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Mental Math Tricks

The Observatory Almanac — Section 19

Every trick here can be learned in about 30 seconds. The goal isn't to replace a calculator — it's to develop numerical intuition: the ability to estimate, check, and sanity-test numbers in your head. That skill is worth more than memorizing multiplication tables.

How to Build These Skills

  1. Read the trick — understand the why, not just the how
  2. Do 3 examples — immediate practice locks it in
  3. Use it in real life — the next bill, the next purchase, the next calculation
  4. Don't use a calculator for small things — that's where practice happens

Part 1: Multiplication Tricks

Multiplying by 11 (any two-digit number)

The trick: For any two-digit number AB, the result is A, (A+B), B — except when A+B ≥ 10, where you carry.

Examples: - 11 × 23: digits are 2 and 3. Middle = 2+3=5. Answer: 253 - 11 × 45: digits are 4 and 5. Middle = 4+5=9. Answer: 495 - 11 × 68: digits are 6 and 8. Middle = 6+8=14. Write 4, carry 1. So: 6+1=7, 4, 8 → 748

Why it works: 11 × AB = 10×AB + AB = the original number shifted plus itself, which creates that middle-digit effect.

Practice: 11×34, 11×72, 11×85, 11×99

Squaring Numbers Ending in 5

The trick: Take the tens digit, multiply it by the next number up, then append 25.

Examples: - 35² → 3 × 4 = 12 → append 25 → 1,225 - 65² → 6 × 7 = 42 → append 25 → 4,225 - 85² → 8 × 9 = 72 → append 25 → 7,225 - 15² → 1 × 2 = 2 → append 25 → 225

Why it works: (10n+5)² = 100n² + 100n + 25 = 100n(n+1) + 25. The 100n(n+1) part is n×(n+1) with two zeros, and you add 25.

Practice: 25², 45², 75², 95², 105²

Multiplying Any Number by 9

Trick 1 (by 9): Multiply by 10, then subtract the original number. - 9 × 47 = 470 − 47 = 423 - 9 × 83 = 830 − 83 = 747

Trick 2 (digit sum check): The digits of any multiple of 9 always sum to 9 (or a multiple of 9). Use this to verify answers. - 423: 4+2+3 = 9 ✓

Multiplying by 5

Divide by 2, then multiply by 10 (or just divide by 2 and move the decimal).

  • 5 × 46 = 46/2 × 10 = 23 × 10 = 230
  • 5 × 73 = 73/2 = 36.5 × 10 = 365
  • 5 × 128 = 128/2 × 10 = 64 × 10 = 640

Multiplying by 4

Double it, then double it again.

  • 4 × 23 = 23×2 = 46 → 46×2 = 92
  • 4 × 37 = 37×2 = 74 → 74×2 = 148

Multiplying Two Numbers Near 100

The trick (Vedic: Base Method) For numbers near 100, find each number's deviation from 100.

Example: 97 × 94 - 97 is 3 below 100; 94 is 6 below 100 - Cross-subtract: 97−6 = 91 (or 94−3 = 91, same result) → first two digits - Multiply deviations: 3 × 6 = 18 → last two digits - Answer: 9,118

Example: 98 × 96 - Deviations: 2 and 4 - Cross: 98−4 = 94 - Multiply: 2×4 = 08 - Answer: 9,408

Works for numbers above 100 too (deviations are positive, add instead of subtract).

The Doubling/Halving Method

When multiplying two numbers, you can double one and halve the other repeatedly until you reach an easy calculation.

  • 32 × 15 = 16 × 30 = 8 × 60 = 4 × 120 = 480
  • 24 × 25 = 12 × 50 = 6 × 100 = 600
  • 18 × 35 = 9 × 70 = 630

Useful for: Any time one number is near a round number after halving/doubling.

Finger Multiplication for 6–10 (The Ancient Trick)

Hold both hands in front of you, fingers spread. Number the fingers from 6 (thumb) to 10 (pinky) on each hand.

To multiply 7 × 8: 1. Touch the "7" finger on one hand to the "8" finger on the other 2. Count the touching fingers plus all fingers below them: that count × 10 = tens digit 3. Multiply the remaining (raised) fingers on each hand together = units digit

Example: 7 × 8 - Touching fingers: 7+8 = 3 touching pairs (the two touching fingers + those below) = 5 total down fingers → 5 × 10 = 50 - Raised fingers: 3 on the 7-hand, 2 on the 8-hand → 3 × 2 = 6 - Answer: 50 + 6 = 56

This works for all combinations from 6×6 to 10×10.

Part 2: Percentage Calculations

The 10% Foundation

Everything in percentage math starts with 10%.

To find 10%: Move the decimal one place left. - 10% of 340 = 34 - 10% of 78 = 7.80 - 10% of 2,500 = 250

Building Other Percentages

  • 5% = Half of 10%
  • 15% = 10% + 5%
  • 20% = 10% × 2
  • 25% = Divide by 4
  • 30% = 10% × 3
  • 1% = Move decimal two places left

Tip Math

Standard tips: 15%, 18%, 20%

For a $47 bill: - 10% = $4.70 - 15% = $4.70 + $2.35 = $7.05 - 20% = $4.70 × 2 = $9.40 - 18% = 20% − 2% = $9.40 − $0.94 = $8.46

Quick 20% tip: Just double the 10% figure.

