How to Improve Mental Counting Skills

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This article was co-written by Daron Cam. Daron Cam is a tutoring teacher and founder of Bay Area Tutors, Inc., a company providing tutoring services in the San Francisco Bay Area in subjects such as math, science, and general academics. . Daron has more than eight years of teaching math in schools and more than nine years of individual tutoring experience. He teaches all levels of math including calculus, pre-algebra, I algebra, geometry, math prep for the SAT/ACT. Daron holds a bachelor’s degree from the University of California, Berkeley and a math teaching certificate from the University of St. Mary.

There are 8 references cited in this article that you can see at the bottom of the page.

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There will come a time when you will need to solve a math problem without a calculator at hand. Visualizing paper and pen in your head is also not a good way because it doesn’t help much. Fortunately, there are faster and easier ways to do mental math without a calculator – and these methods often break down the problem in a way that makes more sense than what you learned in school. Everyone is going to learn something, whether it’s a student who’s scratching their head with homework or a math wizard looking for tricks to do math faster.

Table of Contents

Divide addition and subtraction problems into parts

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Add the hundreds, tens, and units numbers separately. Treat each group as a separate problem:

712 + 281 → “700 + 200,” “10 + 80,” and “2 + 1”
700 + 200 = 9 00, followed by 10 + 80 = 9 0, then 2 + 1 = 3
900 + 90 + 3 = 993 .
Calculating “hundreds” or “tens” instead of odd numbers makes it easier to remember if the sum of the odd numbers is greater than 10. For example, for the problem 37 + 45, we would calculate “30 + 40 = 70” and “7 + 5 = 12”. Then calculate 70 + 12 to get the answer of 82.

Change the calculation to round numbers

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Change the calculation to include round numbers, then correct it when you’re done. Most people calculate round numbers much faster. Keep in mind the numbers you changed to get the correct answer in the end. ^{[1] X Research Source} Example:

Addition : For the 596 + 380 problem, we know that we can add 4 to 596 to round the number up to 600, then calculate 600 + 380 to get 980. Return the rounded number by subtracting 4. from 980 to get 976 .
Subtraction : For problem 815 – 521 , we divide the calculation into parts 800 – 500, 10 – 20, and 5 – 1. To turn the awkward “10 – 20” into “20 – 20”, let’s add 10 to 815 to get 825. Solve this calculation, we get 304, then return the rounded number by subtracting 10 to get 294 .
Multiplication : For the 38 x 3 problem, we can add 2 to the number 38 so that the calculation becomes 40 x 3, which gives the result 120. Since the 2 previously added has already been multiplied by 3, we will have to multiply. return the rounded number by subtracting 2 x 3 = 6 to get 120 – 6 = 114 .

Learn how to add multiple numbers at once

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Rearrange the order of the numbers for easy addition. The addition problem won’t change no matter what order you solve it in. Find the numbers that add up to 10 or some round number:

For example, the calculation 7 + 4 + 9 + 13 + 6 + 51 can be rearranged as (7 + 13) + (9 + 51) + (6 + 4) = 20 + 60 + 10 = 90.

Multiply from left to right

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Notice where the hundreds, tens, and units numbers are. When doing calculations on paper, most people multiply the units first, then multiply from right to left. However, when doing mental calculations, it’s easier to multiply in a different way:

For the 453 x 4 problem, let’s start with 400 x 4 = 1600, followed by 50 x 4 = 200, then 3 x 4 = 12. Add it all up, and you have 1812 .
If both numbers have more than one digit, you can divide them into parts. Each digit must be multiplied by each other, so it will be difficult for you to remember them all. 34 x 12 = (34 x 10) + (34 x 2) , which can be further broken down to (30 x 10) + (4 x 10) + (30 x 2) + (4 x 2) = 300 + 40 + 60 + 8 = 408 .

Try quick multiplication tips for numbers 11 to 19

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Try the method of turning one difficult problem into two easier problems. This is another way to break the problem down into parts. It’s a little hard to remember at first, but once you get used to it you can do multiplication much faster. Learning how to multiply two numbers that are both between 11 and 19 is the easiest, but you can also use it to solve other problems: ^{[2] X Research Source}

Take for example numbers close to 10, such as 13 x 15 . Take the second number minus 10, then add the result with the first number, we have: 15 – 10 = 5, then 13 + 5 = 18.
Multiplying the above result by 10, we have: 18 x 10 = 180.
Next, subtract both sides by 10 and multiply the two results together: 3 x 5 = 15.
Add these two results together to get the final answer: 180 + 15 = 195 .
Be careful with smaller numbers! With the calculation of 13 x 8, we will start with “8 – 10 = -2”, then “13 + -2 = 11”. If you find it difficult to do mental arithmetic with negative numbers, try another method for solving these kinds of problems.
For larger numbers, it’s easier to use a “base number” like 20 or 30 instead of 10. If you try this, don’t forget to use that number in place of all the places where 10 is used above. ^{[3] X Research Source} For example, for the 21 x 24 problem, we start by adding 21 + 4 to get 25. Multiply 25 by 20 (instead of 10) to get 500, then add 1 x 4 = 4 for an answer of 504.

