GCSE Maths: Two-way tables and frequency trees (Higher)
Part of the GCSE Maths probability chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Two-way tables and frequency trees
When people (or items) are sorted by two things at once, such as year group and how they travel, a two-way table keeps the information tidy. One category goes down the side and the other goes across the top. Each cell shows how many are in both categories, and the totals are written at the ends of the rows and the bottom of the columns.
| Walk | Bus | Total | |
|---|---|---|---|
| Boys | 12 | 6 | 18 |
| Girls | 9 | 8 | 17 |
| Total | 21 | 14 | 35 |
Completing a table. Every row adds up to its row total and every column adds up to its column total. So look for a row or column with only one number missing and work it out by adding (if the total is missing) or by subtracting the known numbers from the total (if a cell is missing). Then repeat. Reading probabilities. If one person is chosen at random from the whole table, P(event) = number in the event ÷ grand total. For example P(girl) = 17/35 and P(boy who walks) = 12/35.
Frequency trees. A frequency tree looks like a tree diagram but shows numbers of people instead of probabilities. The first box holds the total. Each branch splits the number into groups, and each group splits again. The numbers coming out of a box must add up to the number in the box. If you are given a fraction or a percentage, multiply the number in the box by it. A frequency tree turns probabilities into expected numbers, so it is a friendly way to see what a tree diagram means: the probability of a route is its number divided by the grand total.
Key ideas
- A two-way table has row totals and column totals. Every row and column adds up to its total.
- To complete it, find a row or column with only one missing number.
- P(event) from a table = number in the event ÷ total in the group chosen from.
- A frequency tree shows numbers, not probabilities. Numbers leaving a box add up to the number in it.
- Take a fraction (or percentage) of the number in the previous box, not of the original total.
| Walk | Bus | Total | |
|---|---|---|---|
| Boys | 12 | ? | 18 |
| Girls | ? | 8 | ? |
| Total | 21 | ? | 35 |
Method. In a two-way table every row and every column adds up to its total. Find a row or column with only one missing number, work it out, then repeat.
- Find the missing number in the Boys row, Bus column. The numbers in a row add up to the row total, so subtract the known numbers from the total.18 − 12 = 6
- Find the missing number in the Total row, Bus column. The numbers in a row add up to the row total, so subtract the known numbers from the total.35 − 21 = 14
- Find the missing number in the Girls row, Walk column. The numbers in a column add up to the column total, so subtract the known numbers from the total.21 − 12 = 9
- Find the missing number in the Girls row, Total column. The numbers in a column add up to the column total, so subtract the known numbers from the total.35 − 18 = 17
- Write the completed table. Check that every row and every column adds up to its total.
Walk Bus Total Boys 12 6 18 Girls 9 8 17 Total 21 14 35
Check. Check the grand total: the row totals add to 35 and the column totals add to 35, and both equal 35. ✓
Watch out. Start with a row or column that has only one missing number.
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Read the numbers from the completed table. The bottom of each fraction is the grand total, 35.
- Find the grand total. It is the bottom right number.35 students
- Girls who walk. Where the Girls row meets the Walk column.9, so P(girl who walks) = 935
- Students who take the bus. Use the Bus column total.14, so P(bus) = 1435 = 25
Check. P(walk) + P(bus) = 21/35 + 14/35 = 1. ✓
Watch out. Use the column total, not just one cell, for “the student takes the bus”.
Method. Work out each number on the tree by finding a fraction of the number in the box before it.
- Start with the total. The first box of a frequency tree holds everyone (or everything) in the survey.Students: 80
- Work out the first branches. Multiply the total by the fraction (or use the number given). The numbers on the first branches must add up to the total.Girls: 35 of 80 = 80 ÷ 5 × 3 = 48
Boys: 25 of 80 = 80 ÷ 5 × 2 = 32
Check: 48 + 32 = 80 - Work out the second branches. Each second branch is a fraction of the number on the branch before it, not of the total.Girls, Left: 18 of 48 = 6
Girls, Right: 78 of 48 = 42
Boys, Left: 14 of 32 = 8
Boys, Right: 34 of 32 = 24 - Draw the frequency tree. Write each number in its box. The numbers coming out of a box must add up to the number in it.
Check. The numbers from each box add up to the number in that box. ✓
Watch out. Take the fraction of the number on the previous branch, not of the original total.
