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12 Math Tricks to Help You Solve Problems Without a Calculator

Working it out in your head.

Andrew Jamieson

Towards Data Science

Addition + Subtraction

1. addition.

The first trick is to simplify your problem by breaking it into smaller pieces. For example, we can rewrite

It’s often easier to work with adding a smaller number, so instead of 131 + 858, swap the numbers

2. Subtraction

Using the complement of a number can help make subtraction easier. The complement is the difference between the original number and a round number — say 100, 1000.

Here are some examples with the number and its complement compared with 100:

Notice that the second digits add up to 10, and the first digit adds up to 9.

Here is how this is helpful

Written by Andrew Jamieson

Andrew has an analytics background with over 20 years of experience in various industries, working with world-leading brands.

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How to Improve Mental Math Skills

Last Updated: May 9, 2024 Approved

This article was co-authored by Daron Cam . Daron Cam is an Academic Tutor and the Founder of Bay Area Tutors, Inc., a San Francisco Bay Area-based tutoring service that provides tutoring in mathematics, science, and overall academic confidence building. Daron has over eight years of teaching math in classrooms and over nine years of one-on-one tutoring experience. He teaches all levels of math including calculus, pre-algebra, algebra I, geometry, and SAT/ACT math prep. Daron holds a BA from the University of California, Berkeley and a math teaching credential from St. Mary's College. There are 8 references cited in this article, which can be found at the bottom of the page. wikiHow marks an article as reader-approved once it receives enough positive feedback. In this case, 84% of readers who voted found the article helpful, earning it our reader-approved status. This article has been viewed 319,528 times.

Eventually, you'll find yourself in a situation where you'll have to solve a math problem without a calculator. Trying to imagine a pen and paper in your head often doesn't help much. Fortunately there are faster and easier ways to do calculations in your head—and they often break down a problem in a way that makes more sense than what you learned in school. Whether you're a stressed-out student or a math wizard looking for even faster tricks, there's something for everyone to learn.

Break addition and subtraction problems into parts.

• 712 + 281 → "700 + 200," "10 + 80," and "2 + 1"
• 700 + 200 = 9 00, then 10 + 80 = 9 0, then 2 + 1 = 3
• 900 + 90 + 3 = 993 .
• Thinking in "hundreds" or "tens" instead of single digits will make it easier to keep track when digits sum to more than ten. For example, for 37 + 45, think "30 + 40 = 70" and "7 + 5 = 12". Then add 70 + 12 to get 82.

Change the problem to make round numbers.

• Addition : For 596 + 380 , realize that you can add 4 to 596 to round it to 600, then add 600 + 380 to get 980. Undo the rounding by subtracting 4 from 980 to get 976 .
• Subtraction : For 815 - 521 , break it up into 800 - 500, 10 - 20, and 5 - 1. To turn the awkward "10 - 20" into "20 - 20", add 10 to 815 to get 825. Now solve to get 304, then undo the rounding by subtracting 10 to get 294 .
• Multiplication : For 38 x 3 , you can add 2 to 38 to make the problem 40 x 3, which is 120. Since the 2 you added got multiplied by three, you need to undo the rounding by subtracting 2 x 3 = 6 at the end to get 120 - 6 = 114 .

Learn to add many numbers at once.

• For example, 7 + 4 + 9 + 13 + 6 + 51 can be reorganized to (7 + 13) + (9 + 51) + (6 + 4) = 20 + 60 + 10 = 90.

Multiply from left to right.

• For 453 x 4 , start with 400 x 4 = 1600, then 50 x 4 = 200, then 3 x 4 = 12. Add them all together to get 1812 .
• If both numbers have more than one digit, you can break it into parts. Each digit has to multiply with each other digit, so it can be tough to keep track of it all. 34 x 12 = (34 x 10) + (34 x 2) , which you can break down further into (30 x 10) + (4 x 10) + (30 x 2) + (4 x 2) = 300 + 40 + 60 + 8 = 408 .

Try a fast multiplication trick best for numbers 11 through 19.

