Mathematical Operations
🔒 Log in to trackInterchanging signs and numbers
🔒 Log in to trackTwin of substitution, but the change is a symmetric swap: 'interchange + and ×' (and sometimes two numbers, e.g. 2 and 8). Wherever one symbol stood, the other now stands, and vice versa. Apply every swap in a single rewrite, then BODMAS.
The fix-the-equation variant gives a false equation and asks which two numbers (or signs), if interchanged, make it correct. This is a sweep, not an insight: try the pairs in a fixed order (say, always from the left), evaluate each rewritten equation in writing, and stop at the pair that balances. With four numbers there are at most six pairs — under a minute of arithmetic, with zero guessing.
Detailed notes
What "interchange" means
To interchange two things means to swap them: wherever the first stood, the second now stands, and the other way round. "Interchange + and ×" turns 3 × 4 + 5 into 3 + 4 × 5. It is like two students exchanging seats: both move. This is different from substitution, where the change may go in one direction only.
Type 1: swap two signs, then find the value
Rewrite the line with the two signs exchanged, then use BODMAS. Example: interchange + and ×; find 3 × 4 + 5 − 6. New line: 3 + 4 × 5 − 6 = 3 + 20 − 6 = 17.
Type 2: swap two numbers (and maybe two signs too)
"Interchange 2 and 8" means every 2 becomes 8 and every 8 becomes 2. Do all swaps in one rewrite, then calculate. Example: interchange + and × and also 2 and 8 in 2 + 8 × 3 − 5. New line: 8 × 2 + 3 − 5 = 16 + 3 − 5 = 14. Careful: swap whole numbers only. If the swap is 2 and 5, the number 25 does not change.
Type 3: which two numbers must be swapped to make the equation true?
You get a wrong equation, for example 4 + 2 × 6 − 3 = 20, and four pairs in the options. Do a sweep: try each option pair, evaluate the new left side, and stop when it equals the right side.
- (4, 2): 2 + 4 × 6 − 3 = 23 ✗
- (2, 3): 4 + 3 × 6 − 2 = 4 + 18 − 2 = 20 ✓
With 4 numbers there are only 6 possible pairs; with 5 numbers there are 10. You normally test only the 4 option pairs.
Type 4: which two signs must be swapped?
Same sweep, but you swap two operators. Example: 12 + 6 ÷ 3 − 2 × 4 = 18. Try + and −: 12 − 6 ÷ 3 + 2 × 4 = 12 − 2 + 8 = 18 ✓.
Type 5: after a given swap, which equation is correct?
The swap is stated first, and four equations follow. Apply the swap to each left side, evaluate, and compare with its right side. Only one balances.
Shortcuts that save time
- Swapping the two numbers around a single × (like 8 × 4 to 4 × 8) changes nothing, because order does not matter in multiplication. Skip such pairs.
- Size check: if the target is large (like 81) you need a big product; if it is small, look for a division or subtraction to be created.
- Last-digit check: 8 × 6 ends in 8; if the target ends in 5 you may drop that option at once.
- Division must usually come out exact: a swap that creates 7 ÷ 5 rarely gives a whole-number answer, so test it last.
Traps
- Swapping only one direction (changing + to × but leaving the old × as it was).
- Forgetting a second occurrence of the same sign or number.
- Evaluating the swapped line left to right instead of by BODMAS.
Quick revision
- Interchange = two-way swap; every occurrence of both items changes.
- Apply all swaps (signs and numbers) in ONE rewrite, then BODMAS.
- "Which pair to swap": sweep the option pairs in a fixed order and stop at the one that balances.
- Swapping the two numbers of one product changes nothing: skip it.
- Use size and last-digit checks to cut options fast.
Types of questions asked
Every way this subtopic shows up in exams — how to recognise it, the formula or logic to use, and a solved example.
Type 1: Interchange two signs, then evaluatecommon3 practice Q
'If + and ÷ are interchanged, find the value of ...'
