PSLE Mathematics Learning Guide · Guide 52
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An average changes only because the total value, the number of items, or both have changed.
That gives a reliable way to solve replacement and before-and-after average problems:
rebuild the total first, update the total and count according to what changed, then calculate the new average.
Guide 4 owns the basic relationship average = total ÷ number of items. This page takes the next narrower job: what happens when an item is added, removed or replaced, or when the average itself changes and you must work backwards to the missing value.
The MOE Primary Mathematics syllabus updated October 2025 provides curriculum context. All examples below are original eduKate teaching material.
Average is a compressed statement about total and count
Five scores have average 18.
Total = 5 × 18 = 90.
The average does not tell you the five individual scores, but it tells you their combined total exactly.
If one score changes while the number of scores remains five, the new average depends entirely on how much the total changed.
Replacing one item keeps the count fixed
Five scores average 18, so total = 90.
A score of 14 is replaced by 24.
The total increases by 24 − 14 = 10.
New total = 100.
Count remains 5.
New average = 100 ÷ 5 = 20.
The replacement changed the total but not the number of items.
A replacement can be solved through total change
If one item rises by 10 and there are 5 items, the average rises by 10 ÷ 5 = 2.
Original average 18 → new average 20.
This shortcut works because the count stays fixed. It should be understood through the total, not memorised without conditions.
Find a missing replacement value from the average change
Six numbers average 25. One number 19 is replaced, and the new average becomes 28.
Original total = 6×25 = 150.
New total = 6×28 = 168.
Total increased by 18.
The replacement value is 19 + 18 = 37.
Check: 150 − 19 + 37 = 168.
Adding an item changes both total and count
Four numbers average 15.
Original total = 60.
A fifth number, 25, is added.
New total = 85.
New count = 5.
New average = 85 ÷ 5 = 17.
You cannot use the replacement shortcut because the number of items changed.
Find the added item from the new average
Five numbers average 16. A sixth number is added and the average becomes 18.
Original total = 5×16 = 80.
New total = 6×18 = 108.
Added number = 108 − 80 = 28.
Removing an item changes total and count
Six scores average 20.
Total = 120.
A score of 10 is removed.
New total = 110.
New count = 5.
New average = 110 ÷ 5 = 22.
Removing a value below the original average causes the average to rise in this example.
Find the removed item from two averages
Eight numbers average 14. One number is removed and the remaining seven average 15.
Original total = 8×14 = 112.
Remaining total = 7×15 = 105.
Removed number = 112 − 105 = 7.
Use the original average as a direction benchmark
If an added value is greater than the current average, the new average rises.
If it is less than the current average, the new average falls.
If it equals the current average, the average stays unchanged.
The same logic works in reverse for removal:
- remove a value above average → average tends to fall;
- remove a value below average → average tends to rise;
- remove a value equal to average → average stays the same.
This is a reasonableness check before exact calculation.
An added value equal to the current average leaves the average unchanged
Seven values average 12. Add another value 12.
Original total = 84.
New total = 96.
New count = 8.
New average = 12.
The added value fits exactly at the existing balance point.
A change in average corresponds to a total change
Ten items average 24. The average rises to 27 while the count remains ten.
Original total = 240.
New total = 270.
Total increase = 30.
Equivalently, average rose by 3 across 10 items, so total rose by 3×10 = 30.
Several replacements can be combined as net total change
Five values average 20. A value 12 is replaced by 22, and a value 30 is replaced by 25.
First replacement changes total by +10.
Second replacement changes total by −5.
Net change = +5.
Original total = 100.
New total = 105.
New average = 21.
Combining two groups requires totals, not averaging the averages blindly
Group A has 4 scores averaging 18. Total A = 72.
Group B has 6 scores averaging 22. Total B = 132.
Combined total = 204.
Combined count = 10.
Combined average = 20.4.
(18 + 22) ÷ 2 = 20 would be wrong because the groups are different sizes.
Averaging two averages works only when group sizes are equal
Two groups each contain 5 items. Their averages are 16 and 24.
Combined average = (16 + 24) ÷ 2 = 20.
This works because both averages carry equal weight: each represents the same number of items.
If the group sizes differ, rebuild the totals.
Find a group’s average from a combined average
Group A has 4 items averaging 15. Group B has 6 items. Together all 10 items average 21.
Total A = 4×15 = 60.
Combined total = 10×21 = 210.
Total B = 150.
Average B = 150 ÷ 6 = 25.
Track both count and total explicitly in multi-stage problems
Five scores average 18.
Add 30, then remove 12.
Initial total = 90, count = 5.
After adding 30: total 120, count 6.
After removing 12: total 108, count 5.
Final average = 21.6.
Writing the state after each change prevents the wrong count from carrying into the next step.
Use a before-and-after state table
| State | Total | Count | Average |
|---|---|---|---|
| before | 90 | 5 | 18 |
| add 30 | 120 | 6 | 20 |
| remove 12 | 108 | 5 | 21.6 |
The table separates the three quantities so one does not silently change without being updated.
