Numbers are a part of our everyday lives. Do you really need a calculator for squaring numbers or calculating percentages? It turns out that many math problems can be quickly solved using mental math. Below are top 15 tips on how to do mental math fast!
1. Simplifying your problem: addition
The usual way of adding and subtracting numbers on paper is from right to left: the answer is generated backwards which ensures that you carry a number when digits sum to 10 or above. However, working from left to right may be more suitable for doing math in your head. The first five tips describe how different addition, subtraction, and multiplication problems can be greatly simplified and done without writing them down or using a calculator.
Consider the following problem:
47 + 36 = ?
The calculation can be simplified by noticing that 36 = 30 + 6. Starting with 47, we add 30 first (47 + 30 = 77), and then add the remaining 6 (77 + 6 = 83) to get the answer: 83.
This method can be considerably faster than doing the calculation on paper for more complex problems such as the following:
759 + 496 = ?
Since 496 = 400 + 90 + 6, we can first add 400 (796 + 400 = 1159), then add 90 (1159 + 90 = 1249) then add 6 (1249 + 6 = 1255).
2. Addition: rounding up and subtracting the difference
The last problem, 759 + 496 = ?, requires us to carry numbers at each step. Maybe we should look for an alternative method? Indeed, we may notice that 496 = 500 – 4 and so it would be easier to add more than needed and then subtract the difference! Starting with 759, we add 500 (759+500=1259) and subtract 4 (1259-4=1255) to get the same result much quicker.
3. Simplifying your problem: subtraction
The left-to-right method of calculation can be similarly applied to subtraction. For instance, consider the problem:
86 – 29 = ?
Notice that 29=20+9 and thus we can subtract 20 first (86-20=66) and then subtract the remaining 9 (66-9=57).
This problem can also be simplified by noticing that 29=30-1. Starting with 86, we can subtract 30 first to get 86-30=56 and then add back 1 (56+1=57). As a rule of thumb, if a calculation requires borrowing, then we can round the second number up to the nearest multiple of ten. This allows us to subtract the rounded number and then add the difference.
4. Simplifying your problem: multiplication
The idea of doing calculations on rounded numbers first and then adjusting for the difference helps with addition and subtraction, but what about multiplication? Consider the following problem:
It is much easier to multiply by a multiple of 10, so let’s break down 67×8 into (60+7)×8. Multiply 60×8 to get 480, and then add 7×8=56 to get 480+56=536.
5. Multiplication: rounding up and subtracting the difference
Another example illustrates that we can also round numbers up and then subtract the difference if this yields an easier problem:
Instead of calculating this on paper or by breaking down 78 into 70+8, we may round 78 up to 80 to get (80-2)×9. This is a much more feasible problem as we can multiply 80×9 first to get 720 and then subtract the ‘extra’ 2×9=18, resulting in 720-18=702.
Note that we also could have rounded up the second number to get 78×(10-1)=780-78=702.
6. Multiplying by 4
Instead of multiplying a number by 4 using right-to-left multiplication, one may just multiply the number by 2 two consecutive times. For instance, to calculate 35×4 we can double our number to get 35×2=70 and then double the result again to obtain 140.
7. Multiplying by 5
Multiplying a number by 5 can be done in your head as follows. First divide the number by 2 and then multiply the result by 10. The second step is equivalent to moving the decimal point one place to the right.
For example, to find 68×5 we divide 68 by 2 to obtain 34 and then move the decimal point to the right to get 340.
8. Multiplying by 9
The tip about rounding up and subtracting the difference in multiplication problems is especially useful when multiplying a number by 9. To multiply something by 9, first multiply it by 10 and then subtract the initial number.
For example, to calculate 88×9 first multiply 88 by 10 to get 880 and then subtract the initial number 88 to get 880-88=880-80-8=800-8=792.
9. Multiplying by 11
Multiplying a two-digit number by 11 can be done using the following famous trick. First add the digits of your number and then put the result between the two digits of your initial number.
