Mathematical Operations

medium importance~1-2 Q in Tier 15 formulas⚡ 6 shortcuts5 subtopics

Overview Learn (5)Formulas (5)Examples (4)Practice (70)

Hidden-rule and defined operations

⏱ 5 min read🧩 5 question types🎯 12 practice Q

The idea in one minute

A new symbol like ★, # or @ is given its own meaning. Either a formula is stated (a ★ b = 2a + 3b), or solved examples are shown and you must find the rule. Put the first number into a and the second into b. The order matters.

What a defined operation is

The examiner invents a symbol and tells you what it does. "a ★ b = 2a + 3b" is just a short name for a small formula.

To find 5 ★ 4, put a = 5 and b = 4: $2 \times 5 + 3 \times 4 = 10 + 12 = 22$ .

Think of a shop rule: "combo price = 2 × burger + 3 × drink". You just plug in the prices.

The order of a and b matters

The first number is a. The second is b. Swap them and the answer can change.

With a ★ b = 2a + 3b: 5 ★ 4 = 22, but 4 ★ 5 = 8 + 15 = 23. A formula like $a^2 + b^2$ does not care about order. One like 2a + 3b does.

Tip: Write "a = __, b = __" before plugging anything in. It sounds slow; it saves the question.

Nested operations

(7 @ 5) @ 3 means: work out 7 @ 5 first, then feed that answer back in.

With a @ b = (a + b) ÷ (a − b): 7 @ 5 = 12 ÷ 2 = 6. Then 6 @ 3 = 9 ÷ 3 = 3. Brackets decide the order, exactly as in BODMAS. When two symbols are defined, give each bracket its own formula, then combine.

Find the hidden rule from examples

The question shows solved pairs, like 5 # 3 = 34 and 6 # 2 = 40, and asks for 7 # 4. Climb a ladder of common rules and test each on every example.

Try	5 and 3 give	6 and 2 give
a + b, a × b	8, 15	8, 12
$a^2 + b^2$	34 ✓	40 ✓
$a^2 - b^2$	16	32
$(a+b)^2$	64	64

The rule is $a^2 + b^2$ , so 7 # 4 = 49 + 16 = 65.

Rule: A rule that fits only one example is not the rule. Two examples are given so that only one rule survives.

Rules that appear most often

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

Formulas with three parts

Some rules add more than two parts. Take $a \diamond b = ab + a + b$ .

4 ◇ 3 = 12 + 4 + 3 = 19.
6 ◇ 2 = 12 + 6 + 2 = 20.

Work out each part on its own line. Then add. A missed part is the most common slip.

Conditional definitions

The formula can depend on a condition: "a ▲ b = a − b if a > b, and a + b if a ≤ b". Check the condition every time.

9 ▲ 4: since 9 > 4, use a − b, giving 5. Then 5 ▲ 7: since 5 ≤ 7, use a + b, giving 12.

Watch: Students reuse the first case without checking. The second step often falls in the other case.

Keep your answer sane

If every option is a whole number and yours is a fraction, recheck the rule. Negative results are fine only when the options allow them.

Tip: After finding the rule, run it once more on a sample you already know. Two seconds, zero doubt.

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1common2 practice Q

Defined formula: a ★ b = given expression

How to spot it:

The question defines a new symbol with a formula in a and b.

a \star b = 2a + 3b

Method

Write a = first number, b = second number.
Put both into the formula.
Calculate.

Why it works:

The symbol is only a short name for the formula.

Try this

If a ★ b = 2a + 3b, find 5 ★ 4.

Show solution

a = 5, b = 4.
2 × 5 = 10 and 3 × 4 = 12.
10 + 12 = 22.

Answer

Type 2common4 practice Q

Hidden rule from solved examples

How to spot it:

Examples like 5 # 3 = 34, 6 # 2 = 40, then '7 # 4 = ?'.

a \# b = a^2 + b^2

Method

Climb the rule ladder: a+b, ab, a²+b², a²−b², (a±b)², ab+a+b.
Keep the rule that fits EVERY example.
Apply it to the asked pair.

