Last reviewed 22 Sept 2026 · 5 min read
Substitution questions
A table says, for example: means , means , means , means . Then
is read by replacing each symbol first:
Substitute the symbols, then apply BODMAS: Brackets, Of, Division and Multiplication (left to right), Addition and Subtraction (left to right). The substitution changes the symbols, never the priority rules.
Interchange questions
Two forms appear:
- "If and are interchanged, which equation is correct?" — swap those two symbols wherever they appear in each option and test it.
- "Interchange which two signs makes the equation correct?" — try each pair in turn; there are only six pairs among four operators, and usually the first or second works.
A third variant swaps two numbers rather than two signs. Test the options rather than solving in general: it is faster.
Balancing an equation
The question gives an equation with blanks and asks which set of signs fills them, such as . Work backwards from the answer: gives no 10; ✓. Test option by option, and remember BODMAS applies to your filled-in version too.
Defined operations
A rule defines a new operator: , or . Substitute the numbers into the definition exactly as written, taking care with the order — most defined operations are not symmetric, so .
For nested expressions, evaluate the innermost bracket first: means find , then apply with 4.
Inserting symbols
"Insert arithmetic signs to make true." Try division first (it constrains most), then multiplication: ✓.
Worked examples
If means , means , means and means , find .
Solution. Substituting: 64.
If means and means , find .
Solution. 14 — division is done before addition even though it was written second.
If and are interchanged, which is correct? (a) (b)
Solution. Swapping the two symbols in (a): . In (b): ✓ → (b).
Which two signs should be interchanged to make correct?
Solution. As printed the left side is . Test the pairs:
- and : ✗
- and : ✗
- and : ✓
So and are interchanged. With only three pairs to try among three signs, testing is quicker than reasoning.
If , find and .
Solution. 22; 4. The operation is not symmetric.
With , find .
Solution. ; then 6.
Insert signs to make true.
Solution. 5 ✓.
Which two numbers should be interchanged to make correct?
Solution. As printed: ✓ — the equation is already correct, so no interchange is needed. Check the given equation before hunting for a swap.
If means , means , means and means , find .
Solution. 18.