As an example, consider multiplying 35 by 11. Adding the digits yields 3+5=8 and putting this between the digits we get 385:
This trick works even if the two digits add up to a number greater than 10: just carry the number as usual. For instance, consider the following problem:
The sum of the two digits equals 13 but 94×11≠9134. Instead we should carry the 1 and add it to the first digit. This yields 10 as the first digit, 4 as the second, and 3 as the digit to put in between:
10. Multiplying by 15
Multiplication by 15 can be replaced by an addition problem. To multiply a number by 15, first multiply it by 10 and then add half of the result.
For instance, to calculate 64×15 we multiply our number by 10 to get 640 and then add half of that to obtain 640+320=960.
11. Squaring numbers
To find a square of a number, say 41, we can use the following trick. First calculate the distance from 41 to the closest multiple of 10. For 41 the distance equals 1 since 41-1=40. Next, subtract the distance from your number to obtain 41-1=40 and add the distance to your number to obtain 41+1=42. Multiply these two numbers to get 40×42=1680. Finally, add the distance squared to get the result: 41^2=1680+1^2=1681.
The trick is summarised in the next figure:
12. Squaring numbers that end in 5
The above method for squaring is particularly useful for squaring numbers that end in 5. Consider the square of 35:
Since the distance equals 5, we need to multiply (35+5)×(35-5)=40×30=1200. Adding the square of 5 we obtain 1200+5^2=1225.
This produces a rule for squaring any two-digit number that ends in 5: take the first digit, add one to it, and then multiply these two numbers. Finally, add 25. For example, the square of 75 is 7×8=56 followed by 25 which results in 5625.
Everyday math problems often involve percentages. Some are easy to do: to calculate 10% of a number, just divide it by 10! This is the same as moving the decimal point one place to the left. For instance, 10% of 73 is 7.3. Similarly, calculating 50% is equivalent to dividing by 2.
What about 15% of 50? A great tip for doing such problems is that percentages are reversible:
15% of 50=50% of 15
Calculating 50% of 15 is much easier than 15% of 50 since we can just divide 15 by 2 to get 7.5.
This trick can be combined with one of the previous tips for simplifying multiplication problems. For instance, consider solving the following:
15% of 90= ?
It is easier to reverse the percentages in order to do this problem in your head:
15% of 90=90% of 15
Now recall the “rounding up” trick for multiplication: instead of calculating 90% of 15 we can calculate 100% and then subtract the extra 10%! Since calculating 10% is as easy as moving the decimal point one position to left, we get 15-1.5=13.5.
14. Units: Fahrenheit and Celsius
The last two tips illustrate that sometimes it is easier to get an approximate answer. To convert temperature from Celsius to Fahrenheit, one can double the number and then add 30 to obtain an approximate answer:
For example, to convert 20℃ to Fahrenheit we multiply 20 by 2 and add 30 which yields 70℉. The exact answer is 68℉.
Similar rule can be used to convert from Fahrenheit to Celsius: subtract 30 first and then divide the number by two:
Let’s convert 80℉ to Celsius. Subtracting 30 yields 50 and dividing by 2 results in 25℃. The exact answer is 26.7℃.
15. Interest: how long does it take for your money to double/triple?
Another practical problem is that of calculating interest and investment value. If you invest your money with a fixed annual interest rate, how long will it take for your money to double? While one can use a calculator to find an exact answer, it is possible to estimate it in your head!
The rule of thumb here is called the Rule of 70: to obtain the number of years that will double your money, divide 70 by the rate of interest. For example, let’s assume you find an investment that will pay you 7% interest annually. Since 70÷7=10, it will take around 10 years for your money to double.
A similar method exists for finding the number of years that is required for your money to triple. This rule is called the Rule of 110 which suggests you should divide 110 by the rate of interest. For instance, investing at 5% will triple your money in approximately 110÷5=22 years.