Original synthetic exercises · 1.0.0
Try a problem before opening its answer
Choose an independent attempt, open the hints if needed, then compare your reasoning with the worked answer. All three versions use the same numbers. Completing a repeated example does not demonstrate transfer to a new dataset.
Use paper or any calculator. No account, timer or upload is required. State the unit and denominator in your answer. Educator review: Not obtained.
Download the complete educator package and answer key · Download all twelve structured exercises
1. Combine unequal class sizes
Use group size to combine completion rates.
| Group | Completed | Enrolled |
|---|---|---|
| Small | 18 | 20 |
| Large | 54 | 80 |
Assumptions
- Each learner occurs in exactly one group.
- Completion status is known for all 100 learners.
Independent prompt: What percentage of all enrolled learners completed? State the numerator and denominator.
Show hints for exercise 1
- Combine the counts before dividing.
- The two classes have different numbers of learners; an equal average of their percentages changes the weighting.
Show worked answer for exercise 1
72 percent
- Count all completions: 18 + 54 = 72.
- Count all enrolled learners: 20 + 80 = 100.
- Compute 72 / 100 × 100 = 72%.
The population contains 100 learners, so each learner gets equal weight. The group rates, 90% and 67.5%, should not receive equal group weight.
Common wrong answers
- 78.75: This is (90 + 67.5) / 2: it gives a group of 20 the same weight as a group of 80.
2. Average cost per notebook
Weight unit costs by quantities.
| Batch | Notebooks | Cost per notebook (dollars) |
|---|---|---|
| First | 10 | 2 |
| Second | 30 | 4 |
Assumptions
- All notebooks are interchangeable for this calculation.
- Costs include all charges; quantities are fully received.
Independent prompt: What is the average cost per notebook across both batches?
Show hints for exercise 2
- Compute total dollars spent on each batch.
- Divide total dollars by total notebooks, not by the number of batches.
Show worked answer for exercise 2
3.5 dollars per notebook
- First batch cost: 10 × 2 = 20 dollars; second batch cost: 30 × 4 = 120 dollars.
- Total cost is 140 dollars for 40 notebooks.
- 140 / 40 = 3.50 dollars per notebook.
Each notebook, rather than each purchase event, is the unit being averaged.
Common wrong answers
- 3: Averaging the two listed unit prices ignores that three times as many notebooks cost 4 dollars.
3. Count removable exact duplicates
Distinguish excess rows from every row in a duplicate group.
| Learner ID | Workshop |
|---|---|
| A | Mapping |
| A | Mapping |
| B | Charts |
| C | Tables |
| C | Tables |
| C | Tables |
Assumptions
- A duplicate means identical values in both columns.
- Keep one copy of each distinct row; no empty values occur.
Independent prompt: How many rows are removed when one copy of each distinct row is retained?
Show hints for exercise 3
- Count how many copies to retain, not how many rows belong to a repeated group.
- For a group containing n identical rows, remove n − 1.
Show worked answer for exercise 3
3 rows
- There are six input rows and three distinct rows: A/Mapping, B/Charts and C/Tables.
- The A group removes one excess row; the C group removes two.
- Total removed: 1 + 2 = 3 rows.
Five rows belong to duplicate groups, but retaining one from each group means only three are removable.
Common wrong answers
- 5: This counts all rows in repeated groups, including the copies that must remain.
- 2: This counts repeated groups, not excess rows.
4. Count repeated identifiers
Keep repeated keys distinct from identical rows.
| Record key | State |
|---|---|
| K1 | draft |
| K1 | final |
| K2 | draft |
| K3 | draft |
| K3 | final |
| K3 | archived |
Assumptions
- A repeated key is any key appearing more than once.
- Different states are meaningful; do not remove rows based on the key alone.
Independent prompt: How many distinct keys occur more than once?
Show hints for exercise 4
- Group by the first column only.
- Count groups whose size exceeds one; do not sum their row counts.
Show worked answer for exercise 4
2 keys
- K1 appears twice, K2 once and K3 three times.
- Only K1 and K3 meet the more-than-once condition.
- There are two repeated keys; all six full rows are different.
Repeated identifiers can signal multiple versions, not accidental copies. This question counts groups and does not authorize deletion.
Common wrong answers
- 3: This is the number of excess occurrences if one row per key were retained, a different question.
- 0: There are no identical full rows, but there are repeated keys.
5. Average only observed readings
State the missing-value policy and keep genuine zeros.
| Record | Reading |
|---|---|
| R1 | 0 |
| R2 | 10 |
| R3 | null (not observed) |
| R4 | 5 |
| R5 | null (not observed) |
Assumptions
- null means not observed, never zero.
- Compute the mean of observed readings only.
- No claim is made about the mean of all five underlying true values.
Independent prompt: What is the mean of the observed readings, retaining the real zero?
Show hints for exercise 5
- A real zero is a measured value and belongs in the denominator.
- Write both the included values and the number of included values.
Show worked answer for exercise 5
5 reading units
- Observed readings are 0, 10 and 5; the two null entries are excluded.