Why it works:

Two examples are given so that only one rule survives the test.

Try this

If 5 # 3 = 34 and 6 # 2 = 40, find 7 # 4.

Show solution

25 + 9 = 34 ✓ and 36 + 4 = 40 ✓.
The rule is a² + b².
49 + 16 = 65.

Answer

Type 3occasional3 practice Q

Conditional definition

How to spot it:

The symbol does different things depending on a condition, like a > b.

Method

Check the condition for the given pair.
Use the matching formula.
Re-check the condition at every new step.

Why it works:

The second step often falls in the other case.

Try this

a ▲ b = a − b if a > b, and a + b if a ≤ b. Find (9 ▲ 4) ▲ 7.

Show solution

9 > 4, so 9 ▲ 4 = 9 − 4 = 5.
5 ≤ 7, so use a + b.
5 + 7 = 12.

Answer

Type 4common3 practice Q

Nested or two-symbol defined operations

How to spot it:

Brackets combine defined operations: (7 @ 5) @ 3, or two symbols in one line.

a \otimes b = 3a - b

Method

Solve the inner bracket with its own formula.
Write its value clearly.
Use that value as the input to the outer operation.

Why it works:

Brackets decide the order, as in BODMAS.

Try this

If a ⊗ b = 3a − b, find (4 ⊗ 2) ⊗ 5.

Show solution

4 ⊗ 2 = 12 − 2 = 10.
Now 10 ⊗ 5 = 30 − 5.
30 − 5 = 25.

Answer

Type 5occasional

Find a hidden input

How to spot it:

The formula and the full result are given; one of the two input numbers is asked.

Method

Write the equation with the unknown in place.
Undo the arithmetic step by step.
Check by computing the operation fully.

Why it works:

It is the same formula read backwards, one undo at a time.

Try this

If a ★ b = 2a + 3b and 5 ★ b = 28, what is b?

Show solution

2 × 5 + 3b = 28, so 10 + 3b = 28.
3b = 18, so b = 6.
Check: 2 × 5 + 3 × 6 = 28 ✓.

Answer

Formula sheet

Sum of squares

a \# b = a^2 + b^2

A very common hidden rule: 5 # 3 = 34.

Difference of squares

a @ b = a^2 - b^2

Also equals (a+b)(a−b): 8 @ 2 = 60.

Product plus sum

a \diamond b = ab + a + b

Multiply, then add both numbers: 4 ◇ 3 = 19.

Shortcuts that save time

⚡ Rule ladder for hidden operations

Test in this order: a + b, a × b, a² + b², a² − b², (a+b)², (a−b)², ab + a + b. Stop at the first rule that fits ALL examples.

Example

8 @ 2 = 60 and 7 @ 3 = 40. Find 9 @ 4.

Show solution

64 − 4 = 60 ✓ and 49 − 9 = 40 ✓.
The rule is a² − b².
9 @ 4 = 81 − 16 = 65.

Answer

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Swapping a and b in a formula that is not symmetric.

a is the first number, b the second. Write both down before plugging in.

Mistake 02

Accepting a rule after checking one example.

The rule must fit every example given.

Mistake 03

Reusing the first case of a conditional definition.

Check the condition again at every new step.

Mistake 04

Feeding the outer number into an inner bracket.

Finish the inner bracket fully; its value is the new input.

Mistake 05

Keeping a fraction answer when all options are whole numbers.

Recheck which rule fits both examples.

Quick revision

Read this the night before the exam.

Defined symbol = a small formula. Plug a = first number, b = second.
Order matters unless the formula is symmetric.
Nested: inner bracket first; its result becomes the new input.
Hidden rule ladder: a+b, ab, a²+b², a²−b², (a±b)², ab+a+b. Must fit ALL examples.
Conditional: re-check the condition at each step.
Answers must match the options' shape: whole, positive, and so on.

Practice: 12 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 12 questions

Suggested time 9 min · wrong answers go to your mistake notebook automatically.

Continue learning