11 Useful, Genius Math Tricks That Are Actually Easy

By Damian Davila on 18 August 2014 (Updated 27 September 2014) 9 comments

"Pure mathematics is, in its way, the poetry of logical ideas," said Albert Einstein. So learning some basic and impressive math must at least be the limericks of logical ideas.

If you want to provide your math skills a major boost, here are 11 useful tricks that you will make you better at math (or at least fake it 'till you make it!), all of which have kick-butt real world applications.

1. Faster Percentage Calculation

Show off by being the one who doesn't bust out the smartphone to calculate the tip. The quickest way to calculate percentages is to multiply numbers first and worry about the two decimal places later. Remember that a "percent" means a fraction out of 100, which means move the decimal two digits to the left.

  • 20 percent of 70? 20 times 70 equals 1400, so the answer is 14.
  • Notice how 70 percent of 20 is also 14.
  • If you need to calculate the percentage of a number, such as 72 or 29, then round up and down to the nearest multiple (70 and 30 respectively) to get a quick estimate.

Multiplying integers is always faster than multiplying decimals.

2. Easy Rules for Divisibility

If you need to be able to decide quickly if 408 slices of pie can be evenly split by 12 people, here are some useful shortcuts. These rules works for all numbers without fractions and decimals.

  • Divisible by 2 if the number's last digit is divisible by 2 (e.g. 298).
     
  • Divisible by 3 if the sum of the digits of the number are divisible by 3 (501 is because 5 + 0 + 1 equals 6, which is divisible by 3).
     
  • Divisible by 4 if the last two digits of the number are divisible by 4 (2,340 because 40 is a multiple of 4).
     
  • Divisible by 5 if the last digit is 0 or 5 (1,505).
     
  • Divisible by 6 if the rules of divisibility for 2 and 3 work for that number (408).
     
  • Divisible by 9 if the sum of digits of the number are divisible by 9 (6,390 because 6 + 3 + 9 + 0 equals 18, which is divisible by 9).
     
  • Divisible by 12 if the rules of divisibility for 3 and 4 work for that number (e.g. 408).

3. Faster Square Roots

Everybody knows that the square root of 4 is 2, but what about the square root of 85?

Give a quick estimate by:

  1. Finding the nearest square. In this case, the square root of 81 is 9.
     
  2. Determining the next nearest square. In this case, the square root of 100 is 10.
     
  3. The square root of 85 is a value between 9 and 10. Since 85 is closer to 81, the actual value must be 9 point something.

4. The Rule of 72

Want to know how long it will take for your money to double at a certain interest rate? Skip the financial calculator and use the rule of 72 to estimate the effects of compound interest.

  • Just divide the number 72 by your target interest rate, and you get the approximate number of years that it will take for your money to double.
     
  • If you were to invest in a 0.9% CD, it would take about 80 years for your money to double.

On the other hand, if you were to invest in a mutual fund with a 7% return, it would take your original funds about 10.28 years to double.

5. The Rule of 115

If double your money sounds too wimpy and you prefer to up the ante by tripling your money, then use the number 115 instead to estimate the number of years it will take your money to triple. For example, an investment at a 5% growth rate would take about 23 years to triple.

6. Figure Out the Hourly Rate

Sometimes to make an apples to apples comparisons between jobs you need to compare the hourly rate of each jobs. For example, if you are able to work the same amount of hours, which job pays better, one with an annual salary of $58,000 or one with a hourly rate of $31?

Figure out the hourly rate of an annual salary by dropping the three zeros and dividing that number by 2. In this case, the hourly rate would be 58/2 = $29. Keeping all other things equal, the $31/hour gig pays better.

7. Advanced Finger Math

You fingers can do more than plain addition and subtraction. If you have problems remembering the multiplication table of 9, try this finger math trick:

  1. Open both of your hands, extending your fingers, in front of you.
     
  2. To multiply 9 by 5, fold down your fifth finger from the left. To multiply 9 by 6, fold down your sixth finger from the left, and on.
     
