Averages Problems
Question 11 mark
The mean of 6 numbers is 9.
Work out the total of the 6 numbers.
Hint
Mean = total ÷ how many numbers, so work backwards.
Worked solution
The mean of 6 numbers is 9.
Work out the total of the 6 numbers.
- Total = mean \times how many numbers.
- 9\times6=54
- Answer: 54.
Question 21 mark
A group of friends spend a total of £48 on lunch. The mean amount spent by each friend is £6.
How many friends are in the group?
Hint
The total is shared equally to give the mean.
Worked solution
A group of friends spend a total of £48 on lunch. The mean amount spent by each friend is £6.
How many friends are in the group?
- Number of friends = total \div mean.
- 48\div6=8
- Answer: 8 friends.
Question 32 marks
The lengths of three films are
1 hour 40 minutes, 2 hours, 1 hour 50 minutes.
Work out the mean length of the three films. Give your answer in minutes.
Hint
Change each length into minutes before you add.
Worked solution
The lengths of three films are
1 hour 40 minutes, 2 hours, 1 hour 50 minutes.
Work out the mean length of the three films. Give your answer in minutes.
- In minutes: 100,\ 120 and 110.
- Total: 100+120+110=330 minutes.
- 330\div3=110
- Answer: 110 minutes.
Question 42 marks
A set of numbers has a mean of 20 and a range of 8.
The number 5 is added to every number in the set.
Write down the new mean.
1 mark
Write down the new range.
1 mark
Hint
Adding the same amount to every number moves all of them up together.
Worked solution
A set of numbers has a mean of 20 and a range of 8.
The number 5 is added to every number in the set.
Part (a)
Write down the new mean.
- Every number is 5 more, so the total goes up by 5 for each number.
- The mean goes up by 5: 20+5=25.
Part (b)
Write down the new range.
- The largest and smallest numbers both go up by 5.
- The difference between them does not change, so the range is still 8.
Question 52 marks
Sam has taken 5 tests. His mean score is 14.
Sam takes one more test. His mean score for all 6 tests is 15.
What did Sam score in the 6th test?
Hint
Work out the total score before and after the 6th test.
Worked solution
Sam has taken 5 tests. His mean score is 14.
Sam takes one more test. His mean score for all 6 tests is 15.
What did Sam score in the 6th test?
- Total for 5 tests: 5\times14=70.
- Total for 6 tests: 6\times15=90.
- Score in the 6th test: 90-70=20.
- Answer: 20.
Question 61 mark
The mean of a list of numbers is 6.5
Every number in the list is multiplied by 4.
Work out the mean of the new list.
Hint
If every number is 4 times as big, the total is 4 times as big too.
Worked solution
The mean of a list of numbers is 6.5
Every number in the list is multiplied by 4.
Work out the mean of the new list.
- Every number is multiplied by 4, so the total is multiplied by 4.
- How many numbers there are does not change, so the mean is also multiplied by 4.
- 6.5\times4=26
- Answer: 26.
Question 72 marks
Five friends each buy a sandwich. They pay
£3.45, £2.80, £4.15, £3.60, £2.50
Work out the mean amount they pay.
Hint
Add the five amounts, then share the total between 5.
Worked solution
Five friends each buy a sandwich. They pay
£3.45, £2.80, £4.15, £3.60, £2.50
Work out the mean amount they pay.
- Total: 3.45+2.80+4.15+3.60+2.50=16.50
- 16.50\div5=3.30
- Answer: £3.30
Question 81 mark
Here are five numbers.
4, 5, 5, 6, 7
The 7 is changed to 47.
Which of the averages change?
Hint
Work out the mean, the median and the mode before and after the change.
Worked solution
Here are five numbers.
4, 5, 5, 6, 7
The 7 is changed to 47.
Which of the averages change?
- Only the median changes.
- Only the mode changes.
- Only the mean changes.
- The mean, the median and the mode all change.
- Before: mean 27\div5=5.4, median 5, mode 5.
- After: 4,\ 5,\ 5,\ 6,\ 47 has mean 67\div5=13.4, median 5, mode 5.
- The large value 47 changes the total, so only the mean changes.
- Answer: only the mean changes.
Question 92 marks
A hotel cleaner took 4.8 hours to clean 6 rooms.
Work out the mean time taken to clean each room. Give your answer in minutes.
Hint
There are 60 minutes in an hour.
Worked solution
A hotel cleaner took 4.8 hours to clean 6 rooms.
Work out the mean time taken to clean each room. Give your answer in minutes.
