Explain and justify a solution
Show how the answer follows and justify claims such as ‘only’, ‘minimum’ and ‘maximum’.
Section 26-2.2 · 2026–2027 · 2.2 · Find cases satisfying constraints
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Before you begin
- Read tables and instructions in English
- Use arithmetic, ratios and whole-number constraints
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Before calculating, what should you identify?
What you will learn
- Communicate a reproducible chain of reasoning
- Use exclusion or bounds to justify a solution
A result needs a chain of support
Label each value, operation and inference. An explanation describes how you obtained a result; a justification supplies evidence that it satisfies the claim. To justify a maximum, show an attaining case and why the next or any greater case cannot work.
A printer needs twelve minutes per complete booklet with a fixed eight-minute setup in a one-hour session. Four booklets use 8+4×12=56 minutes; five use 68. Since additional booklets take positive time, four is the maximum.
Use the command precisely
‘State’ can require a concise result, while ‘explain’ asks for relationships or reasons and ‘justify’ asks for support. Even when a yes/no answer is requested, include the decisive check if the question requires working. A table can be a proof when it is complete and its entries are explained.
For certainty in a game, consider every permitted reply, not only the reply you hope the opponent chooses. For uniqueness in a search, show why all other permitted cases fail. A list of examples cannot prove a universal claim unless it exhausts the finite domain.
Sources for this lesson
Pause and explain
What else is needed after showing four booklets fit?
Worked example
Justify the maximum number of booklets in sixty minutes.
Show the worked solution
- Four fit in 56 minutes.
- Five require 68 minutes and fail.
- Any larger count takes even longer.
Answer Four is the maximum. The four-booklet construction attains the limit, and the time requirement excludes five or more, completing the justification.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Guided: Explain why ‘four booklets take 56 minutes’ alone is incomplete as a maximum proof.
Give me a hint
It does not exclude more booklets.
Compare my reasoning
- The statement proves four are feasible.
- It says nothing about five.
- Add the 68-minute calculation and monotonicity.
The statement establishes a feasible output only. A maximum claim also needs the exclusion of five or more: five takes 68 minutes, and further booklets add twelve minutes each.
Look for these in your work
- Distinguishes feasibility
- Adds an upper bound
- Explains larger counts
Independent: A van holds at most 45 boxes. Each complete order contains 8 boxes and orders cannot be split. Justify maximum orders.
Give me a hint
Test adjacent whole counts.
Compare my reasoning
- Five orders need 40 boxes.
- Six need 48, exceeding capacity.
- More than six needs still more space.
The maximum is five complete orders. Five fits within 45 boxes; six requires 48 and violates capacity. This proves both attainability and the bound.
Look for these in your work
- Shows a feasible case
- Excludes the next case
- Uses indivisible orders
Application: A two-digit code has different digits from 1,2,3 and must total 4. Someone gives only 13. Justify the full answer.
Give me a hint
Check every allowed first digit.
Compare my reasoning
- First digit 1 gives 13.
- First digit 2 would repeat 2 and is disallowed.
- First digit 3 gives 31.
The complete answer is 13 and 31. The three possible first digits are exhausted, and the middle case fails the different-digit rule; giving 13 alone misses a valid code.
Look for these in your work
- Completes the search
- Rejects the repeated digit
- Explains completeness
Common mistakes
- Writing an unexplained calculator output
- Using examples as a universal proof
- Proving feasibility but not the claimed maximum
What can you explain now?
What else is needed after showing four booklets fit?
Compare with the explanation
Show that five or more cannot fit
The upper-bound argument distinguishes a maximum from a feasible output.
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Make it stick
What else is needed after showing four booklets fit?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.