4x4 Tips for Quick and Easy Mental Math: Boost Your Calculation Skills

Intro

Since my childhood, I’ve always done mental math, especially when handling money. I didn’t do anything special, just simple tricks like adding or removing zeros when dealing with multiples of ten or splitting numbers into smaller parts.

Now, as I continue to do mental math, I’ve decided to refine my skills and learn some new tricks. I find mental division particularly challenging, so I’ll explore this topic more and share my findings with you.

Note that division and multiplication are closely related, and some techniques are the same but reversed. However, for clarity, I’ll separate them into distinct sections.

Let’s start with simple tricks and then move to more complex ones.

Multiplication

Multiplying by Single Digits

Multiplying a number by a single digit is easy. Just do it separately for each digit position and then sum the results. For example, to multiply 23 by 4, multiply 20 by 4 and 3 by 4, then sum the results: 20 * 4 + 3 * 4 = 80 + 12 = 92.

More examples:

  • 56 * 7 = 50 * 7 + 6 * 7 = 350 + 42 = 392
  • 123 * 4 = 100 * 4 + 20 * 4 + 3 * 4 = 400 + 80 + 12 = 492

Know the Multiplication Table up to 19×9

You don’t need to memorize this now, but it may be useful for faster calculations. If you don’t know them, you can always calculate them using the previous trick.

Here’s a simple table for reference:

x   2 3 4 5 6 7 8 9
11   22 33 44 55 66 77 88 99
12   24 36 48 60 72 84 96 108
13   26 39 52 65 78 91 104 117
14   28 42 56 70 84 98 112 126
15   30 45 60 75 90 105 120 135
16   32 48 64 80 96 112 128 144
17   34 51 68 85 102 119 136 153
18   36 54 72 90 108 126 144 162
19   38 57 76 95 114 133 152 171

Knowing this table, you can calculate 17 * 8 = 136 instantly.

For larger numbers, like in the previous example, 123 * 4, you can split it into 12 * 4 and 3 * 4, then sum the results: 12 * 4 + 3 * 4 = 48 + 12 = 60.

Multiplication Table Skills Test

I wrote a simple script to test your knowledge. It will prompt you with a multiplication problem, and you have to answer it.

Decompose a Number into Smaller Parts

Sometimes you can decompose a number into smaller parts and then multiply them. For example:

  • 350 * 6 = 350 * 2 * 3 = 700 * 3 = 2100
  • 35 * 16 = 35 * 2 * 8 = 70 * 8 = 560

Or you can decompose a number into a sum of smaller parts and then multiply them. For example:

  • 350 * 6 = 300 * 6 + 50 * 6 = 1800 + 300 = 2100

Let’s say you have two numbers with two digits:

  • 39 * 12 = 39 * 10 + 39 * 2 = 390 + 78 = 468
  • 48 * 21 = 48 * 20 + 48 * 1 = 960 + 48 = 1008

It’s more convenient to split the smaller number.

Power of 2

When you multiply a number by 2, 8, 16, 32, etc., which are powers of 2, you can just multiply the given number by 2 the appropriate number of times. For example:

  • 23 * 8, multiply 23 three times by 2:
    23 * 8 = 23 * 2 * 2 * 2 = 46 * 2 * 2 = 92 * 2 = 184

  • 23 * 16, multiply 23 four times by 2:
    23 * 16 = 23 * 2 * 2 * 2 * 2 = 46 * 2 * 2 * 2 = 92 * 2 * 2 = 184 * 2 = 368

The same goes for division. Just divide by 2 the appropriate number of times:

  • 184 / 8, divide 184 three times by 2:
    184 / 8 = 184 / 2 / 2 / 2 = 92 / 2 / 2 = 46 / 2 = 23

Multiply by 5

Five and tens are the easiest numbers to multiply. To multiply a number by 5, just multiply it by 10 and then divide by 2 or vice versa. First divide it by 2, then multiply it by 10. For example:

  • 23 * 5 = 23 * 10 / 2 = 230 / 2 = 115
  • 56 * 5 = 56 * 10 / 2 = 560 / 2 = 280

or

  • 23 * 5 = 23 / 2 * 10 = 11.5 * 10 = 115
  • 56 * 5 = 56 / 2 * 10 = 28 * 10 = 280

Multiply by 25

To multiply a number by 25, just multiply it by 100 and then divide by 4 or vice versa. First divide it by 4, then multiply by 100. For example:

  • 23 * 25 = 23 * 100 / 4 = 2300 / 4 = 575
  • 56 * 25 = 56 * 100 / 4 = 5600 / 4 = 1400

or

  • 23 * 25 = 23 / 4 * 100 = 5.75 * 100 = 575
  • 56 * 25 = 56 / 4 * 100 = 14 * 100 = 1400

Adding or Removing Multiplication Factors

Sometimes it can be easier to add or remove multiplication factors. For example:

  • 39 * 12 = 12 * 39 = 12 * 40 - 12 = 480 - 12 = 468

I just added 1 to 39 to get 40, then subtracted 12 from the result.

