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Math · Data Literacy, Two-Step Arithmetic & Evidence
Bar Graph Word Problems: Totals, Differences and Evidence
Use one accurate bar graph to solve two-step word problems, calculate totals and averages, and explain why a chart of book loans cannot prove the number of borrowers.
The idea
The instructional bar chart shows an invented set of books borrowed at a library over five days: Monday 12, Tuesday 18, Wednesday 9, Thursday 15, Friday 21. The numbers are hypothetical and are not a record of a real library. The vertical axis begins at zero and is labelled every 6 books.
A graph question is not always a one-step question. You might first add Monday and Tuesday, then compare that total with Wednesday through Friday. Name each subtotal before subtracting: 12 + 18 = 30, 9 + 15 + 21 = 45, and 45 − 30 = 15.
Read values between labelled gridlines carefully. Wednesday is 9, not 6 or 12; Thursday is 15, between 12 and 18. The scale marks are reference points, not a rule that every value must be a multiple of six.
An accurate total is 12 + 18 + 9 + 15 + 21 = 75 book loans. To check, combine pairs in a different order: (12 + 18) + (9 + 15 + 21) = 30 + 45 = 75.
Be precise about the measured quantity. Seventy-five book loans do not prove that 75 different people borrowed books. Some people may borrow more than one book. A mathematical answer should include the unit and not invent unseen information.
Worked examples
Combine two groups of days
How many more loans were recorded Wednesday–Friday than Monday–Tuesday?
- Read Mon 12, Tue 18, Wed 9, Thu 15, Fri 21 from the graphic.
- First group: 12 + 18 = 30 loans.
- Second group: 9 + 15 + 21 = 45 loans.
- Compare: 45 − 30 = 15 loans.
15 more book loans on Wednesday–Friday
Read a bar between ticks
How many more books were borrowed on Thursday than Wednesday?
- The chart ticks increase by six, but bars can stop between ticks.
- Read Thursday as 15 and Wednesday as 9.
- 15 − 9 = 6; check with 9 + 6 = 15.
6 more book loans
Add all five days
What is the five-day total?
- Read every value: 12, 18, 9, 15, 21.
- Make convenient groups: (12 + 18) + (9 + 15 + 21).
- 30 + 45 = 75. Check that no day was counted twice.
75 book loans
Calculate an average
What was the mean daily number of loans in this example?
- The total is 75 loans across 5 days.
- Divide 75 by 5 to find the equal-share amount.
- This mean is a daily average, not the actual count for every day.
15 loans per day on average
Stay inside the evidence
Does a total of 75 loans prove 75 different people visited?
- Read the chart title carefully: books borrowed.
- The chart does not include visitor identities, visits or unique borrowers.
- One person can borrow multiple books; we cannot infer how many distinct people visited.
No: 75 is a count of loans, not unique people
Common mistakes
These are tempting because they use familiar operations. Check what the question is actually measuring.
Why this fails: Ticks show a scale; a bar between 12 and 18 can represent 15.
Why this fails: The two days are already included. That would double count them.
Why this fails: Readers need to know whether the answer means books, people, days or something else.
Why this fails: The chart only records example counts of books borrowed per day.
Why this fails: This graph explicitly describes illustrative data, not an actual library's records.
Foundation · Read values correctly
Foundation · Read values correctly practice
Use the existing instructional graph: Mon 12, Tue 18, Wed 9, Thu 15, Fri 21 books borrowed.
1. According to the chart, how many books were borrowed on Monday?
2. How many books were borrowed on Wednesday?
3. Which day shows 21 book loans?
4. What does one labelled step of the vertical scale represent?
5. What is the difference between Friday and Wednesday?
6. How many loans happened on Tuesday and Thursday together?
7. Which day had the fewest book loans?
8. How many more books were borrowed on Tuesday than Monday?
Apply · Combine and compare
Apply · Combine and compare practice
Choose whether the question needs addition, subtraction, or both.