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Read the numbers off the frequency tree. The bottom is 80.
- Left-handed girls. Follow the Girls branch then Left.6 students, so P = 680 = 340
- All left-handed students. Add the left-handed girls and the left-handed boys.6 + 8 = 14
- Write the probability. Left-handed students over all students.1480 = 740
Check. P(right-handed) = 3340, and the two add to 1. ✓
Watch out. “Left-handed” includes both the girls and the boys.
Watch out
In a two-way table, take care which total goes on the bottom. For “a girl who walks” use the grand total (35). In a frequency tree, a second-stage fraction is a fraction of the number on its own branch, not of the original total: 18 of the 48 girls is 6, not 18 of 80.Exercise 23K Two-way tables
Non-calculator| Zoo | Museum | Total | |
|---|---|---|---|
| Boys | 14 | 6 | 20 |
| Girls | 9 | 11 | 20 |
| Total | 23 | 17 | 40 |
Method. Find where the Girls row meets the Zoo column.
- Find the Girls row. Rows go across.Girls: 9, 11, 20
- Find the Zoo column. Columns go down.Zoo: 14, 9, 23
- Read where they meet. 9 girls voted for the zoo
Check. 9 + 11 = 20, which is the Girls total. ✓
Watch out. Do not read a total by mistake.
Method. This is the Museum column total.
- Find the Museum column. Columns go down.Museum: 6, 11, 17
- Read the total at the bottom. The bottom number of a column is its total.17 students voted for the museum
Check. 23 + 17 = 40, the grand total. ✓
Watch out. Use the total, because the question asks about all students.
Method. This is the Boys row total.
- Find the Boys row. Rows go across.Boys: 14, 6, 20
- Read the total at the end. The last number of a row is its total.20 boys
Check. 14 + 6 = 20. ✓
Watch out. A row total is the last number in the row.
| Swim | Run | Total | |
|---|---|---|---|
| Adults | ? | 11 | 30 |
| Children | 8 | ? | ? |
| Total | ? | 23 | 50 |
Method. In a two-way table every row and every column adds up to its total. Find a row or column with only one missing number, work it out, then repeat.
- Find the missing number in the Adults row, Swim column. The numbers in a row add up to the row total, so subtract the known numbers from the total.30 − 11 = 19
- Find the missing number in the Total row, Swim column. The numbers in a row add up to the row total, so subtract the known numbers from the total.50 − 23 = 27
- Find the missing number in the Children row, Run column. The numbers in a column add up to the column total, so subtract the known numbers from the total.23 − 11 = 12
- Find the missing number in the Children row, Total column. The numbers in a column add up to the column total, so subtract the known numbers from the total.50 − 30 = 20
- Write the completed table. Check that every row and every column adds up to its total.
Swim Run Total Adults 19 11 30 Children 8 12 20 Total 27 23 50
Check. Check the grand total: the row totals add to 50 and the column totals add to 50, and both equal 50. ✓
Watch out. Start with a row or column that has only one missing number.
| Tea | Coffee | Total | |
|---|---|---|---|
| Men | 16 | ? | 35 |
| Women | ? | 11 | ? |
| Total | ? | ? | 60 |
Method. In a two-way table every row and every column adds up to its total. Find a row or column with only one missing number, work it out, then repeat.
- Find the missing number in the Men row, Coffee column. The numbers in a row add up to the row total, so subtract the known numbers from the total.35 − 16 = 19
- Find the missing number in the Total row, Coffee column. The total at the bottom of a column is the sum of the numbers in the column.19 + 11 = 30
- Find the missing number in the Women row, Total column. The numbers in a column add up to the column total, so subtract the known numbers from the total.60 − 35 = 25
- Find the missing number in the Women row, Tea column. The numbers in a row add up to the row total, so subtract the known numbers from the total.25 − 11 = 14
- Find the missing number in the Total row, Tea column. The numbers in a row add up to the row total, so subtract the known numbers from the total.60 − 30 = 30
- Write the completed table. Check that every row and every column adds up to its total.
