Before removal
Count5 scores
Average72 points
Total5 × 72 = 360
Chapter 3 · Lesson 3.4
Add or Remove Values: Track the Total and the Count
增加或删去数据：同时追踪总数与个数
When a score, measurement, or judge’s mark enters or leaves a set, include or exclude that value in the total and update the number of values. Keep both in view, and the new average becomes manageable.
Change the total. Change the count. Rebuild the average.
By the end of this lesson, I can…
- update both the total and the count when values are added or removed;
- recover a missing, added, or removed value from two averages;
- find how many values were originally present after an average changes;
- recover highest and lowest scores from trimmed averages;
- find a high–low difference without knowing every score.
Average = updated total ÷ updated count
Update both parts of the fraction, even when adding or removing 0 leaves the total unchanged. At least one value must remain to calculate a mean.
Mission 1
Not completed
Two quantities move together
An average depends on a total and a count. Removing a positive value reduces the total and the count. Adding a positive value increases both. Adding or removing 0 changes the count but leaves the total unchanged. In this lesson, given averages are exact.
Total44
Count4
Average11
Core rule: when one value is removed, subtract it from the total and subtract 1 from the count. When one value is added, add it to the total and add 1 to the count.
Mission 2
Not completed
Use a before-and-after ledger
A ledger prevents the most common mistake: changing the total but forgetting that the count also changed.
After one score is removed
Count4 scores
Average70 points
Total4 × 70 = 280
Removed score = old total − new total = 360 − 280 = 80
Mission 3
Not completed
Added or removed value machine
Choose whether one value is added or removed. The machine updates the count first, rebuilds both totals, and finds the added or removed value. The five practice questions below always use eight scores averaging 74, followed by a ninth score that makes the mean 76, even if you change the machine inputs.
Old total592
New count9
New total684
Added value92
Mission 4
Not completed
Recover the original count
Now the removed value is known, but the original number of values is not. We can use how far the removed value was above the old average and how far the remaining average fell.
original problem: Distinct positive integers have an average of 100. One value is 108. After 108 is removed, the remaining average is 99. How many values were there originally?
108
8 above 100
old average 100
new average 99
−1
−1
−1
−1
−1
−1
−1
−1
1
Find the removed value’s excess.
108 − 100 = 8.
108 − 100 = 8.
2
Find the drop in the remaining average.
100 − 99 = 1.
100 − 99 = 1.
3
Spread the lost excess.
An 8-point excess disappearing makes each remaining share fall by 1, so 8 values remain.
An 8-point excess disappearing makes each remaining share fall by 1, so 8 values remain.
4
Restore the removed value.
8 remaining values + 1 removed value = 9 original values.
8 remaining values + 1 removed value = 9 original values.
Remaining count = (removed value − old average) ÷ (old average − new average). This division applies when the two averages differ.
Why “distinct positive integers” is not used yet: those conditions matter for the largest-value part of the worked example, which belongs in Lesson 3.5. The count can be found without them.
Optional algebra route
If the original count is n, then the original total is 100n. After removal, the total is 99(n − 1). Therefore 100n − 108 = 99(n − 1), which gives n = 9.
Mission 5
Not completed
Find the highest and lowest scores
A trimmed average tells us the total that remains after one extreme score is removed. Subtract that remaining total from the original total to recover the removed score.
original problem: Six judges’ scores have a mean of 9.6. Consider two separate removals from the original six scores: remove the highest and the remaining five average 9.4; instead remove the lowest and the remaining five average 9.8. No maximum score of 10 is assumed in this problem.
Highest?
MiddleM₁
MiddleM₂
MiddleM₃
MiddleM₄
Lowest?
All six6 × 9.6 = 57.6original total
Highest removed5 × 9.4 = 47lowest + four middle
Lowest removed5 × 9.8 = 49highest + four middle
Highest = 57.6 − 47 = 10.6 · Lowest = 57.6 − 49 = 8.6
Mission 6
Not completed
Remove both extremes
Start again with the original six scores from Mission 5, then remove both the highest and the lowest. In this original example, the two extremes are equally far from the original average, so their effects balance.