9. Find the total books borrowed on all five days.
10. How many books were borrowed Monday through Wednesday?
11. How many books were borrowed Wednesday through Friday?
12. How many more books were borrowed Wed–Fri than Mon–Tue?
13. Find Monday and Friday together, then subtract Wednesday.
14. Tuesday and Thursday total 33. How many fewer is that than all five days?
15. If five identical bags can each hold 15 books, can all 75 recorded loans be counted as five full bags of books?
16. What is the average number of daily loans across the five days?
Think · Reject a misleading interpretation
Think · Reject a misleading interpretation practice
Explain what the graph does and does not establish.
17. A student says the graph proves 75 different people borrowed a book. Is this justified?
18. A student adds all five bars and gets 66. Identify a valid check.
19. A student reads Wednesday as 12 because the nearest labelled tick is 12. What was overlooked?
20. Tuesday's 18 is twice Wednesday's 9. Does that mean twice as many borrowers visited?
21. A student says 21 − 12 = 11 for Friday compared with Monday. Find the error.
22. Can this chart alone tell whether any one book was borrowed on two days?
23. Which day is closest to the five-day average, and by how much?
24. If a class needs a statement supported by the chart, which is better: 'Friday had most loans' or 'Friday had most visitors'?
Challenge · Plan two or three steps
Challenge · Plan two or three steps practice
Build a number sentence and then state units and limits.
25. If Saturday had the same number of loans as Wednesday, what would the six-day total be?
26. If Friday's 21 loans were shared equally between three imaginary displays, how many would each display get?
27. How many extra Tuesday loans would be needed to match Friday?
28. If Thursday's loans increased by 5 in a hypothetical second week, what would Thursday's count be?
29. What fraction of the five-day total happened on Friday? Give the simplest fraction.
30. Suppose Monday and Tuesday were combined into one 30-loan bar. Would the five-day total change?
31. A statement says 'Friday was busier than every other day.' Rewrite it with accurate measurement language.
Answer key
- 1. 12 books
The Monday bar reaches 12 on the vertical scale. - 2. 9 books
The Wednesday bar reaches halfway between the 6 and 12 gridlines. - 3. Friday
Friday is the tallest bar and has the value 21. - 4. 6 books
The chart labels the vertical axis 0, 6, 12, 18, 24. - 5. 12 books
21 − 9 = 12. - 6. 33 books
18 + 15 = 33. - 7. Wednesday
Wednesday's 9 is the smallest of the five counts. - 8. 6 books
18 − 12 = 6. - 9. 75 books
12 + 18 + 9 + 15 + 21 = 75. - 10. 39 books
12 + 18 + 9 = 39. - 11. 45 books
9 + 15 + 21 = 45. - 12. 15 books
Wed–Fri = 45; Mon–Tue = 30; difference = 15. - 13. 24 books
12 + 21 − 9 = 24. - 14. 42 books
75 − 33 = 42. - 15. Yes, numerically 5 × 15 = 75.
This is an arithmetic grouping of the recorded total; the graph does not show actual bags. - 16. 15 books per day
The total 75 divided by the 5 days equals 15. - 17. No
The bars count book loans, not unique people. One person might borrow several books. - 18. Add again to obtain 75
12 + 18 = 30; 9 + 15 + 21 = 45; 30 + 45 = 75. - 19. The bar is between gridlines and represents 9.
A scale interval may contain values between labelled ticks. - 20. No
Only book loans are shown; borrowers or visitors were not measured. - 21. The correct difference is 9 books.
21 − 12 = 9; inverse check: 12 + 9 = 21. - 22. No
The chart has counts by day, not titles or individual borrowing histories. - 23. Thursday, exactly equal.
The average is 15 and Thursday is 15. - 24. Friday had most loans.
The data measure loans; visitor counts are unknown. - 25. 84 loans
The five-day total is 75 and an assumed extra 9 gives 84. - 26. 7 loans per display
21 ÷ 3 = 7, as a hypothetical grouping. - 27. 3 loans
21 − 18 = 3. - 28. 20 loans
15 + 5 = 20. This is hypothetical; the graph shows one set of example counts. - 29. 7/25
Friday is 21 out of 75; divide numerator and denominator by 3. - 30. No, it remains 75.
Combining categories adds their counts but does not add new loans. - 31. Friday recorded more book loans than each of the other four days.
The graph supports comparisons of loans only; it does not measure every form of busyness.