Tea Coffee Total Men 16 19 35 Women 14 11 25 Total 30 30 60
Check. Check the grand total: the row totals add to 60 and the column totals add to 60, and both equal 60. ✓
Watch out. Start with a row or column that has only one missing number.
| Blue | White | Total | |
|---|---|---|---|
| Small | 7 | ? | 18 |
| Medium | ? | 15 | 27 |
| Large | 9 | ? | ? |
| Total | ? | ? | 70 |
Method. In a two-way table every row and every column adds up to its total. Find a row or column with only one missing number, work it out, then repeat.
- Find the missing number in the Small row, White column. The numbers in a row add up to the row total, so subtract the known numbers from the total.18 − 7 = 11
- Find the missing number in the Medium row, Blue column. The numbers in a row add up to the row total, so subtract the known numbers from the total.27 − 15 = 12
- Find the missing number in the Total row, Blue column. The total at the bottom of a column is the sum of the numbers in the column.7 + 12 + 9 = 28
- Find the missing number in the Large row, Total column. The numbers in a column add up to the column total, so subtract the known numbers from the total.70 − 18 − 27 = 25
- Find the missing number in the Large row, White column. The numbers in a row add up to the row total, so subtract the known numbers from the total.25 − 9 = 16
- Find the missing number in the Total row, White column. The numbers in a row add up to the row total, so subtract the known numbers from the total.70 − 28 = 42
- Write the completed table. Check that every row and every column adds up to its total.
Blue White Total Small 7 11 18 Medium 12 15 27 Large 9 16 25 Total 28 42 70
Check. Check the grand total: the row totals add to 70 and the column totals add to 70, and both equal 70. ✓
Watch out. Start with a row or column that has only one missing number.
| Saturday | Sunday | Total | |
|---|---|---|---|
| Adult | 15 | 10 | 25 |
| Child | 5 | 20 | 25 |
| Total | 20 | 30 | 50 |
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total the probability is out of. The Adult row total is the number of adult tickets.25 adult tickets
- Find the number we want from the table. Every ticket is equally likely, so the total goes on the bottom.50 tickets
- Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(an adult ticket) = 2550
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 25. The value does not change but the numbers are smaller.25 ÷ 25 = 1 and 50 ÷ 25 = 2
2550 = 12
Check. The other 25 outcomes do not fit the event, so P(the event does not happen) = 12. Then 2550 + 2550 = 5050 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. The bottom of the fraction is the total of the whole group chosen from (usually the grand total). Read the correct row and column.
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total the probability is out of. The Saturday column total is the number of Saturday tickets.20 Saturday tickets
- Find the number we want from the table. Every ticket is equally likely, so the total goes on the bottom.50 tickets
- Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(a Saturday ticket) = 2050
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 10. The value does not change but the numbers are smaller.20 ÷ 10 = 2 and 50 ÷ 10 = 5
2050 = 25
Check. The other 30 outcomes do not fit the event, so P(the event does not happen) = 35. Then 2050 + 3050 = 5050 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. The bottom of the fraction is the total of the whole group chosen from (usually the grand total). Read the correct row and column.
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total the probability is out of. Where the Child row meets the Sunday column.20 child tickets for Sunday
- Find the number we want from the table. Every ticket is equally likely, so the total goes on the bottom.50 tickets
- Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(a child ticket for Sunday) = 2050
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 10. The value does not change but the numbers are smaller.20 ÷ 10 = 2 and 50 ÷ 10 = 5
2050 = 25
Check. The other 30 outcomes do not fit the event, so P(the event does not happen) = 35. Then 2050 + 3050 = 5050 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. The bottom of the fraction is the total of the whole group chosen from (usually the grand total). Read the correct row and column.
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total the probability is out of. Not adult means child, so use the Child row total.25 child tickets
- Find the number we want from the table. Every ticket is equally likely, so the total goes on the bottom.50 tickets
- Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(not an adult ticket) = 2550
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 25. The value does not change but the numbers are smaller.25 ÷ 25 = 1 and 50 ÷ 25 = 2
2550 = 12
Check. The other 25 outcomes do not fit the event, so P(the event does not happen) = 12. Then 2550 + 2550 = 5050 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. The bottom of the fraction is the total of the whole group chosen from (usually the grand total). Read the correct row and column.
| In band | Not in band | Total | |
|---|---|---|---|
| Boys | 8 | 12 | 20 |
| Girls | 6 | 19 | 25 |
| Total | 14 | 31 | 45 |
Method. Compare the fraction of boys in the band with the fraction of girls in the band, not the raw numbers.
