Mathematical Operations

What a defined operation is

Sometimes the examiner invents a new symbol, like ★, # or @, and tells you what it does. For example, "a ★ b = 2a + 3b". The symbol is just a short name for a small formula. To find 5 ★ 4 you put a = 5 and b = 4 into the formula: 2 × 5 + 3 × 4 = 10 + 12 = 22. It is like a shop rule: "price of a combo = 2 × price of burger + 3 × price of drink" — you just plug in the prices.

Type 1: the formula is given

Put the first number in place of a and the second in place of b. The order matters: with a ★ b = 2a + 3b, 5 ★ 4 = 22 but 4 ★ 5 = 8 + 15 = 23. Always check which number is a.

Common formulas: $a^2 + b^2$, $a^2 - ab + b^2$, $\frac{a+b}{a-b}$, $3a - b$, $ab + a + b$.

Type 2: nested (chained) operations

(7 @ 5) @ 3 means: first work out 7 @ 5, then use that answer as the new a. With a @ b = (a + b) ÷ (a − b): 7 @ 5 = 12 ÷ 2 = 6. Then 6 @ 3 = 9 ÷ 3 = 3. Brackets decide which operation is done first, just like BODMAS. When two symbols are defined, work out each bracket with its own formula, then combine.

Type 3: find the hidden rule from examples

The question shows solved examples such as 5 # 3 = 34 and 6 # 2 = 40, and asks for 7 # 4. You must guess the rule and test it on every example. Try the common rules in this order:

So the rule is $a^2 + b^2$ and 7 # 4 = 49 + 16 = 65. A rule that fits only one example is not the rule. Two different rules can fit one example; the second example decides.

Type 4: conditional definitions

The formula may depend on a condition: "a ▲ b = a − b if a > b, and a + b if a ≤ b". Check the condition every time you use the symbol. 9 ▲ 4: 9 > 4, so 9 − 4 = 5. Then 5 ▲ 7: 5 ≤ 7, so 5 + 7 = 12. Students often use the first case again without checking; that is the trap.

Rules that appear most often

• Sum or difference of squares: $a^2 \pm b^2$.
• Square of sum or difference: $(a \pm b)^2$.
• Product plus sum: $ab + a + b$.
• Weighted sums: $2a + 3b$, $3a - b$.
• Mixed: $a^2 + b$, $a \times b - a$.

How to avoid mistakes

• Write "a = __, b = __" before plugging in.
• In nested questions, finish the inner bracket fully and write its value.
• For hidden rules, test on all examples before answering.
• Negative or fraction results are allowed only if the question allows them; if every option is a whole number and yours is not, recheck.

Quick revision

• Defined symbol = small formula; plug in a (first number) and b (second number).
• Order of a and b matters unless the formula is symmetric.
• Nested: inner bracket first; its result becomes the new input.
• Hidden rule: try a+b, ab, $a^2 \pm b^2$, $(a \pm b)^2$, ab+a+b; it must fit EVERY example.
• Conditional: re-check the condition at each step.