Mental Math – Tips and Tricks for Quick Calculations

This course provides a collection of tips and tricks for performing mental math calculations quickly and efficiently. It is designed for anyone looking to improve their ability to perform arithmetic in their head, focusing on practical methods for (common) mathematical tasks.

Approximating Square Roots

For a rough estimate of the square root of a number, find the nearest perfect squares and use them as reference points.

To estimate $\sqrt{50}$, note that $\sqrt{49} = 7$ and $\sqrt{64} = 8$. $\sqrt{50}$ is closer to 7 since 50 is closer to 49, so in the range of [7, 8] the answer to $\sqrt{50}$ is way closer to 7.

There is a way to approximate the answer on the comma, even for larger numbers.

• Let’s say you want to find the square root of 333.
• We know that the closest round number is 18, because $18^2 = 324$.
• The difference between $333-324=9$.
• Next, you divide the difference (9) by the round square root (18) and you divide this always by 2, so you have $\frac{9}{18} / 2 = \frac{9}{36}$.
• Finally, you add this number on top of the round root (18), so your answer would be $18 + \frac{9}{36} = 18.25$.
• What’s the real answer? It’s 18.248, which is very close to our approximation.

Difference of Consecutive Squares

To subtract squares of consecutive numbers, use the formula

\begin{equation} N^2 – (N-1)^2 = 2N – 1 \end{equation}

$5000^2 – 4999^2 = 2*5000 – 1 = 9999$

Sum All Numbers from A to B

To sum up all numbers from 1 to N, simply use the formula:

\begin{equation} \frac{N*(N+1)}{2} \end{equation}

To sum up all numbers from 1 to 10, we do

\begin{equation} 10*11/2 = 55 \end{equation}

In case you need to sum up all number from 101 to 200, we do

\begin{equation} 200*201/2 – 100*101/2 = 20100 – 5050 = 15050 \end{equation}

Sum Odd Numbers from A to B

To sum up the odd numbers from 1 to N, where N is the last odd number, use the formula:

\begin{equation} (\frac{N+1}{2})^2 \end{equation}

So, what is the sum of all odd numbers from 100 to 200?

Let’s sum up all odd numbers in the range from 0 to 200. The last odd number is 199. To sum up all odd numbers up to 199 is calculated by 

\begin{equation} (\frac{199+1}{2})^2 = 10000 \end{equation}

Second, let’s sum up all numbers in the range from 0 to 100. The last odd number is 99. To sum up all odd numbers up to 99 is calculated by 

\begin{equation} (\frac{99+1}{2})^2 = 2500 \end{equation}

The sum of all odd numbers from 100 to 200 is then given by 10000 – 2500 = 7500.

Multiplying by 9

Use the fact that 9 = 10 – 1. So, for example:

• $9*18 = (10-1)*18 = 180 – 18 = 162$
• $9*37 = (10-1)*37 = 370 – 37 = 333$
• $9*382 = (10-1)*382 = 3820 – 382 = 3438$

The same goes for any number that ends with a nine, for example:

• $29 * 33 = (30-1) * 33 = 990 – 33 = 957$
• $39 * 14 = (40-1) * 14 = 560 – 14 = 546$
• $69 * 26 = (70-1) * 26 = (20*70 + 6*70) – 26 = 1400 + 420 – 26 = 1794$

Multiplying by 11

Use the fact that 11 = 10 + 1. So, for example:

• $11*18=(10+1)*18=180+18=198$
• $11*37=(10+1)*37=370+37=407$
• $11*246=(10+1)*246=2460+246=2706$

The same goes for any number that ends with a one, for example:

• $21*14=(20+1)*14=280+14=294$
• $31*52=(30+1)*52=1560+52=1612$
• $51×87=(50+1)×87=\frac{100*87}{2} + 87 = \frac{8700}{2} + 87 =4.350 + 87 =4437$

Multiplying by 5

To multiply a number by 5, first multiply it by 10 and then divide by 2.

To calculate $39*5$, do $39*10$ and then $390/2=195$

Dividing by 5

To divide a number by 5, multiply it by 2 and then divide by 10.

To calculate $160/5$, multiply the numerator and denumerator by 2, to get $320/10$.

Squaring Numbers Close to 100

For numbers of the form 100 + x, with x being either positive or negative, we have

\begin{equation} (100+x)^2 = 100000 + 200x + x^2 \end{equation}

For example, $97^2$ could be calculated as

\begin{equation} 97^2 = 100000 + 200*(-3) + (-3)^2 \end{equation}

\begin{equation} 97^2 = 100000 – 600 + 9 = 9409 \end{equation}

Percent Tricks

Some percentages should be remembered as follows:

• 1% of a number is just the number divided by 100
• 10% of a number is just the number divided by 10
• 25% of a number is just the number divided by 4
• 50% of a number is just the number divided by 2

From here, every other number can be easily constructed:

• 15% of 200 = 10% of 200 plus halve of 10% of 200 = 20 + 20/2 = 30
• 75% of 400 = 3 times 25% of 400 = 3*100 = 300
• 30% of 600 = 3 times 10% of 600 = 3*60 = 180

Quick Fraction Division to Integer Multiplication

When a number is divided by a fractions, it’s good to remember that:

• if a number is divided by $\frac{1}{2}$, it’s the same as multiplying it by 2
• if a number is divided by $\frac{1}{4}$, it’s the same as multiplying it by 4
• if a number is divided by $\frac{1}{X}$, it’s the same as multiplying it by X
• if a number is divided by $\frac{Y}{X}$, it’s the same as multiplying it by X and dividing it by Y

For example:

• $4 / \frac{1}{2} = 4*2 = 8$
• $8 / \frac{2}{7} = 8*7/2 = 28$