• Let's look at numbers close to 10, like 13 x 15 . Subtract 10 from the second number, then add your answer to the first: 15 - 10 = 5, and 13 + 5 = 18.
• Multiply your answer by ten: 18 x 10 = 180.
• Next, subtract ten from both sides and multiply the results: 3 x 5 = 15.
• Add your two answers together to get the final answer: 180 + 15 = 195 .
• Careful with smaller numbers! For 13 x 8, you start with "8 - 10 = -2", then "13 + -2 = 11". If it's hard to work with negative numbers in your head, try a different method for problems like this.
• For larger numbers, it will be easier to use a "base number" like 20 or 30 instead of 10. If you try this, make sure you use that number everywhere that 10 is used above. [3] X Research source For example, for 21 x 24, you start by adding 21 + 4 to get 25. Now multiply 25 by 20 (instead of ten) to get 500, and add 1 x 4 = 4 to get 504.

Simplify problems with numbers ending in zero.

• Addition : If all numbers have zeroes at the end, you can ignore the zeroes they have in common and restore them at the end. 85 0 + 12 0 → 85 + 12 = 97, then restore the shared zero: 97 0 .
• Subtraction works the same way: 10 00 - 7 00 → 10 - 7 = 3, then restore the two shared zeroes to get 3 00 . Notice that you can only remove the two zeroes the numbers have in common, and must keep the third zero in 1000.
• Multiplication : ignore all the zeroes, then restore each one individually. 3 000 x 5 0 → 3 x 5 = 15, then restore all four zeroes to get 15 0 , 00 0 .
• Division : you can remove all shared zeroes and the answer will be the same. 60, 000 ÷ 12, 000 = 60 ÷ 12 = 5 . Don't add any zeroes back on.

Easily multiply by 4, 5, 8, or 16.

• To multiply by 5, instead multiply by 10, then divide by 2.
• To multiply by 4, instead double the number, then double it again.
• For 8, 16, 32, or even higher powers of two, just keep doubling. For example, 13 x 8 = 13 x 2 x 2 x 2, so double 13 three times: 13 → 26 → 52 → 104 .

Memorize the 11s trick.

• What is 7 2 x 11?
• Add the two digits together: 7 + 2 = 9.
• Put the answer in between the original digits: 7 2 x 11 = 7 9 2 .
• If the sum is more than 10, place only the final digit and carry the one: 5 7 x 11 = 6 2 7 , because 5 + 7 = 12. The 2 goes in the middle and the 1 gets added to the 5 to make 6.

Turn percentages into easier problems.

• 79% of 10 is the same as 10% of 79. This is true of any two numbers. If you can't find the answer to a percentage problem, try switching it around.
• To find 10% of a number, move the decimal one place to the left (10% of 65 is 6.5). To find 1% of a number, move the decimal two places to the left (1% of 65 is 0.65).
• Use these rules for 10% and 1% to help you with more difficult percentages. For example, 5% is ½ of 10%, so 5% of 80 = (10% of 80) x ½ = 8 x ½ = 4 .
• Break percentages into easier parts: 30% of 900 = (10% of 900) x 3 = 90 x 3 = 270 .

Memorize advanced multiplication shortcuts for specific problems.

• For problems like 84 x 86 , where the tens place is the same and the ones place digits sum to exactly 10, the first digits of the answer are (8 + 1) x 8 = 72 and the last digits are 4 x 6 = 24, for an answer of 7224 . That is, for a problem AB x AC, if B + C = 10, the answer starts with A(A+1) and ends with BC. This also works for larger numbers if all digits besides the ones place are identical. [6] X Research source
• You can rewrite the powers of five (5, 25, 125, 625, ...) as powers of 10 divided by an integer (10 / 2, 100 / 4, 1000 / 8, 10000 / 16, ...). [7] X Research source So 88 x 125 becomes 88 x 1000 ÷ 8 = 88000 ÷ 8 = 11000 .

Memorize squares charts.

• Memorize the squares from 1 to 20 (or higher, if you're ambitious). (That is, 1 x 1 = 1; 2 x 2 = 4; 3 x 3 = 9, and so on.)
• To multiply two numbers, first find their average (the number exactly between them). For example, the average of 18 and 14 is 16.
• Square this answer. Once you've memorized the squares chart, you'll know that 16 x 16 is 256.
• Next, look at the difference between the original numbers and their average: 18 - 16 = 2. (Always use a positive number here.)
• Square this number as well: 2 x 2 = 4.
• To get your final answer, take the first square and subtract the second: 256 - 4 = 252 .

Find useful ways to practice your mental math.

