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PSLE Mathematics · Primary 5 and 6

Average

The average of a set of numbers is the total shared out equally among all the items. In upper primary you find averages, use the average to find a total, work out a missing value, and see how the average changes when an item is added or removed. One relationship does most of the work: total = average × number of items. Average links closely to speed, because average speed also uses totals. This guide explains each question type with examples, a worked problem and practice.

What the syllabus expects

  • Understand average as the total divided by the number of items
  • Calculate the average of a set of data
  • Find the total given the average and the number of items
  • Find a missing value given the average and the other values
  • Solve problems where items are added to or removed from a set and the average changes

What average means

Average = total ÷ number of items. If four test scores are 72, 85, 90 and 81, the total is 72 + 85 + 90 + 81 = 328, so the average is 328 ÷ 4 = 82. Picture the average as levelling out: if every pupil had scored the same mark, each would have scored 82, and the total would still be 328. The average does not have to be one of the values in the set, and it can be a decimal even when every value is a whole number. It must lie between the smallest and the largest values, which gives you a quick check. A score of 0 still counts as an item, so it adds nothing to the total but still adds 1 to the number of items.

Total = average × number

Turning the formula round gives total = average × number of items, and this is the key to most PSLE average questions. If the average of 5 numbers is 24, their total is 5 × 24 = 120. If four of the numbers are 20, 18, 30 and 25, which add up to 93, the fifth number must be 120 − 93 = 27. You cannot add averages directly, but you can always add totals. So whenever a question gives you an average, multiply it back into a total straight away and work with totals until the last step. Then divide by the correct number of items only if the question asks for an average.

When an item is added or removed

The average mass of 6 pupils is 40 kg, so their total mass is 6 × 40 = 240 kg. A seventh pupil joins and the average becomes 41 kg. The new total is 7 × 41 = 287 kg, so the new pupil has a mass of 287 − 240 = 47 kg. You can check this another way. The new pupil is 47 − 40 = 7 kg above the old average, and sharing those extra 7 kg among all 7 pupils raises the average by exactly 1 kg. Removing an item works the same way: find the old total and the new total, and the difference is the item removed. If every value in a set goes up by the same amount, say 3 marks each, the average also goes up by exactly 3.

Averages of two groups

The average of a combined group is not the average of the two group averages unless the groups are the same size. Class A has 20 pupils with an average score of 70, and Class B has 30 pupils with an average score of 80. The totals are 20 × 70 = 1400 and 30 × 80 = 2400. The combined average is (1400 + 2400) ÷ 50 = 3800 ÷ 50 = 76, not (70 + 80) ÷ 2 = 75. The larger class pulls the combined average towards its own average. The same idea appears in questions about the average number of items sold on weekdays and at weekends, where there are 5 weekdays but only 2 weekend days.

Worked example

The average mass of Amy, Bala and Chitra is 38 kg. Bala is 3 kg heavier than Amy, and Chitra is 3 kg heavier than Bala. When Dev joins them, the average mass of the four children becomes 40 kg. Find (a) Dev's mass (b) Chitra's mass.

  1. Total mass of Amy, Bala and Chitra = 3 × 38 = 114 kg.
  2. Total mass of all four children = 4 × 40 = 160 kg.
  3. Dev's mass = 160 − 114 = 46 kg.
  4. Draw a model: Amy is 1 unit, Bala is 1 unit + 3 kg, Chitra is 1 unit + 3 kg + 3 kg = 1 unit + 6 kg.
  5. 3 units + 9 kg = 114 kg, so 3 units = 105 kg and 1 unit = 35 kg.
  6. Chitra = 35 + 6 = 41 kg.
  7. Check: Amy 35 kg, Bala 38 kg, Chitra 41 kg add up to 114 kg, and 114 + 46 = 160 = 4 × 40.

Answer: (a) Dev's mass is 46 kg (b) Chitra's mass is 41 kg

Common mistakes

  • Averaging two averages when the groups are different sizes. Convert each average into a total, add the totals, then divide by the total number of items.
  • Adding a new value straight onto the old average. Multiply the old average back into a total first, then add the new value.
  • Dividing by the wrong number of items after a change. If one child joins a group of 6, divide the new total by 7, not by 6.
  • Leaving out a value of zero. A pupil who scores 0 still counts as one of the items when you divide.
  • Rounding the average early, then multiplying the rounded value back into a total. Keep exact totals until the final step.

Exam tips

  • Write 'total = average × number' at the top of your working and convert every average into a total before doing anything else.
  • Check that your final average lies between the smallest and largest values in the set. If it does not, you have divided by the wrong number.
  • For problems with differences between people, combine the average with a model, as in the worked example, so the totals and the units are both visible.

Practice questions

1. Easy: Find the average of 15, 22, 18, 27 and 23.

Show answer

21. Working: total = 15 + 22 + 18 + 27 + 23 = 105, and 105 ÷ 5 = 21.

2. Medium: The average of 4 numbers is 36. When one number is removed, the average of the remaining 3 numbers is 33. What number was removed?

Show answer

45. Working: total of 4 numbers = 4 × 36 = 144. Total of 3 numbers = 3 × 33 = 99. The number removed is 144 − 99 = 45.

3. Harder: A bakery sold an average of 45 cakes a day from Monday to Friday, and an average of 66 cakes a day on Saturday and Sunday. What was the average number of cakes sold per day for the whole week?

Show answer

51 cakes. Working: Monday to Friday total = 5 × 45 = 225. Weekend total = 2 × 66 = 132. Week total = 225 + 132 = 357, and 357 ÷ 7 = 51. Note that (45 + 66) ÷ 2 = 55.5 is wrong because the groups have 5 days and 2 days.

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