Mathematical Operations
🔒 Log in to trackBalancing and operator-filling
🔒 Log in to trackAn equation is a scale: the left side must equal the right side. Three question shapes test it. Pick the equation that is already true, fill sign blanks so the line balances, or place a full set of signs (sometimes including =) in order. Evaluate every option in writing; exactly one balances.
What balancing means
An equation is like a weighing scale. The left side and the right side must show the same value.
Here you either check which equation is already balanced, or choose signs that balance it. Nothing is hidden. You need clean BODMAS and a fixed order of testing.
Pick the correct equation
Four full equations are given; exactly one is true. Build a value column: one row per option, left-side value against the right side.
- 24 ÷ 6 − 2 × 3 = 1 → 4 − 6 = −2 ✗
- 5 × 4 + 6 ÷ 2 = 23 → 20 + 3 = 23 ✓
- 9 + 6 ÷ 2 − 3 = 11 → 9 + 3 − 3 = 9 ✗
- 7 × 2 + 8 ÷ 4 = 18 → 14 + 2 = 16 ✗
Tip: A wrong option is often the left-to-right value. 7 + 8 × 2 − 5 is 18 by BODMAS, but 25 left to right. Both 18 and 25 can sit among the options.
Fill the blanks with signs
"Which pair of signs makes 14 ? 2 ? 3 = 21 true?" Put each option pair in and evaluate.
- (+, ×): 14 + 2 × 3 = 20 ✗
- (÷, ×): 14 ÷ 2 × 3 = 21 ✓
- (×, −): 28 − 3 = 25 ✗
The first sign of the pair goes into the first blank, always.
Sequentially replace the * signs
Recent papers write: "Select the combination of signs to sequentially replace the * signs and balance the equation", like 21 * 7 * 5 * 4 * 3 = 27.
"Sequentially" fixes the placement: first sign into the first *, second into the second, and so on. Option ÷, ×, +, × gives 21 ÷ 7 × 5 + 4 × 3 = 15 + 12 = 27 ✓.
When = is one of the signs
The line has no = yet: 12 * 4 * 3 * 16, and one option sign is "=". Place the signs in order, then compare the two sides.
×, ÷, = gives 12 × 4 ÷ 3 = 16, that is 48 ÷ 3 = 16 ✓. Both sides match.
Watch: The = usually sits just before the last number, but not always. Test the placement; examiners move it to catch guessers.
Fill blanks inside brackets
Brackets can hold a blank too: (7 ? 3) ? 5 = 50. The bracket is solved first, so a sign inside it is powerful.
(+, ×) gives (7 + 3) × 5 = 50 ✓, while (×, +) gives only 7 × 3 + 5 = 26.
One full walk-through
Find the pair that makes 30 ? 6 ? 4 = 9 true. The options are (×, −), (÷, +), (+, ÷) and (−, ×).
- (×, −): 30 × 6 − 4 = 180 − 4 = 176. Too big. ✗
- (÷, +): 30 ÷ 6 + 4 = 5 + 4 = 9. ✓
- (+, ÷): 30 + 6 ÷ 4 is not a whole number. ✗
- (−, ×): 30 − 6 × 4 = 30 − 24 = 6. ✗
Only the second row matches, so the answer is (÷, +).
Test in a smart order
- Look at the right side. Big target → you need ×. Small target → look for ÷ or −.
- Test options that create exact divisions first. 72 ÷ 8 = 9 is friendly; 7 ÷ 5 is a dead end.
- Use the last digit. If the target ends in 7 and an option's arithmetic ends in 0, drop it.
- Once one option balances, mark it. In a well-set question only one works.
Tip: Never judge an equation by looks. A four-row value column is faster than staring.
Question types you will see
Each type: how to recognise it, the method step by step, and one question to try.
Pick the arithmetically correct equation
'Which of the following equations is correct?' with four complete equations.
Evaluate each left side with BODMAS.
Write one value per row.
Compare with the right side.
Mark the only match.
Wrong options are usually left-to-right values of a correct-looking line.
Which equation is correct: 24 ÷ 6 − 2 × 3 = 1, 5 × 4 + 6 ÷ 2 = 23, 9 + 6 ÷ 2 − 3 = 11 or 7 × 2 + 8 ÷ 4 = 18?
Show solutionHide solution
Row values: −2, 23, 9, 16.
Right sides: 1, 23, 11, 18.
Only the second row matches.
5 × 4 + 6 ÷ 2 = 23
Fill the blanks with a sign pair
'Which pair of signs will make 14 ? 2 ? 3 = 21 correct?'
