Make simple deductions
Find what must follow from combined facts, and distinguish necessary conditions from sufficient ones.
Section 28-2.3 · 2028–2030 · 2.3 · Deduce from simple relations
Original StudyForA scenarios and formative model answers; no Cambridge marks, predicted grades or certified mastery. Alternative reasoned answers may be valid. Full-length unseen papers, a wider mixed-problem bank and calibrated specialist teacher marking remain outstanding.
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
- Deduce unknown quantities from related totals
- Use conditional rules without reversing them
Compare cases that differ in one feature
If two purchases have the same number of notebooks but differ by one pen, subtracting their totals isolates the pen price. Keep unchanged quantities aligned; a difference in two unknown quantities does not isolate either by itself.
Two notebooks and one pen cost 11 credits; two notebooks and three pens cost 17. The difference of 6 pays for two pens, so each pen costs 3. The notebooks together cost 8, so each costs 4.
A condition is not its converse
‘A pass is required for entry’ makes a pass necessary, but entry might also require an available place. ‘Every gold pass guarantees entry’ makes gold sufficient; a standard pass might also admit someone. If P implies Q, P establishes Q and not-Q excludes P. Q alone does not establish P, and not-P alone does not exclude Q.
Separate ‘must’, ‘may’ and ‘cannot’. A deduction is required by the facts; a plausible guess is only compatible with them. Test a claimed deduction by trying to construct a counterexample satisfying all the original rules.
Sources for this lesson
Pause and explain
If all gold passes admit entry, does entry prove a gold pass was used?
Worked example
Find both item prices from the two purchases.
Show the worked solution
- Subtract 11 from 17 to obtain 6 for two pens.
- Each pen costs 3.
- 11−3=8 pays for two notebooks.
Answer A pen costs 3 credits and a notebook costs 4. Substitution gives 2×4+3=11 and 2×4+3×3=17, so both original totals are satisfied.
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: Check the notebook and pen prices in both purchases.
Give me a hint
Substitute the values into each original total.
Compare my reasoning
- Use notebook 4 and pen 3.
- Calculate 2×4+3.
- Calculate 2×4+3×3.
The totals are 11 and 17 credits respectively. Checking both matters: a pair of prices that matches only one purchase is not a solution to the combined information.
Look for these in your work
- Uses both prices
- Checks both totals
- Explains verification
Independent: Three identical badges and one ribbon cost 13 credits; three badges and two ribbons cost 17. Find the prices.
Give me a hint
Subtract the purchases first.
Compare my reasoning
- The extra ribbon costs 17−13 = 4.
- Three badges cost 13−4 = 9.
- Divide 9 by three.
A ribbon costs 4 credits and a badge costs 3. The checks are 3×3+4=13 and 3×3+2×4=17.
Look for these in your work
- Isolates the difference
- Finds the second price
- Checks both equations
Application: Every visitor must register, but registration does not reserve a seat. Mina is registered. Can you deduce she has a seat?
Give me a hint
Distinguish a necessary step from a guarantee.
Compare my reasoning
- Registration is required for visitors.
- It explicitly does not reserve a seat.
- Look for a separate allocation fact.
No. Mina satisfies the registration condition, but the facts do not establish an allocated seat. Registration is necessary for visiting and is not stated to be sufficient for seating.
Look for these in your work
- Uses the conditional rule
- Avoids its converse
- States the missing evidence
Common mistakes
- Reversing an implication
- Treating a plausible result as necessary
- Failing to substitute a deduction back
What can you explain now?
If all gold passes admit entry, does entry prove a gold pass was used?
Compare with the explanation
No, another admission route may exist
The statement supplies a sufficient route, not an exclusive route.
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Make it stick
If all gold passes admit entry, does entry prove a gold pass was used?
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.