- 4.8 hours =4.8\times60=288 minutes.
- 288\div6=48
- Answer: 48 minutes.
Question 102 marks
Here are four numbers.
6, 13, 8, 17
One more number is added to the list. The mean of the five numbers is the same as the mean of the four numbers.
What number was added?
Hint
Work out the mean of the four numbers first.
Worked solution
Here are four numbers.
6, 13, 8, 17
One more number is added to the list. The mean of the five numbers is the same as the mean of the four numbers.
What number was added?
- Mean of the four numbers: (6+13+8+17)\div4=44\div4=11.
- For the mean of five numbers to be 11, the total must be 5\times11=55.
- 55-44=11
- Answer: 11. Adding a number equal to the mean keeps the mean the same.
Question 113 marks
Ava played 3 rounds of a game. She scored 14, 9 and 19 points.
Ben played 4 rounds of the same game. He scored 18, 27 and 24 points in three of the rounds, and x points in the other round.
Ava's mean score : Ben's mean score =2:3
Work out the value of x.
Hint
Find Ava's mean score, then use the ratio to find Ben's mean score.
Worked solution
Ava played 3 rounds of a game. She scored 14, 9 and 19 points.
Ben played 4 rounds of the same game. He scored 18, 27 and 24 points in three of the rounds, and x points in the other round.
Ava's mean score : Ben's mean score =2:3
Work out the value of x.
- Ava's mean: (14+9+19)\div3=42\div3=14.
- 2 parts =14, so 1 part =7 and Ben's mean is 3\times7=21.
- Ben's total: 4\times21=84.
- 18+27+24=69, so x=84-69=15.
- Answer: x=15.
Question 123 marks
A charity collection box was open for 5 days. The mean amount collected per day was £24.
On the first four days, £18, £30, £21 and £28 were collected.
On the fifth day, all the money collected was in 50p coins.
How many 50p coins were collected on the fifth day?
Hint
Work out the total for all 5 days first.
Worked solution
A charity collection box was open for 5 days. The mean amount collected per day was £24.
On the first four days, £18, £30, £21 and £28 were collected.
On the fifth day, all the money collected was in 50p coins.
How many 50p coins were collected on the fifth day?
- Total for 5 days: 5\times24=120 pounds.
- First four days: 18+30+21+28=97 pounds.
- Fifth day: 120-97=23 pounds.
- There are two 50p coins in each pound: 23\times2=46.
- Answer: 46 coins.
Question 134 marks
Here are six numbers, written in order of size.
2, 5, x, 11, 13, y
The median of the six numbers is 9. The mean of the six numbers is 10.
Work out the value of x.
2 marks
Work out the value of y.
2 marks
Hint
With six numbers in order, the median is halfway between the 3rd and 4th numbers.
Worked solution
Here are six numbers, written in order of size.
2, 5, x, 11, 13, y
The median of the six numbers is 9. The mean of the six numbers is 10.
Part (a)
Work out the value of x.
- The median is halfway between the 3rd and 4th numbers, x and 11.
- So x and 11 add up to 2\times9=18.
- x=18-11=7
Part (b)
Work out the value of y.
- The total of the six numbers is 6\times10=60.
- 2+5+7+11+13=38
- y=60-38=22
Question 143 marks
A van carries 4 small parcels and 6 large parcels.
The mean mass of the 4 small parcels is 1.5 kg.
The mean mass of the 6 large parcels is 4 kg.
Work out the mean mass of all 10 parcels.
Hint
Work out the total mass of each kind of parcel. Do not just find the mean of 1.5 and 4.
Worked solution
A van carries 4 small parcels and 6 large parcels.
The mean mass of the 4 small parcels is 1.5 kg.
The mean mass of the 6 large parcels is 4 kg.
Work out the mean mass of all 10 parcels.
- Total mass of the small parcels: 4\times1.5=6 kg.
- Total mass of the large parcels: 6\times4=24 kg.
- Total mass of all 10 parcels: 6+24=30 kg.
- 30\div10=3
- Answer: 3 kg.
Question 152 marks
Two teams play in a bowling league.
Team A has 5 players. Their mean score is 62.
Team B has 4 players. Their mean score is 60.
A new player joins Team B and scores 75.
Which statement is now correct?
Hint
Work out the total score for Team B with the new player.
Worked solution
Two teams play in a bowling league.
Team A has 5 players. Their mean score is 62.
Team B has 4 players. Their mean score is 60.
A new player joins Team B and scores 75.
Which statement is now correct?
