Tips & Tricks

Problem

Multiplying two-digit numbers by 11 using traditional methods takes time.

Solution

Add the two digits and place the sum between them. If sum > 9, carry the 1.

Benefit

Reduces calculation time by 80% and can be done mentally.

Example

23 × 11:
2 _ 3 → 2 (2+3) 3 → 253

67 × 11:
6 _ 7 → 6 (6+7) 7 → 6(13)7 → 737

89 × 11:
8 _ 9 → 8 (8+9) 9 → 8(17)9 → 979

Problem

Squaring numbers ending in 5 using traditional multiplication is slow.

Solution

Take the first digit(s), multiply by (itself + 1), then append 25.

Benefit

Instant calculation without long multiplication, saves 90% of time.

Example

25² = 2×(2+1) = 2×3 = 6, append 25 → 625

75² = 7×(7+1) = 7×8 = 56, append 25 → 5625

105² = 10×(10+1) = 10×11 = 110, append 25 → 11025

Problem

Memorizing the 9 times table is difficult for many students.

Solution

Hold up 10 fingers. For 9×n, fold down the nth finger. Fingers left of fold = tens, right = ones.

Benefit

Visual method that works for all single-digit multiplications with 9.

Example

9 × 3:
Fold down 3rd finger → 2 fingers left, 7 fingers right → 27

9 × 7:
Fold down 7th finger → 6 fingers left, 3 fingers right → 63

Problem

Multiplying large numbers by 5 requires long multiplication.

Solution

Multiply by 10, then divide by 2. Or divide by 2, then multiply by 10.

Benefit

Converts difficult multiplication into simple division/multiplication by 10.

Example

86 × 5:
86 × 10 = 860
860 ÷ 2 = 430

Or: 86 ÷ 2 = 43
43 × 10 = 430

Problem

Calculating percentages like 15% or 18% requires complex mental math.

Solution

Break percentages into easy parts: 10%, 5%, 1%. Then combine them.

Benefit

Calculate any percentage mentally by breaking it into simple components.

Example

15% of 80:
10% = 8
5% = 4
15% = 8 + 4 = 12

18% of 200:
10% = 20
5% = 10
1% = 2
3% = 6
18% = 20 + 10 + 10 - 6 = 36

Problem

Testing if large numbers are divisible by 3 requires actual division.

Solution

Sum all digits. If the sum is divisible by 3, the number is divisible by 3.

Benefit

Instant divisibility test without performing division.

Example

Is 2,847 divisible by 3?
2 + 8 + 4 + 7 = 21
21 ÷ 3 = 7 ✓
Yes, 2847 is divisible by 3

Is 5,923 divisible by 3?
5 + 9 + 2 + 3 = 19
19 is not divisible by 3 ✗

Problem

Checking divisibility by 9 for large numbers requires long division.

Solution

Sum all digits. If the sum is divisible by 9, the number is divisible by 9.

Benefit

Quick mental check for divisibility by 9 without calculation.

Example

Is 7,362 divisible by 9?
7 + 3 + 6 + 2 = 18
18 ÷ 9 = 2 ✓
Yes, 7362 is divisible by 9

(In fact: 7362 ÷ 9 = 818)

Problem

Finding square roots mentally for non-perfect squares is difficult.

Solution

Find the two perfect squares it falls between, then estimate proportionally.

Benefit

Get accurate estimates (±0.5) of square roots mentally in seconds.

Example

√50:
49 < 50 < 64
7 < √50 < 8
50 is close to 49, so √50 ≈ 7.1
(Actual: 7.07)

√80:
64 < 80 < 81
8 < √80 < 9
80 is very close to 81, so √80 ≈ 8.9
(Actual: 8.94)

Problem

Multiplying numbers like 23×27 or 41×49 requires full multiplication.

Solution

When tens digit is same and ones add to 10: Multiply tens×(tens+1), then ones×ones.

Benefit

Special case shortcut that saves 80% of calculation time.

Example

23 × 27 (same tens 2, ones add to 10: 3+7=10):
Tens: 2 × (2+1) = 2 × 3 = 6
Ones: 3 × 7 = 21
Answer: 621

41 × 49:
Tens: 4 × 5 = 20
Ones: 1 × 9 = 09
Answer: 2009

Problem

Adding fractions requires finding least common denominator which is time-consuming.

Solution

Use cross-multiplication: (a/b + c/d) = (ad + bc) / (bd)

Benefit

Universal method that works for any fraction addition without finding LCD.

Example

2/3 + 3/4:
(2×4 + 3×3) / (3×4)
(8 + 9) / 12
17/12 = 1 5/12

1/5 + 2/7:
(1×7 + 2×5) / (5×7)
(7 + 10) / 35
17/35

Problem

Multiplying by 99 requires long multiplication with carrying.

Solution

Multiply by 100 and subtract the original number.

Benefit

Converts difficult multiplication into simple subtraction.

Example

46 × 99:
46 × 100 = 4600
4600 - 46 = 4554

73 × 99:
73 × 100 = 7300
7300 - 73 = 7227

Problem

Calculating 15% or 20% tips at restaurants requires mental math under pressure.

Solution

10% = move decimal left. 20% = double that. 15% = 10% + half of 10%.

Benefit

Calculate common tip percentages in 3 seconds without calculator.

Example

Bill: $84.50
10% = $8.45
20% tip = $8.45 × 2 = $16.90

15% tip = $8.45 + $4.23 = $12.68

Bill: $42.00
10% = $4.20
15% tip = $4.20 + $2.10 = $6.30

Problem

The exact formula (C × 9/5) + 32 is difficult to calculate mentally.

Solution

Double the Celsius, subtract 10%, then add 32.

