Mental Arithmetic — Rounding, Approximation, Order of Magnitude
The techniques for rapid mental calculation — when to round, how to approximate, and how to reason about the size of an answer before computing it.
The Goal
The goal of mental arithmetic isn’t to replace a calculator. It’s to:
- Quickly check whether an answer is plausible
- Estimate fast enough to make a decision in real time
- Catch errors before they propagate
Precision is for paper and machines. Mental arithmetic is about being right to within a factor of 2 or 3 — fast.
Rounding
Rounding replaces a number with a nearby “cleaner” number to simplify calculation.
Rules
- Look at the first digit being dropped
- If it’s 5 or above, round up; below 5, round down
3.47 → 3.5 (to 1 decimal place)
3.44 → 3.4
2,847 → 2,800 (to nearest hundred)
2,850 → 2,900
Significant figures vs decimal places
- Decimal places: count from the decimal point (3.1416 to 2 d.p. = 3.14)
- Significant figures: count from the first non-zero digit (0.004237 to 2 s.f. = 0.0042)
For estimation, significant figures are more meaningful — 3,714,000 to 2 s.f. = 3,700,000.
Rounding strategy for estimation
Round to 1 significant figure first, then adjust:
47 × 83 ≈ 50 × 80 = 4,000 (actual: 3,901 — within 3%)
When you round one factor up, round another down — errors cancel.
Order of Magnitude
The order of magnitude of a number is the power of 10 closest to it.
350 → order of magnitude 10² (hundreds)
7,000 → order of magnitude 10⁴ (thousands)
0.006 → order of magnitude 10⁻³ (thousandths)
More precisely: the order of magnitude is floor(log₁₀(n)).
Why it matters: before doing any calculation, know roughly what size answer to expect. If you’re computing the number of heartbeats in a lifetime and get 10¹², something’s wrong — the answer should be around 10⁹.
The scale ladder
| Power | Size |
|---|---|
| 10⁰ | 1 — one |
| 10¹ | 10 — tens |
| 10² | 100 — hundreds |
| 10³ | 1,000 — thousands |
| 10⁶ | 1,000,000 — millions |
| 10⁹ | 1,000,000,000 — billions |
| 10¹² | trillions |
Moving between adjacent rows means multiplying or dividing by 1,000 (three orders of magnitude) — the difference between millions and billions, or thousands and millions.
Approximation Techniques
Compatible numbers
Replace hard numbers with nearby numbers that are easier to compute:
97 × 4 ≈ 100 × 4 = 400 (then subtract 3 × 4 = 12 → 388)
198 + 347 ≈ 200 + 350 = 550
Breaking apart (distributive property)
17 × 6 = (10 × 6) + (7 × 6) = 60 + 42 = 102
23 × 8 = (20 × 8) + (3 × 8) = 160 + 24 = 184
Halving and doubling
If one number is awkward, halve it and double the other:
35 × 16 = 35 × 16
= 70 × 8
= 140 × 4
= 280 × 2
= 560
Percentage tricks
- 10% of x: move decimal point left one place
- 5%: halve the 10%
- 15%: 10% + 5%
- 1%: move decimal point two places left
- 20%: double the 10%
15% of 240:
10% = 24
5% = 12
15% = 36
Squaring numbers near 50 or 100
51² = (50+1)² = 2500 + 100 + 1 = 2601
49² = (50−1)² = 2500 − 100 + 1 = 2401
101² = 10201
99² = 9801
Difference of squares: a² − b² = (a+b)(a−b)
97 × 103 = (100−3)(100+3) = 100² − 3² = 10000 − 9 = 9991
Checking Answers
Casting out nines
A quick arithmetic check. The digit sum of a correct calculation should work out. If 7 × 8 = 56, then digit sums: 7 × 8 = 56 → 5+6=11 → 1+1=2. And 7 × 8 mod 9 = 56 mod 9 = 2. ✓
Ballpark check
Estimate the answer first, compute, then verify the computed answer is close to the estimate. This catches large errors (wrong order of magnitude, misplaced decimal point).
Dimensional sanity
Make sure the units of your answer make sense. If you’re computing speed and get km², something went wrong.
Sign and parity
- Odd × odd = odd
- Even × anything = even
- Negative × negative = positive
If your calculation of 7 × 9 produces an even number, it’s wrong.
Estimation Under Uncertainty
When you don’t know a value precisely, bracket it:
"The population of London is somewhere between 5M and 15M.
I'll use 10M."
Use geometric midpoints (√(low × high)) for estimates that span orders of magnitude — not arithmetic midpoints. The midpoint between 1M and 100M isn’t 50.5M; it’s √(10⁸) = 10⁴ × √(10⁴) ≈ 10M.
Compound estimation
When multiplying several estimates, errors multiply too — but if errors are random (some high, some low), they partly cancel. A chain of five estimates each within 2× of the true value can produce an answer within 4–5× of truth. That’s usually good enough.
If all estimates are biased in the same direction (all overestimates), errors compound. Be aware of systematic bias — people consistently underestimate time, overestimate crowds, underestimate distances.
Rules of Thumb
These are the ones worth having ready:
| Calculation | Shortcut |
|---|---|
| Multiply by 5 | Divide by 2, multiply by 10 |
| Multiply by 25 | Divide by 4, multiply by 100 |
| Divide by 5 | Multiply by 2, divide by 10 |
| Square a number ending in 5 | (n)(n+1) followed by 25 — e.g., 35² = 3×4 | 25 = 1225 |
| Multiply by 11 | Add adjacent digits — 43 × 11 = 4(4+3)3 = 473 |
| Convert km to miles | Multiply by 0.6 (or 5/8) |
| Convert °C to °F | (°C × 2) + 30 (quick) or (°C × 1.8) + 32 (accurate) |
| Tip at 15% | 10% + half of 10% |
The goal isn’t to memorise all of these — it’s to have enough tools that you’re never stuck computing something by hand when a pattern applies.