Main worked workshop: sixteen original changing-average tasks
1.
Five values average 12. Find total.
Answer: 60.
2.
Five values average 12. One value rises by 15. New average?
Answer: total rises 15; average rises 3 to 15.
3.
Eight values average 20. One 14 is replaced by 30.
Answer: total rises 16; average rises 2 to 22.
4.
Six values average 18. One value 10 is replaced and new average is 21. Find replacement.
Reasoning: totals 108→126, rise18; replacement=28.
5.
Four values average 17. Add 27. New average?
Answer: total68+27=95; 95÷5=19.
6.
Seven values average 16. Add one value and new average 18. Added value?
Answer: original112; new144; added32.
7.
Six values average 25. Remove 10. New average?
Answer: total150−10=140; 140÷5=28.
8.
Nine values average 14. Remove one and remaining eight average 15. Removed value?
Answer: 126−120=6.
9.
Ten values average 30. Average rises to 32 with count unchanged. Total increase?
Answer: 20.
10.
Five values average 20. Replace 12 by 27.
Answer: total rises15; new average23.
11.
Group A: 3 items average 10. Group B: 7 items average 20. Combined average?
Answer: (30+140)÷10=17.
12.
Two groups each have 6 items, averages 14 and 22. Combined average?
Answer: 18.
13.
A learner averages group averages 12 and 20 to get 16, but groups contain 2 and 8 items.
Repair: totals 24 and160; combined average184÷10=18.4.
14.
A learner removes a low value but keeps the original count when finding new average.
Repair: removal changes both total and count.
15.
Five values average 18. Add 30 and remove 10.
Answer: final total=110, count5, average22.
16.
Four values average 21. A fifth value equal to 21 is added.
Answer: average remains 21.
Replacement has a useful invariant
When exactly one item is replaced by another:
count stays constant.
Therefore:
change in total = change in average × count.
This is the fastest route for many replacement questions.
But do not use it when an item is added or removed, because the count changes.
A new average can reveal the added value without knowing individual originals
You do not need the original list of values. Average and count already determine the original total.
This is an important structural idea: the total carries all the information needed for average changes, even when individual values are unknown.
Why “average of averages” is dangerous
An average gives each underlying item equal weight. If one group has twice as many items, its average should contribute twice as much weight to a combined average.
Rebuilding group totals automatically handles this weighting correctly.
Only when group counts are equal is the simple mean of the group averages valid.
Use direction before arithmetic
If you add a score of 40 to a group averaging 18, the new average must rise but remain below 40.
If you remove a score of 5 from a group averaging 18, the average of what remains must rise.
If a replacement changes 30 to 20, the average must fall.
These direction checks can expose a sign error or wrong operation immediately.
Common changing-average errors
Error 1: work directly with averages without rebuilding totals.
Error 2: forget that adding/removing changes the count.
Error 3: treat replacement like addition and increase the count.
Error 4: average two group averages despite unequal group sizes.
Error 5: find a correct intermediate total but divide by the wrong number of items.
Independent transfer check
- Seven numbers average 24. Find total.
- Seven numbers average 24. One number increases by 14. Find new average.
- Five numbers average 16. One 12 is replaced and new average becomes 20. Find replacement.
- Six numbers average 18. Add 32. Find new average.
- Eight numbers average 15. Add one number and new average becomes 17. Find added number.
- Nine numbers average 22. Remove one number and the remaining eight average 24. Find removed number.
- Group A has 4 items averaging 12; Group B has 6 items averaging 18. Find combined average.
- Explain why replacing one item keeps the count fixed but adding an item does not.
Independent-check answers
1. 168.
2. total182; average26.
3. original80; new100; increase20; replacement32.
4. total108+32=140; count7; average20.
5. original120; new153; added33.
6. original198; remaining192; removed6.
7. totals48+108=156; 156÷10=15.6.
8. Replacement removes one item and inserts one item, so the number of items remains unchanged; addition introduces an extra item.
Parent and tutor guide
Do not begin with a shortcut. Start every early problem by writing the three quantities: total, count, average.
For replacement questions, ask what stays fixed. For addition/removal, ask which two quantities change.
Use direction predictions before calculation: “Should the average rise or fall?” Then compare the exact answer with that prediction.
For combined groups, deliberately use unequal group sizes so the child sees why averaging the two averages can fail.
The learner’s final card
What is the total before the change? How many items are there? Was something added, removed or replaced? Which quantities changed? What should happen to the average directionally? Can the new total and count reproduce the final average?
Complete Batch 13
Use Guide 49: Mixed Numbers and Improper Fractions, Guide 50: Parallelogram and Trapezium Area, and Guide 51: Remainders in Context.
Return to the PSLE Learning Guide Mathematics route.
Sources and boundaries
MOE Primary Mathematics syllabus, updated October 2025; SEAB 2026 PSLE formats.
All values and worked solutions are original eduKate teaching material. This guide deepens the average-total-count relationship without replacing the established Guide 4 owner.