Mathematical Operations

At a glance 10 rules

1. 1Decode firstRewrite the whole expression with the real signs in one pass, then calculate. Never work with the coded signs in your head.
2. 2BODMAS after decodingThe decoded signs obey BODMAS: brackets, then ÷ and × from left to right, then + and − from left to right. Where a sign stood before does not matter.
3. 3Letter codes in optionsWrite the key once (P = ÷, Q = ×, R = +, S = −), then size-check: an option holding a big product rarely lands on a small target, so test the others first.
4. 4Defined operationFor $a \star b = 2a - b$, a is the first number and b the second, so $a \star b \ne b \star a$. Do the inner bracket first; with a two-case rule, check the case at every step.
5. 5Interchange two signsSwap the pair everywhere in the equation, recompute the left side and compare it with the right. Pairs that move × or ÷ change the value most.
6. 6Interchange two numbersSwap the two numbers in place and keep every sign where it is. A number beside × or ÷ moves the total most, so test those pairs first.
7. 7Fill the blanksPut each option's signs in order and evaluate with BODMAS. Order matters: $8 \times 2 \div 4 = 4$ but $8 \div 2 \times 4 = 16$.
8. 8Missing numberDecode, simplify every known part with BODMAS, then undo the operations around ? one at a time to find it.
9. 9Find the codeIf the signs behind symbols are unknown, start with the equation that fixes the most symbols, keep only the signs that fit every equation, then evaluate.
10. 10Largest valueTo make an expression as large as possible, put × between the two biggest neighbours and use ÷ to shrink the term you subtract.