- Find the fraction of boys in the band. Boys in the band over all boys.820 = 25 = 0.4
- Find the fraction of girls in the band. Girls in the band over all girls.625 = 0.24
- Compare. 0.4 is bigger than 0.24.Boys are more likely to be in the band.
- Conclude. Phil is right.
Check. Using a common denominator of 100: boys 40/100, girls 24/100. ✓
Watch out. Compare fractions of each group, because the groups are different sizes (20 boys, 25 girls).
| Snack | No snack | Total | |
|---|---|---|---|
| Hot drink | 27 | 18 | 45 |
| Cold drink | 21 | 54 | 75 |
| Total | 48 | 72 | 120 |
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total the probability is out of. Where the Hot drink row meets the Snack column.27 customers
- Find the number we want from the table. The whole group is 120 customers.120 customers
- Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(a hot drink and a snack) = 27120
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 3. The value does not change but the numbers are smaller.27 ÷ 3 = 9 and 120 ÷ 3 = 40
27120 = 940
Check. The other 93 outcomes do not fit the event, so P(the event does not happen) = 3140. Then 27120 + 93120 = 120120 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. The bottom of the fraction is the total of the whole group chosen from (usually the grand total). Read the correct row and column.
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total the probability is out of. Use the Snack column total.48 customers
- Find the number we want from the table. The whole group is 120 customers.120 customers
- Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(a snack) = 48120
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 24. The value does not change but the numbers are smaller.48 ÷ 24 = 2 and 120 ÷ 24 = 5
48120 = 25
Check. The other 72 outcomes do not fit the event, so P(the event does not happen) = 35. Then 48120 + 72120 = 120120 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. The bottom of the fraction is the total of the whole group chosen from (usually the grand total). Read the correct row and column.
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total the probability is out of. Where the Cold drink row meets the No snack column.54 customers
- Find the number we want from the table. The whole group is 120 customers.120 customers
- Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(a cold drink and no snack) = 54120
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 6. The value does not change but the numbers are smaller.54 ÷ 6 = 9 and 120 ÷ 6 = 20
54120 = 920
Check. The other 66 outcomes do not fit the event, so P(the event does not happen) = 1120. Then 54120 + 66120 = 120120 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. The bottom of the fraction is the total of the whole group chosen from (usually the grand total). Read the correct row and column.
| Friday | Saturday | Total | |
|---|---|---|---|
| Adult | 150 | ? | 330 |
| Child | ? | 120 | ? |
| Total | ? | ? | 600 |
Method. Complete the table, find the relative frequency of Child tickets for Saturday, then multiply by 150.
- Find the missing number in the Adult row, Saturday column. The numbers in a row add up to the row total, so subtract the known numbers from the total.330 − 150 = 180
- Find the missing number in the Total row, Saturday column. The total at the bottom of a column is the sum of the numbers in the column.180 + 120 = 300
- Find the missing number in the Child row, Total column. The numbers in a column add up to the column total, so subtract the known numbers from the total.600 − 330 = 270
- Find the missing number in the Child row, Friday column. The numbers in a row add up to the row total, so subtract the known numbers from the total.270 − 120 = 150
- Find the missing number in the Total row, Friday column. The numbers in a row add up to the row total, so subtract the known numbers from the total.600 − 300 = 300
- Find the relative frequency of Child tickets for Saturday. Child, Saturday over all tickets.120600 = 15
- Multiply by 150. Expected number = probability × number of tickets.15 × 150 = 30
Check. Total check: 150 + 180 + 120 + 150 = 600. ✓
Watch out. Use the relative frequency from the whole table as the probability.
Method. The grand total is the sum of all the cell numbers.
- Add all the cells. Every person is in exactly one cell.6 + 9 + 14 + 11 = 40
- Write the probability of the second cell. Second cell over the grand total.940
- Say what to check. It is already in simplest form because 9 and 40 share no factor.940
Check. All four cell probabilities add up: 6 + 9 + 14 + 11 = 40, so 40/40 = 1. ✓
Watch out. Every item belongs in exactly one cell of a two-way table.
| Jab | No jab | Total | |
|---|---|---|---|
| Under 40 | 3 | 22 | 25 |
| 40 and over | 11 | 14 | 25 |
| Total | 14 | 36 | 50 |
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total the probability is out of. Where Under 40 meets Jab.3 patients
- Find the number we want from the table. The grand total.50 patients
- Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(under 40 and had a jab) = 350
Check. The other 47 outcomes do not fit the event, so P(the event does not happen) = 4750. Then 350 + 4750 = 5050 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. The bottom of the fraction is the total of the whole group chosen from (usually the grand total). Read the correct row and column.