Benefit

Estimates Fahrenheit within ±1 degree using simple mental math.

Example

20°C to °F:
20 × 2 = 40
40 - 4 = 36
36 + 32 = 68°F
(Exact: 68°F)

30°C to °F:
30 × 2 = 60
60 - 6 = 54
54 + 32 = 86°F
(Exact: 86°F)

Problem

Verifying multiplication results requires recalculating the entire problem.

Solution

Use casting out nines: Sum digits of each number repeatedly until single digit, then verify.

Benefit

Quick verification method that catches 90% of calculation errors.

Example

Verify: 234 × 56 = 13,104

234: 2+3+4=9 → 9
56: 5+6=11 → 1+1=2
9 × 2 = 18 → 1+8=9

13,104: 1+3+1+0+4=9 ✓
Answer is likely correct!

Problem

Subtracting from 1000 with borrowing is error-prone.

Solution

Subtract each digit from 9, except the last digit which you subtract from 10.

Benefit

No borrowing needed, eliminates most common subtraction errors.

Example

1000 - 456:
9-4=5
9-5=4
10-6=4
Answer: 544

1000 - 723:
9-7=2
9-2=7
10-3=7
Answer: 277

Problem

Multiplying by 15 requires multiple steps with traditional methods.

Solution

Multiply by 10, then add half of that result.

Benefit

Reduces complex multiplication to simple addition.

Example

24 × 15:
24 × 10 = 240
240 ÷ 2 = 120
240 + 120 = 360

36 × 15:
36 × 10 = 360
360 ÷ 2 = 180
360 + 180 = 540

Problem

Squaring numbers like 48 or 52 requires long multiplication.

Solution

Use (50-a)² = 2500 - 100a + a² or (50+a)² = 2500 + 100a + a²

Benefit

Converts difficult squares into simple arithmetic around 2500.

Example

48² (50-2):
2500 - 200 + 4 = 2304

52² (50+2):
2500 + 200 + 4 = 2704

47² (50-3):
2500 - 300 + 9 = 2209

Problem

Testing divisibility by 4 for large numbers requires division.

Solution

Check only the last two digits. If they form a number divisible by 4, the whole number is.

Benefit

Instantly test any number for divisibility by 4.

Example

Is 5,328 divisible by 4?
Check last 2 digits: 28
28 ÷ 4 = 7 ✓
Yes!

Is 7,862 divisible by 4?
Check last 2 digits: 62
62 ÷ 4 = 15.5 ✗
No!

Problem

Checking divisibility by 6 requires checking both 2 and 3 separately.

Solution

Number must be even AND sum of digits divisible by 3.

Benefit

Combines two simple rules for quick divisibility check.

Example

Is 426 divisible by 6?
Even? Yes (ends in 6)
Digit sum: 4+2+6=12, 12÷3=4 ✓
Yes, divisible by 6!

Is 534 divisible by 6?
Even? Yes
Digit sum: 5+3+4=12, 12÷3=4 ✓
Yes!

Problem

Multiplying by 25 requires complex long multiplication.

Solution

Divide by 4 and multiply by 100 (or multiply by 100 and divide by 4).

Benefit

Converts multiplication into simple division by 4.

Example

32 × 25:
32 ÷ 4 = 8
8 × 100 = 800

48 × 25:
48 ÷ 4 = 12
12 × 100 = 1200

Problem

Figuring out what day of the week a date falls on requires a calendar.

Solution

Use Zeller's congruence simplified: Learn anchor days for each month and count forward/backward.

Benefit

Impress others by calculating any date's day of week mentally.

Example

For 2024 (leap year):
Jan 1 = Monday (anchor)
Feb 1 = Thursday (+31 days = +3 days)
Mar 1 = Friday (+29 days = +1 day)

What day is Feb 14, 2024?
Feb 1 is Thursday
14-1 = 13 days forward
13 ÷ 7 = 1 week + 6 days
Thursday + 6 = Wednesday

Problem

Calculating how long it takes for investment to double requires complex formulas.

Solution

Divide 72 by the interest rate to get approximate years to double.

Benefit

Quick investment doubling time calculation without formulas or calculators.

Example

6% annual return:
72 ÷ 6 = 12 years to double

9% annual return:
72 ÷ 9 = 8 years to double

12% annual return:
72 ÷ 12 = 6 years to double

Problem

Finding cube roots mentally is extremely difficult.

Solution

Memorize cubes 1-10, then find which two cubes the number falls between.

Benefit

Estimate cube roots within ±0.5 by knowing 10 reference points.

Example

Cubes to memorize:
1³=1, 2³=8, 3³=27, 4³=64, 5³=125
6³=216, 7³=343, 8³=512, 9³=729, 10³=1000

∛200:
125 < 200 < 216
5 < ∛200 < 6
200 is closer to 216, so ∛200 ≈ 5.8
(Actual: 5.85)

Problem

Calculating percentage change between two values is confusing and error-prone.

Solution

((New - Old) / Old) × 100. Positive = increase, negative = decrease.

Benefit

Universal formula for all percentage change calculations.

Example

Price increased from $50 to $65:
((65 - 50) / 50) × 100
(15 / 50) × 100 = 30% increase

Price decreased from $80 to $60:
((60 - 80) / 80) × 100
(-20 / 80) × 100 = -25% decrease

Problem

Adding long lists of numbers one at a time is slow and error-prone.

Solution

Look for pairs that sum to 10, group numbers that are easy to add (like 25+75=100).

Benefit

Speeds up mental addition by 50% by reducing the number of operations.

Example

Add: 8 + 14 + 2 + 6 + 25 + 75

Group strategically:
(8 + 2) = 10
(14 + 6) = 20
(25 + 75) = 100

10 + 20 + 100 = 130