The 50-Base Method: Squares of the 40s, 50s and 60s
Level 3 Mental squares · Reviewed by The NumThink editorial team · Last reviewed
Short version: (50 ± d)² = 2,500 ± 100d + d²: 53² = 2,500 + 300 + 9 = 2,809.
This page is part of the NumThink Mental squares course. Read why the method works, watch it run on worked examples, then drill it in the practice box below until it feels automatic.
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No signup · instant feedbackWhy it works
(50 + d)² = 2,500 + 2×50×d + d² = 2,500 + 100d + d². The middle term is just "d hundreds", so the method is: 2,500, add d×100, add d².
For 40s, d is negative: 47² = 2,500 − 300 + 9 = 2,209. Same formula, sign flips.
The steps
- Measure the distance from 50. 53 → d = 3.
- Add d hundreds to 2,500. 2,500 + 300 = 2,800.
- Add the small square. d² = 9 → 2,809.
Worked examples
Easy 51²
Show the thinking
d = 1
2,500 + 100 + 1 = 2,601.
Answer: 2,601
Medium 53²
Show the thinking
d = 3
2,500 + 300 + 9 = 2,809.
Answer: 2,809
Hard 47²
Show the thinking
d = −3
2,500 − 300 + 9 = 2,209.
Answer: 2,209
When to use it
- Squares from 41 to 59 — this is THE tool for that neighborhood.
- The d² part uses squares you know (1–9), so everything stays small.
When NOT to use it
- Numbers ending in 5 — the ending-in-5 rule wins.
- Numbers far from 50 — the correction grows past the benefit; use near-known-squares instead.
Common mistakes
Sign errors in the 40s. 47² answered 2,800 + 9. Fix: Below 50, the middle term subtracts: 2,500 − 300. Only the d² is always added.
Using d instead of d×100. 53² answered 2,503 + 9. Fix: The middle term is d HUNDREDS — 3 means 300.
Check yourself: The answer must sit near 2,500: 53² ≈ 2,800 ✓ and 47² ≈ 2,200 ✓. If you land far from 2,500, the sign or scale slipped.