Average Problems
Use average as a total-value model to recover missing data, combine unequal groups, find required scores, and handle values that enter, leave, or change.
Average Problems
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An average represents a total
The arithmetic mean is not just “add and divide.” It connects three quantities:
average = total ÷ number of valuestotal = average × number of valuesMost UPCAT-style average problems hide the total. Recover the old total, describe what changed, then divide by the correct new count.
Recover a total
Multiply the average by the number of values.
Missing value
Subtract the known values from the required total.
Combined groups
Add group totals, then divide by the combined count.
Value added or removed
Update both the total and the number of values.
Corrected entry
Change the total by corrected value−recorded value.
Average speed
Use total distance÷total time, not usually the mean of two speeds.
Use OLD → CHANGE → NEW
Write three lines:
OLD total = old average × old countNEW total = old total ± changeNEW average = new total ÷ new countFor a required score, compute the target total first and subtract the current total.
Why it works
Averages of averages are not automatically valid. Each group must be weighted by its number of observations.
Five forms you should recognize
Problem: Four quiz scores average 15. Three scores are 12, 14, and 19. Find the fourth.
required total = 4(15) = 60known total = 12 + 14 + 19 = 45missing score = 60−45 = 15Problem: Lia’s first four tests average 82. What fifth score gives her an average of 85?
current total = 4(82) = 328target total = 5(85) = 425required score = 425−328 = 97Problem: A class of 20 averages 78 and a class of 30 averages 84. Find the combined average.
combined total = 20(78) + 30(84) = 4080combined average = 4080÷50 = 81.6Why: The larger class contributes more scores, so 81 is not the answer.
Problem: Six measurements average 24. After one is removed, five average 22. Find the removed value.
original total = 6(24) = 144remaining total = 5(22) = 110removed value = 144−110 = 34Problem: A vehicle travels 120 km at 60 kilometers per hour and another 120 km at 40 kilometers per hour. Find its average speed.
time = 12060 + 12040 = 2 + 3 = 5 hoursaverage speed = 240÷5 = 48 kilometers per hourWhy: Average speed uses total distance divided by total time.
Check before you commit
- Averaging two group averages without checking group sizes
- Using the old count after a value is added or removed
- Subtracting a removed value from the average instead of the total
- Forgetting to correct the total before dividing
- Assuming the required next score equals the desired average
- Taking the simple mean of speeds for equal-distance trips
Do you need the lesson-or just practice?
One original question in each form recommends your next step. It does not yet verify mastery.
Work at the level you need.
Foundations
Build the core procedure with immediate explanations.
Core Practice
Use mixed forms with less scaffolding.
UPCAT-Style Transfer
Apply the competency in unfamiliar representations.
Ready to verify this competency?
A score of 5/5 verifies mastery. An unsuccessful attempt loads a different five-form bank.
Average Problems FAQ
When may I average two averages directly?
When the two groups have equal numbers of observations.
Why can a required score exceed the target average?
It must compensate for earlier scores below the target.
Is weighted mean part of this competency?
Yes when groups or components contribute unequal amounts; multiply each mean or score by its weight.
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