- Swap both signs everywhere (two-way).
- BODMAS on the new line.
Why: interchange is symmetric, so both signs change places.
Example: Interchange + and ×: 3 × 4 + 5 − 6.
3 + 4 × 5 − 6 = 3 + 20 − 6 = 17.
Type 2: Interchange numbers (with or without signs), then evaluatecommon2 practice Q
'If 2 and 8 are interchanged (and + and ×) ...' — numbers are swapped too.
- Do every swap in one rewrite.
- Swap whole numbers only (25 is not changed by swapping 2 and 5).
- BODMAS.
Why: doing swaps one after another invites double swaps.
Example: Interchange + and × and 2 and 8: 2 + 8 × 3 − 5.
8 × 2 + 3 − 5 = 16 + 3 − 5 = 14.
Type 3: Which two numbers to interchange to balancecommon2 practice Q
A false equation and options like '4 and 2', '6 and 3'.
- For each option, swap the two numbers.
- Evaluate the left side; stop when it equals the right side.
Why: only one pair can restore the balance in a well-set question.
Example: Which two numbers should be interchanged: 4 + 2 × 6 − 3 = 20?
Swap 2 and 3: 4 + 3 × 6 − 2 = 20 ✓ → 2 and 3.
Type 4: Which two signs to interchange to balancecommon2 practice Q
A false equation and options like '+ and −', '× and ÷'.
- For each option, swap the two signs everywhere.
- Evaluate; stop at the one that balances.
Why: same sweep idea as number swaps, applied to operators.
Example: Which two signs should be interchanged: 12 + 6 ÷ 3 − 2 × 4 = 18?
Swap + and −: 12 − 2 + 8 = 18 ✓ → + and −.
Type 5: After a given interchange, which equation is correctoccasional2 practice Q
The swap is stated; four equations follow.
- Apply the swap to each left side.
- Evaluate and compare with the right side.
Why: the right sides are plain numbers and do not change.
Example: If + and − are interchanged, is 10 − 4 + 2 × 3 = 8 correct?
Swap: 10 + 4 − 2 × 3 = 14 − 6 = 8 ✓ — correct.
Shortcut tricks
⚡ Swap table in the margin
Write 'a ↔ b', 'op1 ↔ op2' at the left, rewrite the expression once with all swaps applied, evaluate.
Example: Q. Interchange + with × and 2 with 6 in 2 + 6 × 3.
Sol. After swap: 6 × 2 + 3 = 12 + 3 = 15.
One rewrite covering every swap beats applying them piecemeal.
⚡ Fixed-order sweep for fix-the-equation
Sweep pairs left to right: (n1,n2), (n1,n3), (n1,n4), (n2,n3), (n2,n4), (n3,n4). One written value per row; the row matching the RHS is the answer. If no row matches, you mis-evaluated something — redo the row, not the method.
Example: Q. Make 8 × 4 − 6 + 3 = 35 correct by interchanging two numbers.
Sol. Sweep: (8,4) → 4×6−8+3 = 19 ✗; (8,6) → 6×4−8+3 = 19 ✗; (8,3) → 3×4−6+8 = 14 ✗; (4,6) → 8×6−4+3 = 47 ✗; (4,3) → 8×3−6+4 = 22 ✗; (6,3) → 8×4−3+6 = 35 ✓. Swap 6 and 3.
Six written trials, always in the same order — mechanical and safe.
Where students lose marks
Swapping the signs but forgetting the numbers (or the reverse) in two-part interchanges.
Swapping numbers that appear more than once without checking every occurrence.
Abandoning the sweep halfway — the correct pair is often the last one you would try.
Practice sets — 14 questions
Sets of 10, mixed across the question types above. Each answer comes with a step-by-step explanation.
Topic test · 10 questions
Suggested time 8 min · wrong answers go to your mistake notebook automatically.