How to Answer 11+ Balancing Sums Questions

A balancing sums question gives you a sum with a hole in it, like 15 − 6 = 4 + [?], and asks which number fills the hole so that both sides are worth the same. Picture an old-fashioned set of weighing scales: the equals sign is the middle, and your job is to make the two pans balance.

The mistake almost everyone makes at first is treating it like a normal sum — see numbers, start adding. But the missing number isn't "the answer to the sum". It's the piece that makes the right side match the left side. So the winning move is to ignore the hole completely at first: deal with the side that has no hole, get its total, and only then turn to the other side.

Left side first, every time. Here's the routine.

The method

Work out the full left side first. It has no gaps, so nothing can go wrong: 15 − 6 = 9. That 9 is your target.
Work out as much of the right side as you can. In 3 × 5 − [?], you know the 3 × 5 = 15 part before you touch the gap.
Ask what the gap must be to hit the target. "15 minus what makes 9?" — that's a much easier question than the one you started with.
Check by running the whole right side through with your number in the hole. Both sides should now say the same total. If they don't, one of your two sides is wrong — recount the left one first.

Worked examples

Example 1: one step each side

Find the number that replaces [?] so that both sides are equal. 15 − 6 = 4 + [?]

A. 6 B. 4 C. 9 D. 7 E. 5

Left side first: 15 − 6 = 9. Target: 9.
Right side: 4 + [?] must make 9.
4 + 5 = 9, so the gap is 5. (Not 9 — that's the target itself, planted in the options.)

Answer: E. 5

Example 2: two steps on the left

Find the number that replaces [?] so that both sides are equal. 36 ÷ 4 + 3 = 3 × 5 − [?]

A. 12 B. 3 C. 15 D. 5 E. 9

Left side first: 36 ÷ 4 = 9, then 9 + 3 = 12. Target: 12.
Right side: 3 × 5 = 15, so the right side is 15 − [?].
15 minus what makes 12? Three. Check: 15 − 3 = 12, and 12 = 12 — balanced. (Spot the traps in the options: 12 is the target and 15 is the half-finished right side.)

Answer: B. 3

Example 3: the gap in the middle of the sum

Find the number that replaces [?] so that both sides are equal. 56 ÷ 7 × 3 = 6 × [?] − 12

A. 4 B. 8 C. 5 D. 6 E. 12

Left side first: 56 ÷ 7 = 8, then 8 × 3 = 24. Target: 24.
Right side: 6 × [?] − 12 must make 24. Undo the take-away: before losing 12, the 6 × [?] part must have been 24 + 12 = 36.
6 times what is 36? Six. Check the whole side: 6 × 6 − 12 = 36 − 12 = 24 — balanced.

Answer: D. 6

The traps

Giving the target instead of the missing piece. You work out that both sides equal 12 and proudly answer 12 — but the question asked what goes in the hole. The target total is nearly always planted in the options, waiting for exactly this slip.
Doing only half the right side. In 3 × 5 − [?], stopping at 15 and answering 15 is the second-favourite wrong answer. The gap is part of the sum — finish the thought.
Subtracting the easy way round. "15 − [?] = 12" means the gap is 3. Writing 15 − 12 without thinking gives 3 here — but in a rush, children often compute the two visible numbers in whatever order is easiest and answer nonsense. Say the sentence: "15 minus what makes 12?"
An unbalanced check. If your check gives 23 = 24, don't shave the answer by one and hope — one side has a slip in it. Recount the left side first; it's usually a missed × or ÷.

Try these yourself

Find the number that replaces [?] so that both sides are equal. 3 × 4 = 20 − [?]

A. 6 B. 8 C. 10 D. 12 E. 4

Find the number that replaces [?] so that both sides are equal. 7 × 6 = [?] × 3 + 6

A. 42 B. 12 C. 6 D. 13 E. 11

Find the number that replaces [?] so that both sides are equal. 3 × 9 + 13 = 100 ÷ [?] + 20

A. 20 B. 40 C. 4 D. 10 E. 5

Answers

1: B — left side is 12, and 20 − 8 = 12.
2: B — left side is 42; before the + 6 the right side must be 36, and 12 × 3 = 36.
3: E — left side is 27 + 13 = 40; before the + 20 the right side must be 20, and 100 ÷ 5 = 20.

The complete 11+ question-type guide
VR: Letter Sums — the sister type: sums where letters stand for numbers
VR: Letter & Number Sequences — more number patterns to hunt down

All example questions come from the Brainblox question bank — practise this question type inside 15 arcade games, timed mock exams and focused practice. Brainblox is an independent app: not affiliated with, endorsed by or licensed by any exam board, school or assessment publisher, and it contains no past papers.