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Mathematical Operations

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medium importance~1-2 Q in Tier 15 formulas⚡ 6 shortcuts5 subtopics
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Subtopic 4 of 5·← Interchanging signs and numbers·Hidden-rule and defined operations →

Balancing and operator-filling

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Three shapes live here:

  1. Pick the correct equation — four complete equations are offered; exactly one is arithmetically true. Evaluate all four with BODMAS, in writing, and tick the true one. This is the cheapest correct mark in the whole section (four one-line evaluations, ~40 seconds).
  2. Fill the operator blanks — '5 ? 2 ? 3 = 13' with sign-pair options like (×, +). Test each pair in order; the balancing pair wins.
  3. Choose the sign set — three blanks, options are full sign sets such as '÷, +, ×'. Same method: evaluate each set, keep the one that matches the RHS.

The examiner plants one option that would be true if evaluated left to right but fails under BODMAS (or vice versa). If your evaluation disagrees with the key, re-check the order of operations before re-checking the arithmetic.

📖 Notes🧩 4 question types🎯 13 practice Q

Detailed notes

What balancing means

An equation is like a weighing scale: the left side and the right side must have the same value. In these questions you either check which equation is already balanced, or choose the signs that will balance it. Nothing is hidden; you only need clean BODMAS and a fixed order of testing.

Type 1: pick the correct equation

Four full equations are given; exactly one is true. Evaluate each left side and compare. Example options: 16 ÷ 4 × 2 + 6 = 14, and 7 + 8 × 2 − 5 = 25. First one: 4 × 2 + 6 = 14 ✓. Second one: 7 + 16 − 5 = 18, not 25 ✗ (25 is the left-to-right value, a planted trap).

Type 2: fill the blanks with a pair or set of signs

"Which pair of signs makes 14 ? 2 ? 3 = 21 true?" Put each option in and evaluate.

  • ÷ and ×: 14 ÷ 2 × 3 = 7 × 3 = 21 ✓
  • × and −: 28 − 3 = 25 ✗ With three or four blanks the options are full sets like "−, +, ×". Replace the blanks in order from left to right.

Type 3: replace the * signs in order

Recent papers write the question as: "Select the correct combination of mathematical signs to sequentially replace the * signs and balance the equation", e.g. 21 * 7 * 5 * 4 * 3 = 27. "Sequentially" means the first sign of the option goes into the first *, the second into the second *, and so on. Option "÷, ×, +, ×": 21 ÷ 7 × 5 + 4 × 3 = 3 × 5 + 12 = 15 + 12 = 27 ✓.

Type 4: the = sign is one of the options

Here the equation has no = sign yet: 12 * 4 * 3 * 16, and one option sign is "=". After filling, check that the part before = equals the part after =. "×, ÷, =" gives 12 × 4 ÷ 3 = 16 → 48 ÷ 3 = 16 ✓. Tip: the = sign usually sits just before the last number, but always test; examiners move it to catch guessers.

A fast testing order

  1. Look at the right side. Is it big or small compared with the numbers? A big target needs ×; a small one needs ÷ or −.
  2. Test options that create exact divisions first. 72 ÷ 8 = 9 is friendly; 7 ÷ 5 is usually a dead end.
  3. Use the last digit: if the left side must end in 7 and your option gives a product ending in 0 plus 5, drop it.
  4. Once one option balances, still glance at the rest only if time allows; in a correct question only one works.

Common mistakes

  • Applying BODMAS in your head and slipping on the order (× before +).
  • Putting option signs in the wrong blanks (reading the option from the right).
  • In "=" questions, calculating the whole line as one expression instead of comparing both sides.

Quick revision

  • Balanced = left value equals right value; test each option in writing.
  • "Sequentially replace *" = first sign into first *, second into second, and so on.
  • When "=" is among the signs, compare the two sides after filling.
  • Target size tells you whether you need × (big) or ÷/− (small).
  • Exact divisions and last digits cut options quickly.
  • The left-to-right value is the usual trap option.

Types of questions asked

Every way this subtopic shows up in exams — how to recognise it, the formula or logic to use, and a solved example.

Type 1: Pick the arithmetically correct equationvery common2 practice Q
How to spot it:

'Which of the following equations is correct?' with four complete equations.

  1. Evaluate each left side with BODMAS.
  2. Compare with the right side; exactly one matches.

Why: wrong options are usually left-to-right values.

Example: Is 16 ÷ 4 × 2 + 6 = 14 correct?

4 × 2 + 6 = 8 + 6 = 14 ✓ — yes.

Type 2: Fill the blanks with a sign pair or sign setcommon3 practice Q
How to spot it:

'Which pair/set of signs will make 14 ? 2 ? 3 = 21 correct?'

  1. Put each option's signs into the blanks, left to right.
  2. Evaluate; keep the one that equals the right side.

Why: testing is faster and safer than guessing.

Example: 14 ? 2 ? 3 = 21: try ÷ and ×.

14 ÷ 2 × 3 = 7 × 3 = 21 ✓.

Type 3: Sequentially replace the * signsvery common2 practice Q
How to spot it:

21 * 7 * 5 * 4 * 3 = 27 — 'select the combination of signs to sequentially replace *'.

  1. First sign → first *, second sign → second *, and so on.
  2. BODMAS.
  3. Check against the right side.

Why: 'sequentially' fixes the order of placement.

Example: 21 * 7 * 5 * 4 * 3 = 27 with ÷, ×, +, ×.

21 ÷ 7 × 5 + 4 × 3 = 15 + 12 = 27 ✓.

Type 4: Replace * signs when '=' is one of the optionscommon2 practice Q
How to spot it:

12 * 4 * 3 * 16 — there is no = sign yet; options contain '='.

  1. Place the signs in order, including '='.
  2. Evaluate the left part and the right part separately.
  3. They must be equal.

Why: '=' splits the line into two expressions.

Example: 12 * 4 * 3 * 16 with ×, ÷, =.

12 × 4 ÷ 3 = 16 → 16 = 16 ✓.

Shortcut tricks

⚡ Evaluate all, then eliminate

In pick-the-correct-equation, evaluate every option in a written value column. Exactly one row matches its RHS; the column doubles as your final check.

Example: Q. Which is correct? 24 ÷ 6 − 2 × 3 = 1 / 5 × 4 + 6 ÷ 2 = 23 / 9 + 6 ÷ 2 − 3 = 11 / 7 × 2 + 8 ÷ 4 = 18.

Sol. Values: 4−6 = −2 ✗; 20+3 = 23 ✓; 9+3−3 = 9 ✗; 14+2 = 16 ✗. Answer: the second.

A four-row value column settles it; never stop at the first 'plausible' option.

Where students lose marks

  • Scanning options for a 'nice-looking' equation instead of evaluating.

  • Testing only the first blank of a sign pair — the second sign is where the balance usually breaks.

  • Ignoring equal ranks: ÷ then × must be done left to right, not 'multiplication always first'.

Practice sets — 13 questions

Sets of 10, mixed across the question types above. Each answer comes with a step-by-step explanation.

Topic test · 13 questions

Suggested time 11 min · wrong answers go to your mistake notebook automatically.