Simplify calculations with numbers ending in 0

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If there are numbers ending in 0, you can skip the 0 until the end of the calculation:

Addition : If all numbers end in 0, you can omit the zeros that the numbers all have, and then return it back at the end of the calculation. 85 0 + 12 0 → 85 + 12 = 97, then return the number 0: 97 0 .
The subtraction does the same thing: 10 00 – 7 00 → 10 – 7 = 3, then returns the two zeros to get 3 00 . Note that you can only omit the two zeros that all have and must keep the third zero in 1000.
Multiplication : skip all zeros, then return all. 3 000 x 5 0 → 3 x 5 = 15, then redo all four zeros to get the answer 15 0 , 00 0 .
Division : You can remove all the zeros that all numbers have, and the answer remains the same. 60,000 ÷ 12,000 = 60 ÷ 12 = 5 . Do not return any zeros to the answer.

Quickly multiply a number by 4, 5, 8, or 16

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Memorize the tips with the number 11

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You can multiply a two-digit number by 11 without much calculation. Add the two digits of that number together, then put the result between the two original digits: ^{[4] X Research Source}

How much is 7 2 x 11?
Add two digits together: 7 + 2 = 9.
Put the result between the initial two digits: 7 2 x 11 = 7 9 2 .
If the sum of those two digits is greater than 10, we just put the last digit between the first two and remember the remaining digit: 5 7 x 11 = 6 2 7 , because 5 + 7 = 12. Digits 2 is placed between the two digits, and the number 1 is added to 5 to get the number 6.

Turn a percentage problem into an easier calculation

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Know which percent problems are easier to do mental math. There are a few helpful tips you should know: ^{[5] X Research Sources}

79% of 10 will be equal to 10% of 79. This is true for any two numbers. If you can’t find an answer to a percentage problem, try again.
To find 10% of a number, move the comma to the left one row (10% of 65 is 6.5). To find 1% of a number, shift the comma two rows to the left (1% of 65 is 0.65).
Apply these rules to 10% and 1% to make percentage problems more difficult. For example, 5% is ½ of 10%, so 5% of 80 = (10% of 80) x = 8 x ½ = 4 .
Breaking the percentage problem into parts is easier: 30% of 900 = (10% of 900) x 3 = 90 x 3 = 270 .

Memorize advanced multiplication shortcuts to solve specific math problems

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These tips are very effective, but only to a limited extent. They can turn a seemingly impossible mental math task into a simple task, even though only applicable to a very small percentage of problems. Learn these tips if you already have pretty good mental math skills and want to get up to the speed of a “mathematician”:

For a problem like 84 x 86 , when two numbers with the same tens digit and the ones digit add up to exactly 10, the first digits of the answer will be (8 + 1) x 8 = 72, and the last digits of the answer are 4 x 6 = 24, we have a final answer of 7224 . Thus, for the problem AB x AC, if B + C = 10, the answer will have the first digits A(A+1) and the last digits BC. The same rule holds for larger numbers if all the digits next to the units line are the same. ^{[6] X Research Sources}
You can rewrite powers of 5 (5, 25, 125, 625, …) as powers of 10 divided by an integer (10 / 2, 100 / 4, 1000 / 8, 10000 / 16). , …). ^{[7] X Research Source} Thus, 88 x 125 becomes 88 x 1000 ÷ 8 = 88000 ÷ 8 = 11000 .

Memorize the square table

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The squares table gives you a new way of multiplying. The multiplication table from 1 to 9 makes multiplication of single digit numbers automatic. With larger numbers, instead of trying to remember hundreds of answers, it’s more efficient to just remember the squares (each multiplied by itself). With a little effort, you can use these squares to find answers to other problems: ^{[8] X Research Source}

Memorize squared numbers from 1 to 20 (or greater if you’re ambitious). (That’s 1 x 1 = 1; 2 x 2 = 4; 3 x 3 = 9, and so on.)
To multiply two numbers, we first find their average (the number in the middle of the two numbers). For example, the average of 18 and 14 is 16.
Square the above result. Once you have the squares table, you will know immediately that 16 x 16 is 256.
Next, find the difference between the original numbers and their average: 18 – 16 = 2. (Here we always use positive numbers.)
Square this whole number: 2 x 2 = 4.
To get the final answer, subtract the second squared from the first square: 256 – 4 = 252 .

Find useful ways to practice mental arithmetic

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Practice forever to be talented, diligently all good. If you want to increase your confidence and mental speed, you need to make an effort to use these skills at least 2-3 times a day. The following suggestions can help you train more effectively:

The flashcards will be very useful to learn multiplication and division times tables or to familiarize yourself with tips for solving typical math problems. Write the math problem on one side and the answer on the other, and ask yourself questions every day until you get it all right.
Solving math puzzles is another way to test yourself. Look for well-reviewed apps or educational programs websites.
Practice in everyday situations. You can add items when you go shopping or multiply the price of gas by volume by the volume of your car’s gas tank to get the total amount to pay. The more often you practice as a habit, the easier it will be.

Advice

In everyday life, you don’t always have to know the correct answer. If you are at the grocery store and want to add 71,000 + 88,000 + 103,000 VND, you can round the numbers to the nearest number and estimate the total amount 70,000 + 90,000 + 100,000 = 260,000 VND.
Many people find it easier to calculate money than to do subtraction. Instead of counting 100 – 55, try calculating 1 dollar minus a 50¢ coin and a 5¢ coin.

Steps