Even quicker: Round up your bill to a round number first. $47 → $50. 20% of $50 = $10. Close enough.

Discount Math

"30% off $80" — what do you pay?

Method 1: Find 30%, subtract. - 10% of $80 = $8 → 30% = $24 → $80 − $24 = $56

Method 2: Find what percentage you're paying (70%), apply directly. - 70% of $80 = 7 × $8 = $56

Method 2 is faster when the math is clean.

Sales Tax Estimate

If tax is 8%: - 10% of purchase − 2% = approximately 8% - For $60: 10% = $6 → 2% = $1.20 → tax ≈ $4.80 → total ≈ $64.80

The X% of Y = Y% of X Trick

This is surprisingly useful. 8% of 25 is the same as 25% of 8. - 25% of 8 = 2 (easy!) - So 8% of 25 = 2

Use it when the number you want a percentage of is easier to work with as the percentage. - 7% of 50 = 50% of 7 = 3.5 - 4% of 75 = 75% of 4 = 3

Part 3: Divisibility Rules

Check these instantly without dividing:

Divisible by Rule Example
2 Last digit is even 348 ✓ (ends in 8)
3 Sum of digits divisible by 3 612: 6+1+2=9 ✓
4 Last two digits divisible by 4 316: 16÷4=4 ✓
5 Ends in 0 or 5 85 ✓
6 Divisible by both 2 and 3 312: even, 3+1+2=6 ✓
7 Double last digit, subtract from rest; repeat until obvious 203: 20−(3×2)=14 ✓
8 Last three digits divisible by 8 1,312: 312÷8=39 ✓
9 Sum of digits divisible by 9 729: 7+2+9=18 ✓
10 Ends in 0 340 ✓
11 Alternating sum of digits = 0 or multiple of 11 2,915: 2−9+1−5=−11 ✓
12 Divisible by both 3 and 4

The most useful daily: 2, 3, 5, 9. If you split a bill or a group and want a clean split, check divisibility first.

Part 4: Cross-Multiplication (Comparing Fractions)

To compare 3/7 vs. 4/9 without converting to decimals:

Cross-multiply: 3×9 = 27 vs. 4×7 = 28 Since 28 > 27, 4/9 > 3/7.

Real-world use: "Is 3 for $7 or 4 for $9 a better deal?" Same calculation: 3×9=27, 4×7=28. 4/9 is (slightly) more expensive per unit.

Part 5: Estimation Techniques

Order of Magnitude Thinking

Before calculating, ask: "What power of 10 is this answer near?" This catches massive errors.

  • 47 × 62 ≈ 50 × 60 = 3,000 (actual: 2,914 — you knew it was ~3,000)
  • 0.87 × 234 ≈ 1 × 200 = 200 (actual: 203.58)

The Sandwich Method

Bound the answer above and below: - 3.7 × 8.2: must be between 3×8=24 and 4×9=36. Actual: 30.34.

Round to Convenient Numbers

  • $34.87 + $19.13 ≈ $35 + $19 = $54
  • When accuracy matters: $35+$19=$54, then adjust: −$0.13+$0.13 = exactly $54.00

Part 6: Squaring Numbers Near 50

For any number near 50, use: (50+n)² = 2500 + 100n + n²

  • 53² = 2500 + 300 + 9 = 2,809
  • 47² = 2500 − 300 + 9 = 2,209 (note: (50−3)² = 2500 − 100×3 + 9)
  • 51² = 2500 + 100 + 1 = 2,601

Part 7: The Vedic Math "Vertically and Crosswise" Method

For multiplying two 2-digit numbers:

Example: 32 × 41

  1. Multiply units digits: 2 × 1 = 2 (rightmost digit)
  2. Cross-multiply and add: (3×1) + (2×4) = 3+8 = 11 → write 1, carry 1 (middle digit)
  3. Multiply tens digits: 3×4 = 12, plus carry 1 = 13 (leftmost digits)
  4. Answer: 1,312

Another: 23 × 14 1. 3×4 = 12 → write 2, carry 1 2. (2×4)+(3×1) = 8+3 = 11, +1 carry = 12 → write 2, carry 1 3. 2×1 = 2, +1 = 3 4. Answer: 322 ✓

Part 8: Percentage Reverse Calculation

"$34 is 85% of what original price?" (The "original before discount" problem)

Divide the part by the percentage (as a decimal): 34 ÷ 0.85 = ?

Trick: 34/0.85 = 3400/85 ≈ 40. (Actual: exactly 40)

Or use estimation: if $34 is 85%, then 1% ≈ $0.40, so 100% ≈ $40.

Quick Reference Card

Most Used Tricks: - Multiply by 11: A, A+B, B (with carry if needed) - Square ending in 5: n×(n+1) then "25" - 10% trick: move decimal left one - 20% tip: double the 10% - Divisible by 9: digit sum divisible by 9 - Multiply by 9: ×10 then subtract - Multiply by 5: ÷2 then ×10 - Cross-multiply to compare fractions

Daily uses of mental math: - Tipping at restaurants (20% rule) - Checking if a discount is accurate - Splitting bills evenly - Comparing unit prices - Estimating travel time and cost - Checking change received

The goal is intuition, not perfection. A quick estimate that's within 10% is almost always good enough for real-world decisions.