Another example:

  • 41 * 12 = 12 * 41 = 12 * 40 + 12 = 480 + 12 = 492

Division

Decompose a Number into Smaller Parts

Just like with multiplication, you can decompose a number into smaller parts and then divide them. For example:

  • 350 / 6 = 350 / 2 / 3 = 175 / 3 = 58.3

Or you can split the dividend into smaller parts and then divide them. Pay attention that when splitting, you can divide the dividend but not the divisor. For example:

  • 119 / 7 = 119 / 7 = 70 / 7 + 49 / 7 = 10 + 7 = 17

If you have learned the multiplication table up to 19 x 9, you would know this by heart because 7 * 17 = 119.

Round and Adjust

It’s possible to round the dividend and divisor to make the division easier and then adjust the result to get the exact answer.

For example, 92 / 4 can be rounded to 100 / 4 = 25. To adjust the result, subtract the difference between the rounded and the original dividend, 100 - 92 = 8, and divide it by the divisor, 8 / 4 = 2. The final result is 25 - 2 = 23.

More examples:

  • 56 / 5, round to 60 / 5 = 12, adjust 60 - 56 = 4, 4 / 5 = 0.8, 12 - 0.8 = 11.2

Or using different rounding:

  • 56 / 5, round to 55 / 5 = 11, adjust 55 - 56 = -1, -1 / 5 = -0.2, 11 - (-0.2) = 11 + 0.2 = 11.2

Note that if you round up the dividend, you subtract the difference, and if you round down, you add the difference.

Multiply to Get a Better Divisor

Multiply both the dividend and divisor by the same number to get a better divisor. For example, 39 / 5 can be multiplied by 2 to get 78 / 10 = 7.8.

More examples:

  • 56 / 5 = 56 * 2 / 10 = 112 / 10 = 11.2
  • 120 / 15 = 120 * 2 / 30 = 240 / 30 = 24 / 3 = 8

Leverage the Power of 2

When you divide a number by 2, 8, 16, 32, etc., which are powers of 2, you can just divide the given number by 2 the appropriate number of times. For example:

  • 184 / 8, divide 184 three times by 2:
    184 / 8 = 184 / 2 / 2 / 2 = 92 / 2 / 2 = 46 / 2 = 23

Leverage Any Power

You can use the above technique to divide by other powers. For example:

  • 2250 / 25, divide 2250 two times by 5: 2250 / 25 = 2250 / 5 / 5 = 450 / 5 = 90
  • 2250 / 125, divide 2250 three times by 5: 2250 / 125 = 2250 / 5 / 5 / 5 = 450 / 5 / 5 = 90 / 5 = 18
  • 8100 / 18, divide 8100 two times by 9: 8100 / 18 = 8100 / 9 / 9 = 900 / 9 = 100

Squares

Calculating Powers of 2 for Numbers Ending with 5

This is a simple formula for numbers that end with 5 and need to be squared. Multiply the first part of the number by its next number and then add 25 at the end.

For example, 45^2, multiply the first part (4) by its next number (5) and then add 25 at the end:

  • 45^2 = 4 * 5 = 20 and 25 at the end, 2025

More examples:

  • 65^2 = 6 * 7 = 42 and 25 at the end, 4225
  • 85^2 = 8 * 9 = 72 and 25 at the end, 7225

Using Formulas to Calculate the Square of Any Number

This is a more general formula to calculate the square of any number. The formula is (a + b)^2 = a^2 + 2ab + b^2 or (a - b)^2 = a^2 - 2ab + b^2.

The idea is to decompose the number into simpler parts to calculate the square. For example, 42^2 can be decomposed into (40 + 2)^2 and calculated with the formula:

  1. 40^2 = 1600
  2. 2 * 40 * 2 = 160
  3. 2^2 = 4
  4. 1600 + 4 + 160 = 1764

Together, it looks like this:

  • (40 + 2)^2 = 40^2 + 2 * 40 * 2 + 2^2 = 1600 + 160 + 4 = 1764

If you subtract b in the formula, you must subtract 2ab instead of adding it:

  • 39^2 = (40 - 1)^2 = 40^2 - 2 * 40 * 1 + 1^2 = 1600 - 80 + 1 = 1521

Main Pattern: Decomposing and Transforming

The main idea is to simplify calculations by breaking numbers into smaller parts, using rounding and adjusting, leveraging powers, and using multiplication facts. These techniques can make mental math quicker and more efficient.

