + 22 + 22 = 103\)
Quarter 3: \(24 + 59 + 26 = 109\)
Quarter 4: \(54 + 54 + 20 = 128\)
The largest total is \(128\), in Quarter 4.
The largest single number, \(80\), is in Quarter 1, but that quarter's total is smaller.
- (a) M1the three yearly totals 151, 148 and 148
- (a) A12015, the largest total 151
- (b) B1Quarter 4, whose three years total 128, the largest column total
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Either paper 4 marks
Each student in Year 10 and Year 11 chose either history or geography. The two-way table shows their choices.
History Geography Total Year 10 18 17 35 Year 11 15 13 28 Total 33 30 63 (a)One of the students who chose history is chosen at random.
Find the probability that this student is in Year 10.
(b)One of the Year 11 students is chosen at random.
Find the probability that this student chose geography.
Show answer
(a) \(\frac{6}{11}\)
(b) \(\frac{13}{28}\)Only the history column matters here, so the total to divide by is 33, not 63.
18 of those 33 students are in Year 10.
\(P = \frac{ 18 }{ 33 }\)
As a fraction in its simplest form this is \(\frac{6}{11}\).
This time only the Year 11 row matters, so the total to divide by is 28.
13 of those 28 students chose geography.
\(P = \frac{ 13 }{ 28 }\)
As a fraction in its simplest form this is \(\frac{13}{28}\).
- (a) M1denominator is the history total \(33\), not the grand total
- (a) A1probability \(\frac{6}{11}\)
- (b) M1denominator is the Year 11 total \(28\), not the grand total
- (b) A1probability \(\frac{13}{28}\)
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Either paper 4 marks
A group of Year 10 and Year 11 students chose an activity.
There were \(75\) students altogether.
\(32\) of them were Year 10 students.
\(40\) of them chose rowing.
\(17\) of the Year 10 students chose rowing.
(a)Work out how many of the Year 11 students chose climbing.
(b)One of the students is chosen at random.
Find the probability that this student chose rowing.
Give your answer as a fraction in its simplest form.
Show answer
(a) \(20\)
(b) \(\frac{8}{15}\)A two-way table is the quickest way to hold four facts at once. Fill in what you are told, then fill the gaps by subtracting along the rows and down the columns.
There were \(75 - 32 = 43\) Year 11 students altogether.
Of the Year 11 students, \(40 - 17 = 23\) chose rowing.
So the number of Year 11 students who chose climbing is
\(43 - 23 = 20\)
\(40\) of the \(75\) students chose rowing.
\(P = \frac{ 40 }{ 75 } = \frac{8}{15}\)
- (a) M1subtracts to find a missing value, e.g. \(43\) in the second row or \(23\) in the first column
- (a) A1answer \(20\)
- (b) M1fraction \(\frac{ 40 }{ 75 }\)
- (b) A1simplified to \(\frac{8}{15}\)
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Either paper 4 marks
108 people were each asked to choose their favourite type of book.
Each of them chose romance, adventure, horror or thriller.
48 of these people were children. The rest were adults.
14 of the 24 people who chose thriller were adults.
16 of the 27 people who chose romance were children.
10 children chose horror.
The number of children who chose adventure was the same as the number of adults who chose horror.
Work out the number of adults who chose adventure.
You must show how you get your answer.
Show answer
\(23\)Start by finding the number of adults: \(108 - 48 = 60\). So there were 48 children and 60 adults.
Use the column totals you are given. In the thriller column, \(24 - 14 = 10\) were children. In the romance column, \(27 - 16 = 11\) were adults.
The children row has just one cell left to find. Of the 48 children, 10 chose thriller, 16 chose romance and 10 chose horror, so
\(48 - 10 - 16 - 10 = 12\)
So 12 children chose adventure. The number of adults who chose horror is the same, so 12 adults chose horror.
Now the adults row. Of the 60 adults, 14 chose thriller, 11 chose romance and 12 chose horror, so
\(60 - 14 - 11 - 12 = 23\)
So 23 adults chose adventure.
- M1the other row total, \(108 - 48 = 60\)
- M1the missing cells in the thriller and romance columns, \(10\) children and \(11\) adults
- M1\(12\) children who chose adventure, so \(12\) adults who chose horror
- A1\(60 - 14 - 11 - 12 = 23\) adults who chose adventure
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Either paper 3 marks
Every member of a sports club plays either tennis or squash. The two-way table shows information about the members.
Tennis Squash Total Men 20 7 27 Women 15 28 Total 35 20 55 (a)Calculate the number of women who play squash.
(b)One of the members is chosen at random.
Calculate the probability that the member is a man who plays tennis.
Show answer
(a) \(13\)
(b) \(\frac{4}{11}\)Along the Women row: \(28 - 15 = 13\).
Down the Squash column: \(20 - 7 = 13\).
Both routes give 13, which is the check that the table adds up.
20 of the 55 members are men who play tennis.
\(P = \frac{ 20 }{ 55 }\)
As a fraction in its simplest form this is \(\frac{4}{11}\).
- (b) M120 men who play tennis out of 55 members
- (b) A1probability \(\frac{4}{11}\)
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Either paper 5 marks
Every student in Year 9 and Year 10 at a school chose either French or Spanish. The two-way table shows some information about their choices.
French Spanish Total Year 9 10 25 Year 10 12 Total 19 46 (a)Complete the two-way table.
(b)One of these students is chosen at random.
Find the probability that the student is in Year 10 and chose French.
Show answer
(a) Year 9 Spanish 15, Year 10 French 9, Year 10 total 21, Spanish total 27
(b) \(\frac{9}{46}\)Work along each row and down each column, using the total that is already there.
Year 9 row: \(25 - 10 = 15\) chose Spanish.
French column: \(19 - 10 = 9\) Year 10 students chose French.
Total column: \(46 - 25 = 21\) students are in Year 10.
Spanish column: \(46 - 19 = 27\) students chose Spanish.
Check the Year 10 row: \(9 + 12 = 21\), which agrees.
From the completed table, 9 of the 46 students are in Year 10 and chose French.
\(P = \frac{ 9 }{ 46 }\)
As a fraction in its simplest form this is \(\frac{9}{46}\).
- (a) M1subtracts a known entry from its total, e.g. \(25 - 10\)
- (a) A1Year 9 Spanish \(15\) and Year 10 French \(9\)
- (a) A1Year 10 total \(21\) and Spanish total \(27\)
- (b) M1Year 10 French \(9\) as the numerator over total students \(46\)
- (b) A1probability \(\frac{9}{46}\)
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Either paper 4 marks
Everyone at a leisure centre one morning did exactly one activity. The two-way table shows what they did.
Swimming Gym Tennis Total Adults 12 8 12 32 Children 12 7 7 26 Total 24 15 19 58 (a)One of these people is chosen at random.
Work out the probability that the person went swimming.
(b)Work out the probability that the person is a child who played tennis.
Show answer
(a) \(\frac{12}{29}\)
(b) \(\frac{7}{58}\)The swimming column totals \(12 + 12 = 24\), out of 58 people altogether.
\(P(\text{swimming}) = \frac{ 24 }{ 58 }\)
As a fraction in its simplest form this is \(\frac{12}{29}\).
One cell of the table answers this: 7 children played tennis.
\(P = \frac{ 7 }{ 58 }\)
As a fraction in its simplest form this is \(\frac{7}{58}\).
- (a) M1swimming total \(24\) over everyone \(58\)
- (a) A1probability \(\frac{12}{29}\)
- (b) M1child tennis cell \(7\) over everyone \(58\)
- (b) A1probability \(\frac{7}{58}\)