  3. Get the answer to 9 by 5 by counting your fingers on either side of the bent finger and combining them: 4 and 5 makes 45 and 5 and 4 makes 54.

Now you can quickly figure out the multiplication table of 9 all the way up to 9 times 10.

8. Fast Multiplication by 4

To multiply any number times 4 at lightning speeds: First double the number and then double it again. Let's use this shortcut with 1,223 times 4: double 1,223 is 2,446, and double 2,446 is 4,892.

9. Balanced Average Approach

Instead of using the average formula, you can use the balanced average approach. Think of an average as a target that all items in a list are aiming for and you are trying to balance them out to match that target. For example, let's say that you have 5 exams in your history class and you want to get at least a 92 out of 100. Here are your grades so far:

  • First exam = 81
  • Second exam = 98
  • Third exam = 90
  • Fourth exam = 93

What grade would you need to get on the fifth exam to get a 92 average? Let's add up how much you exceeded or missed your target on every attempt: - 11 + 6 - 2 + 1 equals - 6. To balance your average you need to make up for those - 6 points by making +6 points on top of your target. You need to make 98 on your fifth exam to reach your target grade of 92. Better start studying!

10. Ballpark Fractions

Estimate fractions faster by using easy benchmarks, such as ¼, ⅓, ½, and ¾. For example, 3050 is close to 3060. Since 3060 is ½ and has a bigger denominator than 3050, 3050 must be a little bit bigger than 0.50. (The actual value is 0.60.)

11. The Always-3 Trick

Now here is a party trick in which you can pretend to be Will Hunting.

  • Ask somebody to pick a number.
  • Tell them to double that number.
  • Then, ask them to add 9.
  • Subtract 3.
  • Divide by 2.
  • And finally, to subtract the original number.

No matter whether you use 1, 10, 25, 70, or any other other number, the answer is always 3! Putting your fingers on the side of your head like X-Men's Professor Charles Xavier is highly recommended for dramatic effect.

What is your favorite math trick? Please share in comments!

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Guest's picture
Guest

Thanks for some new tips!

Trick #2, dividing by 12: I think
you mean 408.

Lars Peterson's picture

Good catch, guest. Thanks!

The copy has been corrected.

Damian Davila's picture

Thanks for catching that, Guest! As Lars has indicated, we have updated the article.

Guest's picture
James Davis

Regarding #11:

(2x + 9 - 3)/2 - x = y

(2x + 6)/2 - x = y

x + 3 - x = y

y = 3

Damian Davila's picture

Hi James, nice algebraic explanation of #11!

Guest's picture

Pertaining to number 3, I can provide an additional tip (with explanation) as to how to obtain a more accurate answer than merely "9 point something".

Using the same example of root 85, we must still find the nearest integer root, which is 9. 9 squared is 81, which means 85 is 4 greater than 9 squared.

Now, add half of the difference between 85 and 81 divided by 9. That is, add 2/9 to 9. Squaring this as a check, the answer comes within 0.06%.

This works because of the following:

Root 85 > 9

Let h<0.5

(9+h)^2 = 81 + 18h + h^2

Because h^2 is rather difficult to calculate and necessarily small, we neglect its presence in our approximation. This leaves

81+18h = 85
18h = 4
h = 4/18 = 2/9

As you can see, h is the difference between the number being rooted (85) and the nearest square integer (81) difference by twice the nearest root (which is 2 times 9)

Damian Davila's picture

Nice trick Brendan, thanks for adding it to the list.

Guest's picture
Guest

Regarding number 11; even if they picked a fraction, decimal, irrational number, complex number etc. the trick would still work.

Damian Davila's picture

Great catch Guest! I ran the math one more time and you're 100% correct.
The trick does work with fractions and decimals as well.
Using algebra we can prove it:
Take a number: Let x = the number
Double it: 2x
Add 9: 2x + 9
Subtract 3: 2x + 6
Divide by 2: x + 3
Subtract your original number: 3
That's why your answer is always 3.