You can use different techniques that suit you best, here’s an expression solved with two different methods:

  • 56 * 5 = 56 * 10 / 2 = 560 / 2 = 280
  • 56 * 5 = 50 * 5 + 6 * 5 = 250 + 30 = 280

Which one do you prefer?

Practice

Try to practice with any number that comes into your mind or try this simple script to test your knowledge:

Summary

Multiplication Recap

  • Multiply by Single Digits

    • 56 * 7 = 50 * 7 + 6 * 7 = 350 + 42 = 392
    • 123 * 4 = 100 * 4 + 20 * 4 + 3 * 4 = 400 + 80 + 12 = 492

  • Know Multiplications up to 19 x 9

    • 17 * 8 = 136
    • 12 * 4 + 3 * 4 = 48 + 12 = 60

  • Decompose a Number into Smaller Parts

    • 350 * 6 = 350 * 2 * 3 = 700 * 3 = 2100
    • 35 * 16 = 35 * 2 * 8 = 70 * 8 = 560

  • Power of 2

    • 23 * 8 = 23 * 2 * 2 * 2 = 46 * 2 * 2 = 92 * 2 = 184
    • 23 * 16 = 23 * 2 * 2 * 2 * 2 = 46 * 2 * 2 * 2 = 92 * 2 * 2 = 184 * 2 = 368

  • Multiply by 5

    • 23 * 5 = 23 * 10 / 2 = 230 / 2 = 115
    • 56 * 5 = 56 / 2 * 10 = 28 * 10 = 280

  • Multiply by 25

    • 23 * 25 = 23 * 100 / 4 = 2300 / 4 = 575
    • 56 * 25 = 56 / 4 * 100 = 14 * 100 = 1400

  • Adding or Removing Multiplication Factors

    • 39 * 12 = 12 * 39 = 12 * 40 - 12 = 480 - 12 = 468
    • 41 * 12 = 12 * 41 = 12 * 40 + 12 = 480 + 12 = 492

Division Recap

  • Decompose a Number into Smaller Parts

    • 350 / 6 = 350 / 2 / 3 = 175 / 3 = 58.3
    • 119 / 7 = 70 / 7 + 49 / 7 = 10 + 7 = 17

  • Round and Adjust

    • 92 / 4, round to 100 / 4 = 25, adjust: 25 - 2 = 23
    • 56 / 5, round to 60 / 5 = 12, adjust: 12 - 0.8 = 11.2
    • 56 / 5, round to 55 / 5 = 11, adjust: 11 + 0.2 = 11.2

  • Multiply to Get a Better Divisor

    • 39 / 5 = 78 / 10 = 7.8
    • 56 / 5 = 112 / 10 = 11.2
    • 120 / 15 = 240 / 30 = 8

  • Leverage the Power of 2

    • 184 / 8 = 184 / 2 / 2 / 2 = 23

  • Leverage Any Power

    • 2250 / 25 = 2250 / 5 / 5 = 90
    • 2250 / 125 = 2250 / 5 / 5 / 5 = 18
    • 8100 / 18 = 8100 / 9 / 9 = 100

Powers Recap

  • Calculate Powers of 2 for Numbers Ending with 5

    • 45^2 = 4 * 5 = 20 and 25 at the end, 2025
    • 65^2 = 6 * 7 = 42 and 25 at the end, 4225
    • 85^2 = 8 * 9 = 72 and 25 at the end, 7225

  • Use Formulas to Calculate the Square of Any Number

    • (a + b)^2 = a^2 + 2ab + b^2
      • 42^2 = (40 + 2)^2 = 40^2 + 2 * 40 * 2 + 2^2 = 1764
    • (a - b)^2 = a^2 - 2ab + b^2
      • 39^2 = (40 - 1)^2 = 40^2 - 2 * 40 * 1 + 1^2 = 1521

Final Thoughts

It’s just a little fun skill to have. I bet you have your own math tips, and if you are good at math, you probably have even more advanced tricks up your sleeve.

For me, this was a fun exercise, and I think I will probably come back to my own notes, especially for the division part, which I find less intuitive.

Share your tricks!

P.S.: Tips are not 4x4, but I like the sound of it.