Key words. An outcome is one possible result of an experiment. An event is the outcome (or group of outcomes) we are interested in. Equally likely outcomes have the same chance. Probability = number of favourable outcomes ÷ total number of equally likely outcomes, and it is always between 0 and 1.
Method. Probability = favourable outcomes ÷ total outcomes, when all the outcomes are equally likely. Count the total first, then count the outcomes we want.
- Find the total the probability is out of. Use the No jab column total.36 patients
- Find the number we want from the table. The grand total.50 patients
- Write the probability as a fraction. Probability = number of favourable outcomes ÷ total number of equally likely outcomes.P(no jab) = 3650
- Simplify the fraction. Divide the top and the bottom by their highest common factor, 2. The value does not change but the numbers are smaller.36 ÷ 2 = 18 and 50 ÷ 2 = 25
3650 = 1825
Check. The other 14 outcomes do not fit the event, so P(the event does not happen) = 725. Then 3650 + 1450 = 5050 = 1, as the probabilities of an event happening and not happening must add up to 1. ✓
Watch out. The bottom of the fraction is the total of the whole group chosen from (usually the grand total). Read the correct row and column.
Method. In a two-way table every row and every column adds up to its total. Find a row or column with only one missing number, work it out, then repeat.
- Find the missing number in the Row 1 row, Column 2 column. The numbers in a row add up to the row total, so subtract the known numbers from the total.50 − 28 = 22
- Find the missing number in the Row 2 row, Column 1 column. The numbers in a column add up to the column total, so subtract the known numbers from the total.45 − 28 = 17
- Find the missing number in the Row 2 row, Column 2 column. The numbers in a column add up to the column total, so subtract the known numbers from the total.35 − 22 = 13
- Write the completed table. Check that every row and every column adds up to its total.
Column 1 Column 2 Total Row 1 28 22 50 Row 2 17 13 30 Total 45 35 80
Check. Check the grand total: the row totals add to 80 and the column totals add to 80, and both equal 80. ✓
Watch out. Start with a row or column that has only one missing number.
Exercise 23L Frequency trees
Calculator allowedMethod. Build the frequency tree from the fractions, then add the vegetarian branches.
- Work out the adults and the children. Multiply the total 200 by each fraction.Adults: 35 of 200 = 120
Children: 200 − 120 = 80 - Work out the vegetarians in each group. Take the fraction of each group, not of 200.Adults: 14 of 120 = 30
Children: 12 of 80 = 40 - Add the vegetarians. Add the two vegetarian branches.30 + 40 = 70
- Show the tree. Write the numbers in the boxes.
Check. Not vegetarian: 90 + 40 = 130, and 70 + 130 = 200. ✓
Watch out. Take the fraction of the number on that branch.
Method. Find 60% of 150, then 40% of that.
- Find the smartphone owners. 60% of 150: find 10% then multiply by 6.10% of 150 = 15, so 60% = 6 × 15 = 90
- Find the owners who use the app. 40% of 90: find 10% (9) then multiply by 4.4 × 9 = 36
- State the answer. People without a smartphone are not asked or are not counted, so only these 36.36 people
- Draw the tree. Boxes for owners and non-owners and for app users.
Check. 36 ÷ 150 = 0.24 = 24%, and 60% × 40% = 0.6 × 0.4 = 0.24. ✓
Watch out. Take 40% of the 90 owners, not of all 150 people.
Method. Work out the faulty and good items, then the flagged items in each group, and add.
- Faulty items. 3% of 5000.5000 × 0.03 = 150
- Good items. The rest.5000 − 150 = 4850
- Faulty items flagged. 90% of 150.0.9 × 150 = 135
- Good items flagged. 2% of 4850.0.02 × 4850 = 97
- Add the flagged items. Total flagged = faulty flagged + good flagged.135 + 97 = 232
Check. Recheck 2% of 4850: 1% = 48.5, so 2% = 97. ✓
Watch out. The test flags items from both groups, so add both.