Put the first option pair into the blanks, in order.
Evaluate with BODMAS.
Compare with the right side.
Repeat until one pair works.
Testing is faster and safer than guessing the signs.
Which pair of signs makes 14 ? 2 ? 3 = 21 correct: (+, ×), (÷, ×), (×, −) or (−, +)?
Show solutionHide solution
(+, ×): 14 + 2 × 3 = 20 ✗.
(÷, ×): 14 ÷ 2 × 3 = 7 × 3 = 21 ✓.
(×, −): 28 − 3 = 25 ✗. (−, +): 14 − 2 + 3 = 15 ✗.
÷ and ×
Sequentially replace the * signs
A line like 21 * 7 * 5 * 4 * 3 = 27. 'Select the combination of signs to sequentially replace the * signs.'
Send the first sign into the first *, second into the second, and so on.
Evaluate with BODMAS.
Check against the right side.
'Sequentially' fixes the placement, so each option makes exactly one line.
Which signs balance 21 * 7 * 5 * 4 * 3 = 27 when placed in order: ÷, ×, +, ×?
Show solutionHide solution
Line: 21 ÷ 7 × 5 + 4 × 3.
21 ÷ 7 = 3, then 3 × 5 = 15. Also 4 × 3 = 12.
15 + 12 = 27 ✓.
Yes: ÷, ×, +, × balances it
Replace * signs when = is one of them
A line like 12 * 4 * 3 * 16 has no = yet; the options contain '='.
Place the signs in order, including the =.
Evaluate the part before the =.
Evaluate the part after the =.
The two values must match.
The = splits the line into two separate expressions.
Where do ×, ÷ and = go in 12 * 4 * 3 * 16 to balance it?
Show solutionHide solution
Try ×, ÷, = in order: 12 × 4 ÷ 3 = 16.
Left: 48 ÷ 3 = 16.
Right: 16. They match.
12 × 4 ÷ 3 = 16
Fill blanks inside brackets
The blank sits inside a bracket, like (7 ? 3) ? 5 = 50.
Solve the bracket with the first option's inside sign.
Apply the outside sign.
Compare with the right side.
Repeat with the other options.
The bracket is solved first, so the sign inside it shapes the whole line.
Which pair of signs makes (7 ? 3) ? 5 = 50 correct: (+, ×), (×, +), (−, ×) or (÷, +)?
Show solutionHide solution
(+, ×): (7 + 3) × 5 = 50 ✓.
(×, +): 7 × 3 + 5 = 26 ✗.
(−, ×): (7 − 3) × 5 = 20 ✗. (÷, +): 7 ÷ 3 + 5 is not whole ✗.
- and ×
Shortcuts that save time
List the four options in a column. Evaluate each left side in one written line. Exactly one row matches its right side. The column doubles as your final check.
Which is correct: 24 ÷ 6 − 2 × 3 = 1, 5 × 4 + 6 ÷ 2 = 23, 9 + 6 ÷ 2 − 3 = 11 or 7 × 2 + 8 ÷ 4 = 18?
Show solutionHide solution
24 ÷ 6 − 2 × 3 = 4 − 6 = −2 ✗.
5 × 4 + 6 ÷ 2 = 20 + 3 = 23 ✓.
9 + 6 ÷ 2 − 3 = 9 + 3 − 3 = 9 ✗.
7 × 2 + 8 ÷ 4 = 14 + 2 = 16 ✗.
5 × 4 + 6 ÷ 2 = 23
Mistakes to avoid
Where most students lose marks on this subtopic.
Scanning the options for a nice-looking equation.
Evaluate every option in a written value column.
Putting the option signs into the blanks from the right.
First sign into the first blank, left to right.
Treating ÷ as always after ×.
Equal rank: ÷ and × go left to right.
Computing the whole line when = is one of the signs.
The = splits the line. Compare the two sides separately.
Stopping at the first option that looks close.
Finish each row's arithmetic. Only an exact match counts.
Quick revision
Read this the night before the exam.
Balanced means left value equals right value. Test each option in writing.
Sign pairs: first sign into the first blank, always.
'Sequentially replace *': first sign to first *, in order.
When = is among the signs, compare the part before it with the part after it.
Brackets with blanks are solved first, so signs inside them are powerful.
Target size, exact division and last digit cut options fast.
Practice: 13 questions
Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.
Topic test · 13 questions
Suggested time 11 min · wrong answers go to your mistake notebook automatically.