- Their total is 15 and their count is three.
- 15 / 3 = 5 reading units.
The observed mean uses three readings. Treating unknown values as zero invents observations; deleting the genuine zero also changes the question.
Common wrong answers
- 3: Dividing 15 by all five records silently treats the missing readings as zero.
- 7.5: Dropping the real zero leaves only 10 and 5 and overstates the observed mean.
6. Report a bound with unknown statuses
Distinguish a lower bound from a completed-data estimate.
| Learner | Status |
|---|---|
| L1 | complete |
| L2 | complete |
| L3 | complete |
| L4 | complete |
| L5 | complete |
| L6 | incomplete |
| L7 | incomplete |
| L8 | null (not observed) |
| L9 | null (not observed) |
| L10 | null (not observed) |
Assumptions
- Every row is a distinct enrolled learner.
- null means completion status unknown.
- Known statuses are correct; unknown learners may be complete or incomplete.
Independent prompt: What is the smallest possible completion percentage among all ten enrolled learners?
Show hints for exercise 6
- Use every enrolled learner in the denominator.
- A lower bound is a possible minimum, not a claim that unknown statuses are actually incomplete.
Show worked answer for exercise 6
50 percent
- Five of ten learners are definitely complete.
- The minimum occurs if none of the three unknown learners is complete.
- 5 / 10 × 100 = 50%. The maximum would be 8 / 10 × 100 = 80%.
Completion lies between 50% and 80% under these assumptions. The observed-status rate of 5/7 does not identify the whole group's rate.
Common wrong answers
- 71.428571: This approximately describes only the seven known statuses, not the minimum among all ten learners.
- 80: This is the upper bound if all three unknown statuses are complete.
7. Separate points from relative growth
Subtract rates when the requested unit is percentage points.
| Period | Completed | Enrolled |
|---|---|---|
| Before | 20 | 100 |
| After | 25 | 100 |
Assumptions
- The periods contain comparable groups for this descriptive calculation.
- No causal effect is inferred.
Independent prompt: How many percentage points did the completion rate increase?
Show hints for exercise 7
- Express both rates as percentages first.
- Use subtraction without dividing by the baseline percentage.
Show worked answer for exercise 7
5 percentage points
- The rates are 20 / 100 = 20% and 25 / 100 = 25%.
- Subtract the percentages: 25 − 20 = 5.
- The increase is 5 percentage points; the relative increase is a different quantity.
A change from 20% to 25% is 5 percentage points. Its 25% relative increase uses another denominator and unit.
Common wrong answers
- 25: This is relative percentage growth, not the percentage-point difference.
8. Find relative growth from a baseline
Use the original count as the denominator of relative change.
| Period | Completed projects |
|---|---|
| Before | 20 |
| After | 25 |
Assumptions
- These are counts across equally long periods.
- The original count is nonzero.
Independent prompt: What is the percentage increase in completed projects relative to the original count?
Show hints for exercise 8
- Subtract to find the increase in projects.
- The baseline is the original period, not the final period.
Show worked answer for exercise 8
25 percent
- The absolute increase is 25 − 20 = 5 projects.
- Divide the increase by the original 20 projects: 5 / 20 = 0.25.
- Multiply by 100 to express 25% relative growth.
Relative growth compares the change with the original amount. Counts are not rates, so an absolute increase of five projects is not five percentage points.
Common wrong answers
- 20: Dividing the increase by the final 25 uses the wrong baseline.
- 5: This is the increase in projects, not its percentage relative to the original count.
9. Interpret a flagged document
Use the requested conditional denominator.
| Actual document state | Flagged | Not flagged |
|---|---|---|
| Has formatting error | 8 | 2 |
| No formatting error | 32 | 58 |
Assumptions
- All 100 documents were checked and the actual state was verified.
- Each document appears in exactly one cell.
- This is a synthetic document example, not a model accuracy claim.
Independent prompt: Among flagged documents, what percentage actually has a formatting error?
Show hints for exercise 9
- The words 'among flagged' determine which column is the denominator.
- Count both correct and incorrect flags.
Show worked answer for exercise 9
20 percent
- There are 8 + 32 = 40 flagged documents.
- Eight of those 40 actually have an error.
- 8 / 40 × 100 = 20%.
The 80% rate of flagging erroneous documents (8/10) answers a different conditional question. Both quantities can be true at once.
Common wrong answers
- 80: This is the percentage of erroneous documents that were flagged, reversing the condition.
- 8: This divides by all 100 documents instead of the 40 flagged documents.
10. Choose the requested middle value
Compute a median without substituting a mean.
| Task | Minutes |
|---|---|
| A | 4 |
| B | 5 |
| C | 100 |
| D | 5 |
| E | 6 |
Assumptions
- All five observations are genuine and equally weighted.
- The question asks for a median, not a prediction of a future task.
Independent prompt: What is the median task duration?
Show hints for exercise 10
- Sort the values numerically.
- Choose the middle position; do not average all values.
Show worked answer for exercise 10
5 minutes
- Order the observations: 4, 5, 5, 6, 100.