- Team B has the higher mean, because its new mean is 67.5
- Team A still has the higher mean, because 62 is more than 60.
- Team B has the higher mean, because its new mean is 63
- The two teams now have the same mean.
- Team B's total before: 4\times60=240.
- With the new player: 240+75=315 for 5 players.
- Team B's new mean: 315\div5=63, which is more than 62.
- Answer: Team B now has the higher mean, 63. (Finding the mean of 60 and 75, 67.5, is wrong: Team B has 5 players, not 2.)
Question 16Challenge4 marks
Three classes took the same test.
Class P has 20 students. Their mean mark was 66.
Class Q has 30 students. Their mean mark was 58.
Class R has 25 students.
The mean mark for all 75 students was 61.
Work out the mean mark for Class R.
Hint
Find the total mark for all 75 students and for Classes P and Q.
Worked solution
Three classes took the same test.
Class P has 20 students. Their mean mark was 66.
Class Q has 30 students. Their mean mark was 58.
Class R has 25 students.
The mean mark for all 75 students was 61.
Work out the mean mark for Class R.
- Total for all 75 students: 75\times61=4575.
- Total for Class P: 20\times66=1320.
- Total for Class Q: 30\times58=1740.
- Total for Class R: 4575-1320-1740=1515.
- 1515\div25=60.6
- Answer: 60.6.
Question 17Challenge4 marks
Ruby ran a 5 km route 8 times. She worked out that her mean time was 24 minutes.
Then she found a mistake. One of her times was written down as 31 minutes, but it should have been 21 minutes.
Work out her correct mean time for the 8 runs.
2 marks
Ruby runs the route a 9th time. Her correct mean time for all 9 runs is 23 minutes. Work out her time for the 9th run.
2 marks
Hint
Work with totals, not means.
Worked solution
Ruby ran a 5 km route 8 times. She worked out that her mean time was 24 minutes.
Then she found a mistake. One of her times was written down as 31 minutes, but it should have been 21 minutes.
Part (a)
Work out her correct mean time for the 8 runs.
- Total she used: 8\times24=192 minutes.
- The correct total is 10 minutes less: 192-10=182 minutes.
- 182\div8=22.75 minutes.
Part (b)
Ruby runs the route a 9th time. Her correct mean time for all 9 runs is 23 minutes. Work out her time for the 9th run.
- Total for 9 runs: 9\times23=207 minutes.
- 207-182=25 minutes.
Question 18Challenge3 marks
Three whole numbers have
a median of 7
a mean of 8
a range of 7
Work out the three numbers.
Hint
The middle number is the median. The mean tells you the total of the three numbers.
Worked solution
Three whole numbers have
a median of 7
a mean of 8
a range of 7
Work out the three numbers.
- The median is the middle number, 7. The total of the three numbers is 3\times8=24, so the smallest and largest add to 24-7=17.
- The range is 7, so the largest is 7 more than the smallest. Subtract this difference from their sum: 17-7=10. This is twice the smaller number, so the smaller number is 10\div2=5.
- The largest number is 5+7=12. The three numbers are 5,\ 7,\ 12. Check: their mean is 24\div3=8 and their range is 12-5=7.
Question 19Challenge3 marks
A swimming club has 10 members. Their mean age is 14 years.
Some new members join the club. Each new member is 17 years old.
The total age of all the members is now 225 years.
How many new members joined?
2 marks
Work out the mean age of all the members now.
1 mark
Hint
Work out the total age of the 10 members first.
Worked solution
A swimming club has 10 members. Their mean age is 14 years.
Some new members join the club. Each new member is 17 years old.
The total age of all the members is now 225 years.
Part (a)
How many new members joined?
- Total age of the 10 members: 10\times14=140 years.
- The new members add 225-140=85 years.
- 85\div17=5, so 5 new members joined.
Part (b)
Work out the mean age of all the members now.
- There are now 10+5=15 members.
- 225\div15=15 years.
Question 20Challenge3 marks
There are 7 boxes of pencils. The mean number of pencils in a box is 12.
Every box has at least 10 pencils in it.
Work out the greatest possible number of pencils in one box.
Hint
To make one box as full as possible, make the other six boxes as small as they can be.
Worked solution
There are 7 boxes of pencils. The mean number of pencils in a box is 12.
Every box has at least 10 pencils in it.
Work out the greatest possible number of pencils in one box.
- Total number of pencils: 7\times12=84.
- The other six boxes have at least 10 pencils each: 6\times10=60.
- The largest box can have at most 84-60=24 pencils.
- Answer: 24.