Score 77
(272 + 77) ÷ 5 = 69.8
Below the target.
Chapter 3 · Lesson 3.3
Work Backward from an Average
从平均数倒推：寻找缺少的量
Turn an average into a required total, compare it with what is already known, and use the difference to uncover a missing score, amount, month, or digit.
Reveal the total. Remove the known part. The missing part appears.
By the end of this lesson, I can…
- turn a current average into a current total;
- turn a target average into a required total;
- subtract known totals to find a missing value;
- interpret at least as a minimum threshold;
- reconstruct hidden leading digits using place value.
Missing part = required total − known total
The average tells you which total is required.
Mission 2
Not completed
Compare the current total with the target total
The student wants an average of at least 70 after five tests. Test scores here are whole numbers from 0 to 100, and the given averages are exact. The target average describes all five tests, not only the new test.
Tests 1–4 together272known points
Target share70per test
Target count5tests
Required total350points
Fifth score?missing part
0 pointsTarget: 350 points
272 known
78 needed
Minimum fifth score: 78. A score of 78 gives exactly 70 on average; any greater score gives an average above 70.
Hint
Build both totals first: 68×4 and 70×5. The new score is the gap between them.
Mission 3
Not completed
“At least” means a minimum threshold
Move the fifth-score slider. Watch how the final average changes and find the first score that reaches the target.
Known total272
New total350
Five-test average70
0
Target reached exactly
The final average is 70.
Score 78
(272 + 78) ÷ 5 = 70
The first score that reaches it.
Mission 4
Not completed
Use the backward-average machine
Enter a current count, an exact current average, and a target average after one more value. This machine finds the value that gives exactly the target average. It allows any value from 0 to your chosen maximum, including decimals; it does not restrict scores to whole numbers.
Current total272
New count5
Required total350
Missing value78
Possible: 78 is no greater than the allowed maximum of 100.
Feasibility check: Mathematics may produce a missing score greater than 100. That tells us the requested target cannot be reached under a 100-point scoring limit.
Practice case: Six quizzes average 74. What seventh score gives an average of 76?
Hint
Six quizzes contain 6×74 points. Seven quizzes averaging 76 require 7×76 points.
Mission 5
Not completed
Reach a target average over time
For the first five months, a family saved an average of 420 yuan per month. Starting in June, it saves 600 yuan every month. When does the average from January onward first reach at least 500 yuan?
Table method
Check June, July, August, and September in order until the running average reaches 500.
450 → 471.43 → 487.5 → 500
Difference-from-target method
The first five months average 80 yuan below the 500-yuan monthly target, so their combined shortage is 5 × 80 = 400 yuan. Individual months need not each have saved 420 yuan. Each new month at 600 contributes 100 above 500.
400 ÷ 100 = 4 months
Mission 6
Not completed
Use the average to rebuild hidden leading digits
Four whole numbers have damaged leading digits. Each box is one digit; different boxes may contain different digits, and a multi-digit number cannot begin with zero. Their average is 2010. The visible endings and place values are enough to reconstruct every number.
a
b4
c54
d184
Place-value shortcut: The unknown part is not four ordinary digits added together. It is d thousands, c hundreds, b tens, and a ones.
Hint
8040 − (4 + 54 + 184) = 7798. Read that as 7 thousands, 7 hundreds, 9 tens, and 8 ones.
Mission 7
Not completed
Try a new hidden-digit puzzle
This new example follows the same structure: each box is one digit, boxes need not match, and a multi-digit number cannot begin with zero. Three numbers are □, □6, and □37. Their average is 214.
Build the total
3 × 214 = 642
Remove visible endings
642 − (0 + 6 + 37) = 599
Read place values
599 = 500 + 90 + 9
9 + 96 + 537 = 642642 ÷ 3 = 214
Mission 8
Not completed
Error detective
Choose the best diagnosis for each incorrect solution. A strong mathematician can explain not only what is wrong, but which quantity was misunderstood.
A
“The target average is 70, so the fifth score is 70.”
B
“The target total is 70 × 4 = 280.”
C
“Five months average 420 and June is 600, so the six-month average is (420 + 600) ÷ 2 = 510.”
D
“The hidden digits in 7798 are a=7, b=7, c=9, d=8.”
Mission 9
Not completed
Backward-reasoning workshop
For each problem, write only the requested answer. Use a separate sheet for working if needed.
1Quiz target
Six quizzes average 74. What seventh score gives a seven-quiz average of 76?
Hint
7×76 − 6×74.
2Missing day
Eight daily values average 15. The first seven total 101. Find the eighth value.
Hint
Required total: 8×15.
3Raise the average
Three tests average 82. What fourth score gives an average of 85?
4Check feasibility
Four tests average 90. What fifth score would be required for an average of 93?
5Possible or impossible?
If every test is out of 100, can the target in Question 4 be reached?
6Savings target
January through May average 420 yuan saved per month; from June onward, 600 yuan is saved each month. At the end of which month does the mean from January onward first reach at least 500 yuan?
7Place-value remainder
For □, □4, □54, □184 with average 2010, what remains after subtracting the visible endings from the total?
8Largest rebuilt number
For □, □6, □37 with average 214, what is the three-digit number?
Extra practice: When every value is counted three times
Four numbers are given. Leaving out each number in turn gives four groups of three, whose averages are 45, 60, 65, and 70. Find the average of the original four numbers.
Hint
Write ABC, ABD, ACD, BCD. How many times does each letter appear?
Worked solution — open after trying
The group totals sum to 3 × (45 + 60 + 65 + 70) = 720. Each original number appears three times, so the original total is 720 ÷ 3 = 240. Its average is 240 ÷ 4 = 60.
Extra practice: Restore the worked example’s four hidden digits
The numbers □, □9, □99, and □998 have average 2012. Each box is one leading digit. Rebuild the numbers.
Hint
Subtract 9 + 99 + 998 from 4 × 2012, then read the remaining place values.
Worked solution — open after trying
8048 − 1106 = 6942. The hidden thousands, hundreds, tens, and ones are 6, 9, 4, and 2. The original numbers are 2, 49, 999, and 6998. Check: their sum is 8048 and their mean is 2012.
Mission 10
Not completed
Exit ticket
Complete these five questions without opening the hints above.
Keep working backward
Review the current total, target total, and missing difference.
Certificate of Mastery
Backward-Average Detective
This certifies that
can reveal totals from averages, work backward to missing values, interpret minimum targets, test feasibility, and reconstruct hidden digits with place value.
Teaching notes
The interactive threshold slider, backward-average machine, parallel hidden-digit example, feasibility checks, misconception tasks, added practice, and objective assessment are instructional scaffolds created to make the lesson fully self-contained.