Supposition — the assumption method — replaces guess-and-check with something that finishes in three lines and never depends on luck. It works whenever there are two types of something, you know how many in total, and you know a total value.
The method
- Suppose every item is the same type. Work out what the total would be.
- Compare that with the real total. Find the gap.
- Divide the gap by the difference one swap makes.
Step 3 is where marks are won or lost, so it’s worth being precise about what “one swap” costs.
Worked example
A quiz has 20 questions. Each correct answer scores 3 marks, and 1 mark is deducted for each wrong answer. Jun answered every question and scored 44 marks. How many did he get correct?
Step 1 — suppose all 20 were correct.
20 × 3 = 60 marks
Step 2 — compare with the real score.
60 − 44 = 16 marks too many
Step 3 — what does one wrong answer cost?
This is the step people get wrong. Changing a correct answer to a wrong one does two things: you lose the 3 marks you would have gained, and you lose 1 more as a penalty.
3 + 1 = 4 marks per wrong answer
So:
Number wrong = 16 ÷ 4 = 4
Number correct = 20 − 4 = 16
Jun got 16 correct.
Check: 16 × 3 = 48, minus 4 for the wrong ones = 44 ✓
The classic without penalties
Not every question has a deduction, and then one swap costs only the difference between the two values.
A farmer has chickens and goats — 30 animals with 84 legs in total. How many goats?
Suppose all 30 are chickens: 30 × 2 = 60 legs. Gap: 84 − 60 = 24 legs. One goat instead of one chicken adds 4 − 2 = 2 legs. Goats = 24 ÷ 2 = 12.
Check: 12 goats (48 legs) + 18 chickens (36 legs) = 84 ✓
Common mistakes
- Forgetting the penalty in the swap. Using 3 instead of 3 + 1 above gives 16 ÷ 3, which isn’t even a whole number — a useful warning sign.
- Answering the wrong group. You calculated the number wrong; the question asked how many were correct.
- Assuming everyone answered everything. If questions can be left blank there are three categories, and plain supposition won’t work.
- Not checking. Substituting back takes fifteen seconds and catches almost every slip.
Quick check
A test has 25 questions. 4 marks are given for each correct answer and 2 marks deducted for each wrong one. Priya attempted all of them and scored 64. How many did she get wrong?
Answer
Suppose all 25 correct = 25 × 4 = 100
Actual score = 64
Gap = 100 − 64 = 36
One wrong answer costs = 4 + 2 = 6
(4 not gained, 2 deducted)
Wrong = 36 ÷ 6
6 questions wrong (so 19 correct).
Check: 19 × 4 = 76, minus 2 × 6 = 12, gives 64 ✓