Mathematical Operations
🔒 Log in to trackInterchanging signs and numbers
🔒 Log in to trackInterchange means a two-way swap: 'interchange + and ×' puts + wherever × stood and × wherever + stood. Sometimes two numbers are swapped instead, or both. Apply every swap in one rewrite, then use BODMAS. The fix-the-equation variant asks which pair, when swapped, makes a false equation true — sweep the pairs in order.
What interchange means
To interchange two things is to swap them, both ways at once. "Interchange + and ×" turns 3 × 4 + 5 into 3 + 4 × 5.
Think of two students exchanging seats: both move. This differs from sign substitution, where the mapping can go one way.
Swap two signs, then find the value
Rewrite the line with the two signs exchanged, then use BODMAS.
Interchange + and × in 3 × 4 + 5 − 6. New line: 3 + 4 × 5 − 6 = 3 + 20 − 6 = 17.
Rule: Every occurrence of both signs changes. A sign that appears twice is swapped twice.
A swap of × and ÷
This swap changes the order of work too, so go slowly.
Interchange × and ÷ in 20 ÷ 4 × 5. New line: 20 × 4 ÷ 5. Left to right: 20 × 4 = 80, then 80 ÷ 5 = 16.
The original line gives 20 ÷ 4 × 5 = 5 × 5 = 25. That value is the usual wrong option.
Tip: After any swap, rewrite the line and read it fresh. Do not reuse the old order of work.
Swap numbers too
"Interchange 2 and 8" means every 2 becomes 8 and every 8 becomes 2. Do all swaps in one rewrite, then calculate.
Interchange + and ×, and also 2 and 8, in 2 + 8 × 3 − 5. New line: 8 × 2 + 3 − 5 = 16 + 3 − 5 = 14.
Watch: Swap whole numbers only. If the swap is 2 and 5, a number like 25 does not change at all.
Fix the equation: sweep the pairs
You get a false equation, for example 4 + 2 × 6 − 3 = 20, and pairs in the options. Do a sweep: swap each pair, evaluate, stop at the one that balances.
- (4, 2): 2 + 4 × 6 − 3 = 23 ✗
- (4, 6): 6 + 2 × 4 − 3 = 11 ✗
- (2, 3): 4 + 3 × 6 − 2 = 20 ✓
With 4 numbers there are only 6 possible pairs. You normally test just the 4 option pairs.
Which two signs to swap
Same sweep, but on operators. 12 + 6 ÷ 3 − 2 × 4 = 18. Try + and −: 12 − 6 ÷ 3 + 2 × 4 = 12 − 2 + 8 = 18 ✓.
Work in a fixed order: (+, −), (+, ×), (+, ÷), (−, ×), (−, ÷), (×, ÷). One written value per row.
After a given swap, which equation is correct?
The swap is stated first; four equations follow. Apply the swap to each left side, evaluate, and compare with its right side.
Interchange + and −, then test 10 − 4 + 2 × 3 = 8. Swapped: 10 + 4 − 2 × 3 = 14 − 6 = 8 ✓.
Checks that save time
- Skip pointless pairs: swapping the two numbers of one product, like 8 × 4 to 4 × 8, changes nothing.
- Size check: a big target like 81 needs a big product. A small target needs ÷ or −.
- Last-digit check: 8 × 6 ends in 8. If the target ends in 5, drop that pair at once.
- Exact division: a swap creating 7 ÷ 5 rarely gives a whole number. Test it last.
Tip: If no row balances, you mis-evaluated a row — redo the row, not the method.
Question types you will see
Each type: how to recognise it, the method step by step, and one question to try.
Interchange two signs, then evaluate
'If + and ÷ are interchanged, find the value of …'
Swap both signs everywhere, two-way.
Rewrite the line once.
Evaluate with BODMAS.
Interchange is symmetric, so both signs move.
The signs + and × are interchanged. What is the value of 3 × 4 + 5 − 6?
Show solutionHide solution
New line: 3 + 4 × 5 − 6.