Original set
Total57.6
Count6
Average9.6
Highest and lowest removed
Middle total57.6 − 10.6 − 8.6 = 38.4
Middle count6 − 2 = 4
Middle average38.4 ÷ 4 = 9.6
Do not overgeneralize: removing the highest and lowest does not always keep the average unchanged. It works here because 10.6 is 1 above 9.6 and 8.6 is 1 below 9.6.
Mission 7
Not completed
Find the high–low difference quickly
Sometimes the original average is not given. Two trimmed averages can still reveal the difference between the highest and lowest scores because the middle scores appear in both totals and cancel.
Extra practice: Five judges score an athlete. In two separate cases, start with the same five scores: remove the highest to get a four-score mean of 9.46; instead remove the lowest to get a four-score mean of 9.66. How much greater is the highest than the lowest?
Highest removed
LowestM₁M₂M₃
Total: 4 × 9.46 = 37.84
Lowest removed
HighestM₁M₂M₃
Total: 4 × 9.66 = 38.64
Highest − lowest = 38.64 − 37.84 = 0.80
High–low difference = remaining count × (larger trimmed average − smaller trimmed average)
Mission 8
Not completed
Error detective
Each student below lost track of either the total, the count, or the direction of subtraction.
Error A
Six scores average 80. After one is removed, five average 78. A student says the removed score is 80 − 78 = 2.
Error B
After one value is removed from six values, a student divides the new total by 6.
Error C
To find the highest score, a student subtracts the original total from the total after the highest was removed.
Error D
Six scores lose both the highest and lowest. A student divides the remaining total by 5.
Mission 9
Not completed
Add–remove workshop
Write the requested answer. Build old and new totals before deciding what to add, subtract, or divide.
1Remove one value
Seven values average 18. Remove 24. What is the new average?
Hint
7×18 − 24, then divide by 6.
2Recover the removed score
Six scores average 82. After one is removed, five average 80. Find the removed score.
3Recover the added score
Eight scores average 74. After one is added, nine average 76. Find the added score.
4Remove two values
Ten values average 50. Two removed values total 80. What is the average of the remaining eight?
5Recover a count
A set averages 40. Removing 46 makes the remaining average 39. How many values were there originally?
6Find the highest
Seven judges’ scores average 8.8. After removing the highest, six average 8.5. Find the highest score. No 10-point maximum is assumed.
7Quick spread
Five judges: remove highest → average 9.46; remove lowest → average 9.66. Find highest − lowest.
8Middle-four mean
Use Mission 5’s six-judge data: original total 57.6, highest 10.6, and lowest 8.6. What is the mean after removing both extremes?
Extra practice: Recover the mean before trimming
Six judges give scores. Removing the highest score, 9.8, lowers the average by 0.05. Then the lowest score, 9.42, is removed too. Find the mean of the four remaining scores.
Hint
Removing the highest makes each of five remaining shares fall by 0.05. How far above the original mean was that highest score?
Worked solution — open after trying
The highest score was 5 × 0.05 = 0.25 above the old mean. The old mean was 9.8 − 0.25 = 9.55. After removal it becomes 9.50, so five scores total 47.5. Remove 9.42: (47.5 − 9.42) ÷ 4 = 9.52.
Mission 10
Not completed
Exit ticket
Complete these five questions without reopening the worked examples.
Keep tracking both parts
Review how totals and counts change together.
Certificate of Mastery
Total-and-Count Tracker
This certifies that
can update totals and counts, recover added and removed values, determine original counts, and reason with highest, lowest, and trimmed averages.
Teaching notes
The opening card model, before-and-after ledger, generic change machine, added examples, misconception work, workshop, and objective assessment are instructional scaffolds created to make the lesson self-contained.The distinct-positive-integer maximum problem from the second part of Example 3 is reserved for Lesson 3.5.