• Flashcards are great for memorizing multiplication and division tables, or for getting used to tricks for specific kinds of problems. Write the problem on one side and the answer on the other, and quiz yourself daily until you get them all right.
• Online math quizzes are another way to test your ability. Look for a well-reviewed app or website made by an educational program.
• Practice in everyday situations. You could add together the total of items you buy as you shop, or multiply the gas cost per volume by your car's tank size to find the total cost. The more of a habit this becomes, the easier it will be.

Joseph Meyer

Exercise your mental math muscles. Improve your math skills by solving daily math problems without using calculators, paper, or counting aids. By solely using your mind and getting into math discussions with your classmates, you will refine your skills and discover new approaches to problem-solving.

Practice Problems and Answers

Community Q&A

• In the real world, you don't always need to know the exact answer. If you're at the grocery store and trying to add 7.07 + 8.95 + 10.09, you could round to the closest whole numbers and estimate that the total is roughly 7 + 9 + 10 = 26. Thanks Helpful 12 Not Helpful 3
• Some people find it easier to think in money than abstract numbers. Instead of 100 - 55, try thinking of a dollar minus a 50¢ coin and a 5¢ coin. Thanks Helpful 6 Not Helpful 9

You Might Also Like

• ↑ http://gizmodo.com/10-tips-to-improve-your-mental-math-ability-1792597814
• ↑ https://www.youtube.com/watch?v=Rgw9Ik5ZGaY
• ↑ https://www.youtube.com/watch?v=SV1dC1KAl_U
• ↑ https://www.youtube.com/watch?v=1JW9BA57aR8
• ↑ http://www.wired.co.uk/article/master-mental-maths
• ↑ https://www.youtube.com/watch?v=YCBTw8KAqkw
• ↑ https://www.scientificamerican.com/article/5-tips-faster-mental-multiplication/
• ↑ Daron Cam. Academic Tutor. Expert Interview. 29 May 2020.

About This Article

One way to improve your mental math skills is to memorize your multiplication and division tables, so you always have the answer to those problems instantly. If you have trouble memorizing the numbers, try creating your own flash cards with blank notecards and asking a friend to help you practice. Another good way to practice your mental math skills is to add up the prices of your items when you’re at the store, and check to make sure you added correctly once the cashier rings you up. You can also try downloading a mental math app like Luminosity to keep your math skills sharp. To learn how to visualize an equation in your head, read on! Did this summary help you? Yes No

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Mental Math: 5 Tricks You Can Do in Your Head Without a Calculator

• by [email protected]
• Published: 06.18.2021

Quick, just off the top of your head, what’s 84% of 25?

If you have to answer this type of question without a calculator, chances are you’re not going to be quick about it all. Chances are, you might also experience some math anxiety that makes you feel overwhelmed, apprehensive, or as if you’re having a mental block that prevents you from solving the exercise.

Whether you’re a parent or a student, you can easily gain some confidence and boost your math skills using a few mental tricks for solving math problems in your head at lightning speed. But first, let’s see why mental math is important and how it helps expedite the problem-solving process.

Why Is Mental Math Important?

Doing basic mental math using low numbers or amounts is a reality of everyday life for most people. For some people, this is almost second nature and it is very useful when shopping, paying a cashier, counting items, or scheduling something.

But beyond these simple applications in daily life, mental math can help solve more complex problems that many students fear.

Knowing a few mental math tricks helps students gain more confidence and overcome math anxiety when they are called to solve problems on the blackboard at school.

Also, there are several other benefits that mental math provides, including:

• Solving math problems faster , especially during competitions or exams when calculators are not allowed and time is limited;
• Increasing mental agility , by keeping the brain engaged in exercises and problem-solving rather than relying on technology for every answer;
• Enhancing number sense , which allows students to mentally manipulate numbers in different ways to successfully solve a problem.

To enable you to make the most of these benefits, here are a few mental math tricks that you can practice every day without a calculator.

Tips and tricks you should know

The simple rules of divisibility.

When you want to evenly split things, knowing these simple divisibility rules can help you quickly solve problems before you even have a chance to whip out a calculator.

• All multiples of 2 are divisible by 2 and they always end in 0, 2, 4, 6, or 8.
• All multiples of 3 are divisible by 3 and the digits always add up to 3 or multiples of 3 like 6, 9, 12, 15, etc.
• All multiples of 4 are divisible by 4 and the last two digits are always divisible by 4 and pass the rule for multiples of 2.
• All multiples of 5 are divisible by 5 and always end in 5 or 0.
• All multiples of 6 are divisible by 6 and pass the same rules for both 2 and 3.
• All multiples of 9 are divisible by 9 and the sum of the digits is always divisible by 9.
• All multiples of 10 are divisible by 10 and the number always ends in a 0.
• All multiples of 12 are divisible by 12 and they pass the divisibility rules for 3 and 4.