4 × 5 = 20.
3 + 20 − 6 = 17.
17
Interchange numbers with or without signs
'If 2 and 8 are interchanged (and + and ×)…' Numbers are swapped too.
Do every swap in one rewrite.
Swap whole numbers only.
Evaluate with BODMAS.
Doing swaps one after another invites double swaps.
Interchange + and ×, and also 2 and 8. What is 2 + 8 × 3 − 5?
Show solutionHide solution
New line: 8 × 2 + 3 − 5.
8 × 2 = 16.
16 + 3 − 5 = 14.
14
Which two numbers make the equation true
A false equation, with pairs like '4 and 2', '6 and 3' in the options.
Swap the first option pair.
Evaluate the left side with BODMAS.
Compare with the right side.
Repeat until one pair balances.
Only one pair restores the balance in a well-set question.
Which two numbers should be interchanged to make 4 + 2 × 6 − 3 = 20 correct?
Show solutionHide solution
Swap 4 and 2: 2 + 4 × 6 − 3 = 23 ✗.
Swap 4 and 6: 6 + 2 × 4 − 3 = 11 ✗.
Swap 2 and 3: 4 + 3 × 6 − 2 = 20 ✓.
2 and 3
Which two signs make the equation true
A false equation, with sign pairs like '+ and −', '× and ÷' in the options.
Swap the first option pair everywhere.
Evaluate the left side.
Compare with the right side.
Repeat until one pair balances.
The same sweep works on operators as on numbers.
Which two signs should be interchanged to make 12 + 6 ÷ 3 − 2 × 4 = 18 correct?
Show solutionHide solution
Try + and −: 12 − 6 ÷ 3 + 2 × 4.
6 ÷ 3 = 2 and 2 × 4 = 8.
12 − 2 + 8 = 18 ✓.
- and −
After a given interchange, which equation is correct
The swap is stated first; four complete equations follow.
Apply the swap to each left side.
Evaluate with BODMAS.
Compare with the right side.
Mark the only one that balances.
The right sides are plain numbers and do not change.
The signs + and − are interchanged. Is 10 − 4 + 2 × 3 = 8 correct?
Show solutionHide solution
Swapped line: 10 + 4 − 2 × 3.
2 × 3 = 6.
10 + 4 − 6 = 8 ✓ — correct.
Yes, it is correct
Shortcuts that save time
Write the swaps on the left, like + ↔ × and 2 ↔ 8. Then rewrite the expression once with all swaps applied. Evaluate. Piececemeal swapping invites double swaps.
Interchange + with × and 2 with 6 in 2 + 6 × 3.
Show solutionHide solution
Both swaps at once: 6 × 2 + 3.
6 × 2 = 12, then 12 + 3 = 15.
15
Sweep number 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 right side is the answer.
Make 8 × 4 − 6 + 3 = 35 correct by interchanging two numbers.
Show solutionHide solution
(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 ✓.
Interchange 6 and 3
Mistakes to avoid
Where most students lose marks on this subtopic.
Swapping one way only (+ becomes ×, the old × stays).
Interchange is two-way. Both signs change places, everywhere.
Swapping digits inside a number (changing 25 when told 2 and 5).
Swap whole numbers only.
Applying two swaps in two separate rewrites.
One rewrite with every swap applied at once.
Guessing which pair balances the equation.
Sweep the option pairs in a fixed order and evaluate each in writing.
Evaluating the swapped line left to right.
BODMAS still rules after any swap.
Quick revision
Read this the night before the exam.
Interchange = two-way swap; every occurrence of both items changes.
Apply all swaps, signs and numbers, in ONE rewrite, then BODMAS.
Fix-the-equation: sweep pairs in a fixed order; stop at the row that balances.
Swapping the two numbers of one product changes nothing. Skip it.
Size, last digit and exact division cut option pairs fast.
Numbers and signs can be swapped in the same question.
Practice: 14 questions
Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.
Topic test · 10 questions
Suggested time 8 min · wrong answers go to your mistake notebook automatically.