- With five observations, the third is the middle value.
- The median is 5 minutes; the mean is 120 / 5 = 24 minutes.
The median is defined by position. The unusually long task affects the mean much more, but it is not automatically an error to delete.
Common wrong answers
- 24: This is the arithmetic mean, not the median.
- 100: This is the middle row of the unsorted display; a median requires ordering the values.
11. Compare with explicit subgroup weights
Separate an aggregate comparison from an equally weighted subgroup comparison.
| Method | Difficulty | Successful | Attempted |
|---|---|---|---|
| A | Easy | 9 | 10 |
| A | Hard | 30 | 100 |
| B | Easy | 80 | 100 |
| B | Hard | 2 | 10 |
Assumptions
- Success means the same outcome in every row.
- Compute a descriptive standardized comparison giving Easy and Hard equal 50% weights for each method.
- These data do not establish a causal benefit or justify a universal population weighting.
Independent prompt: Under equal Easy/Hard weights, how many percentage points higher is method A's success rate than method B's?
Show hints for exercise 11
- Compute one rate per method and difficulty.
- Use the specified 50/50 weights, not the observed numbers of attempts.
Show worked answer for exercise 11
10 percentage points
- Method A subgroup rates: 9/10 = 90% and 30/100 = 30%; their equal-weight average is 60%.
- Method B subgroup rates: 80/100 = 80% and 2/10 = 20%; their equal-weight average is 50%.
- The standardized difference is 60 − 50 = 10 percentage points. Raw aggregate rates would rank B above A because the subgroup mixes differ.
Equal subgroup weights are a chosen comparison policy. The raw aggregates, 39/110 and 82/110, answer a different question; neither automatically estimates a causal effect.
Common wrong answers
- -39.090909: This approximately subtracts raw aggregate rates, ignoring the explicitly requested equal subgroup weights.
- 20: This adds the two ten-point subgroup advantages instead of averaging them.
12. Count a union once
Remove the overlap when combining two memberships.
| Membership | Learners |
|---|---|
| Spreadsheet workshop | 30 |
| Chart workshop | 22 |
| Both workshops | 12 |
Assumptions
- Counts refer to distinct learners within each workshop.
- The twelve learners in both are already included in each of the first two counts.
- No other workshops are part of this question.
Independent prompt: How many distinct learners attended at least one of the two workshops?
Show hints for exercise 12
- A learner attending both contributes one to each workshop count.
- For 'at least one', keep the overlapping learners once.
Show worked answer for exercise 12
40 learners
- Adding the workshop counts gives 30 + 22 = 52 memberships.
- The twelve learners in both were counted twice.
- Subtract them once: 52 − 12 = 40 distinct learners.
The union includes single-workshop and both-workshop learners. Subtracting the intersection twice would count only people who attended exactly one workshop.
Common wrong answers
- 52: This counts memberships rather than distinct learners.
- 28: Subtracting the overlap twice counts exactly one workshop, excluding learners who attended both.
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Original exercise content is CC BY 4.0; the downloadable package states its scope and attribution. Arithmetic checks do not establish teaching outcomes or accessibility in every delivery platform.
Download the standalone practice kit
Make the counting policy explicit
A calculation can be arithmetically correct while answering the wrong question. These twelve original problems ask you to identify the observation, counting rule and denominator before computing a result. Every table is synthetic; none describes a real learner, customer, model or vendor.
The examples cover weighted completion and cost, excess duplicate rows and repeated keys, observed-only averages and completion bounds, percentage points and relative growth, flagged-document precision, median, subgroup weights and overlapping memberships. Use paper or an optional calculator; no account or paid software is needed.
Choose the amount of support
Start with an independent prompt and write your calculation plus one sentence naming the denominator. Open the hints only if you need them, then compare your reasoning with the worked answer and the common wrong answers. The downloadable educator package contains every table, assumption, hint and key.
Worked, partially supported and independent variants use the same numbers. A successful repeat attempt is practice, not evidence that you can solve an unseen problem. For a teaching pilot, ask an educator to review the task and design a changed-number follow-up before claiming transfer.
Preserve what missing and duplicate mean
A real zero is an observation. A null in these tables means not observed, so its treatment depends on the stated question. An observed-only mean excludes missing readings; a completion lower bound keeps all enrolled learners in the denominator and leaves unknown statuses unknown.
Duplicate rows and duplicate keys also answer different questions. Keeping one copy of an exact row does not authorize deleting different records that happen to share a key. Write the policy before copying a formula into a spreadsheet.
Reuse and review boundaries
The original exercise text, synthetic tables and explanatory answer keys are available under CC BY 4.0. Credit Skills Path / Editorial Desk Media, link the material and license, and identify adaptations. This scope excludes existing site software, other publications, unrelated downloads, trademarks and linked third-party material. [1]
Arithmetic is recomputed by automated tests. Educator review has not been obtained, no teaching uses have been observed, and no learning-outcome claim is made. The text-only package includes accessibility adaptation notes; inspect the actual teaching platform and learner needs before relying on them.
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