mental_math_tricks.md

Mental Math Mastery: Quick Calculation Tricks & Practice

This guide covers mental math tricks designed to help you solve 2-digit and 3-digit multiplications, percentages, and quick divisions under time pressure, as required in many quantitative interviews.

1. 2-Digit and 3-Digit Multiplication Tricks

Trick 1: Multiplying by 11

How it works: To multiply a 2-digit number by 11, write the first digit, the sum of the two digits, and the last digit. If the sum is greater than 9, carry over the 1 to the first digit. Example: $35 \times 11 \rightarrow 3 \ | \ 3+5 \ | \ 5 \rightarrow 385$ Example with carry: $87 \times 11 \rightarrow 8 \ | \ 8+7 \ | \ 7 \rightarrow 8 \ | \ 15 \ | \ 7 \rightarrow 957$

Practice Questions:

  1. $42 \times 11 = ?$
  2. $75 \times 11 = ?$
  3. $98 \times 11 = ?$

Trick 2: Multiplying Numbers Close to 100

How it works: Find how much less each number is from 100. Subtract the difference of one number from the other number. This gives the first two digits. Multiply the differences together to get the last two digits. Example: $96 \times 93$

  • $100 - 96 = 4$
  • $100 - 93 = 7$
  • First part: $96 - 7 = 89$ (or $93 - 4 = 89$)
  • Second part: $4 \times 7 = 28$
  • Answer: $8928$

Practice Questions:

  1. $98 \times 94 = ?$
  2. $92 \times 95 = ?$
  3. $88 \times 99 = ?$

Trick 3: Squaring a Number Ending in 5

How it works: Take the first part of the number (everything except the 5). Multiply it by the next higher integer. Append 25 to the end. Example: $65^2$

  • First part is 6.
  • Multiply 6 by 7 (the next integer): $6 \times 7 = 42$
  • Append 25: $4225$

Practice Questions:

  1. $35^2 = ?$
  2. $85^2 = ?$
  3. $115^2 = ?$

Trick 4: Vertically and Crosswise (The "Criss-Cross" Method for 2x2)

How it works: Useful for any 2-digit multiplication (e.g., AB x CD).

  1. Multiply the right digits (B x D). Write down the unit digit, carry over the tens.
  2. Cross multiply and add (A x D + B x C). Add the carry. Write down the unit digit, carry over the rest.
  3. Multiply the left digits (A x C). Add the carry. Write it down. Example: $23 \times 14$
  • Right: $3 \times 4 = 12$ (Write 2, carry 1)
  • Cross: $(2 \times 4) + (3 \times 1) = 8 + 3 = 11$. Add carry (1) = 12. (Write 2, carry 1)
  • Left: $2 \times 1 = 2$. Add carry (1) = 3.
  • Answer: $322$

Practice Questions:

  1. $34 \times 21 = ?$
  2. $57 \times 42 = ?$
  3. $83 \times 49 = ?$

2. Percentage Tricks

Trick 1: The "Switching" Rule

How it works: $X%$ of $Y$ is exactly the same as $Y%$ of $X$. Example: What is 16% of 25?

  • Calculate 25% of 16 instead.
  • 25% is $1/4$.
  • $1/4$ of $16 = 4$.

Practice Questions:

  1. 14% of 50 = ?
  2. 32% of 25 = ?
  3. 4.8% of 125 = ?

Trick 2: Breaking Down Percentages (Building Blocks)

How it works: Break a complex percentage into easy chunks: 10%, 5%, and 1%.

  • 10%: Move the decimal point one place left.
  • 5%: Half of 10%.
  • 1%: Move the decimal point two places left. Example: What is 16% of 450?
  • 10% of 450 = 45
  • 5% of 450 = 22.5
  • 1% of 450 = 4.5
  • Add them up: $45 + 22.5 + 4.5 = 72$

Practice Questions:

  1. 15% of 640 = ?
  2. 21% of 850 = ?
  3. 9% of 320 = ?

3. Quick Division Tricks

Trick 1: Dividing by 5

How it works: Multiply the number by 2, then move the decimal point one place to the left (divide by 10). Example: $234 \div 5$

  • Multiply by 2: $234 \times 2 = 468$
  • Move decimal: $46.8$

Practice Questions:

  1. $642 \div 5 = ?$
  2. $138 \div 5 = ?$
  3. $987 \div 5 = ?$

Trick 2: Dividing by 25

How it works: Multiply the number by 4, then move the decimal point two places to the left (divide by 100). Example: $850 \div 25$

  • Multiply by 4: $850 \times 2 = 1700$, $1700 \times 2 = 3400$
  • Move decimal: $34$

Practice Questions:

  1. $625 \div 25 = ?$
  2. $1440 \div 25 = ?$
  3. $330 \div 25 = ?$

Trick 3: Divisibility Rules (For Simplifying Fractions)

  • By 2: Ends in an even number.
  • By 3: Sum of the digits is divisible by 3. (e.g., 144: 1+4+4=9)
  • By 4: The last two digits form a number divisible by 4. (e.g., 316: 16 is divisible by 4)
  • By 9: Sum of the digits is divisible by 9.

Practice Questions (Simplify quickly):

  1. Is 4,236 divisible by 3?
  2. Is 7,128 divisible by 9?

4. General Speed Tips & Drills

  1. Memorize Squares up to 30: E.g., $24^2 = 576$, $25^2 = 625$.
  2. Memorize Cubes up to 12: E.g., $11^3 = 1331$.
  3. Fractions to Percentages: Memorize common equivalents:
    • $1/6 \approx 16.67%$
    • $1/7 \approx 14.28%$
    • $1/8 = 12.5%$
    • $1/9 \approx 11.11%$
    • $1/12 \approx 8.33%$

Practice Under Time Pressure: Set a timer for 1 minute and try to solve as many of the practice questions as you can out loud!

Answer Key

Multiplication (x11): 1) 462, 2) 825, 3) 1078 Multiplication (Close to 100): 1) 9212, 2) 8740, 3) 8712 Squaring (Ending in 5): 1) 1225, 2) 7225, 3) 13225 Criss-Cross: 1) 714, 2) 2394, 3) 4067 Switching Percentages: 1) 7 (50% of 14), 2) 8 (25% of 32), 3) 6 (125% of 4.8 = 4.8 + 1.2) Building Blocks: 1) 96, 2) 178.5, 3) 28.8 Dividing by 5: 1) 128.4, 2) 27.6, 3) 197.4 Dividing by 25: 1) 25, 2) 57.6, 3) 13.2 Divisibility: 1) Yes (4+2+3+6=15), 2) Yes (7+1+2+8=18)