Think-of-a-number tricks

If you want to make an impression, think-of-a-number tricks are a great way to show your math skills and make any mental math practice fun. As a student, you can try them with your peers or parents, and as a parent, you can definitely make your child more interested in math with these captivating tricks. For example, here’s one such trick:

• Think of a number.
• Multiply it by 3.
• Divide this number by 3.
• Subtract the original number from the result obtained after dividing by 3.
• The answer is 2.

Here’s another cool think-of-a-number trick you can practice and use to impress your friends:

• Double that.
• Subtract 4.
• Cut that in half.
• Subtract your original number.
• Your result is 1.

Adding 9 to any number

Adding 10 to any number is easier than adding 9. So do exactly that and then subtract 1. For example, if you have to solve 9 + 8, you can add 1 to 9 and make it 10, and then subtract 1 from 8 and make it 7. So instead of solving 9 + 8, you’re solving 10 + 7 which has the same result.

You can apply this trick to bigger numbers as well. For example, you can transform 87 + 99 into 86 + 100 and get the same results. The same goes when you have to add any number that ends in 9.

Multiplying numbers that end in zero

This is a great trick, especially when you deal with bigger numbers. First, you multiply the numbers without the zeros. Next, you add the zeros at the end for the final result. 

For example, if you want to calculate 300 times 500, you start by multiplying 3 times 5, which is 15. Now, all you have to do is add all the zeros you left out. In this case, your answer would be 150,000.

Calculating percentages

If calculating X% of Y doesn’t come easy, try swapping those numbers and calculate Y% of X. For example, if 84% of 25 seems harder to solve, try solving 25% of 84. In this case, you would divide 84, which represents 100% by 4, which represents 25% and get 21.

To make calculating percentages even easier, you can memorize the basic fractions they equal. For example, 10% is 1/10, 12.5% is 1/8, 20% is 1/5, 25% is 1/4, 50% is 1/2, and 75% is 3/4.

For more mental math tricks that can help your child solve problems fast without a calculator, contact us about our math tutoring program .

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Title: gpt can solve mathematical problems without a calculator.

Abstract: Previous studies have typically assumed that large language models are unable to accurately perform arithmetic operations, particularly multiplication of >8 digits, and operations involving decimals and fractions, without the use of calculator tools. This paper aims to challenge this misconception. With sufficient training data, a 2 billion-parameter language model can accurately perform multi-digit arithmetic operations with almost 100% accuracy without data leakage, significantly surpassing GPT-4 (whose multi-digit multiplication accuracy is only 4.3%). We also demonstrate that our MathGLM, fine-tuned from GLM-10B on a dataset with additional multi-step arithmetic operations and math problems described in text, achieves similar performance to GPT-4 on a 5,000-samples Chinese math problem test set. Our code and data are public at this https URL .

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• \mathrm{Kira\:went\:for\:a\:drive\:in\:her\:new\:car.\:She\:drove\:for\:142.5\:miles\:at\:a\:speed\:of\:57\:mph.\:For\:how\:many\:hours\:did\:she\:drive?}
• \mathrm{The\:sum\:of\:two\:numbers\:is\:249\:.\:Twice\:the\:larger\:number\:plus\:three\:times\:the\:smaller\:number\:is\:591\:.\:Find\:the\:numbers.}
• \mathrm{If\:2\:tacos\:and\:3\:drinks\:cost\:12\:and\:3\:tacos\:and\:2\:drinks\:cost\:13\:how\:much\:does\:a\:taco\:cost?}
• \mathrm{You\:deposit\:3000\:in\:an\:account\:earning\:2\%\:interest\:compounded\:monthly.\:How\:much\:will\:you\:have\:in\:the\:account\:in\:15\:years?}
• How do you solve word problems?
• To solve word problems start by reading the problem carefully and understanding what it's asking. Try underlining or highlighting key information, such as numbers and key words that indicate what operation is needed to perform. Translate the problem into mathematical expressions or equations, and use the information and equations generated to solve for the answer.
• How do you identify word problems in math?
• Word problems in math can be identified by the use of language that describes a situation or scenario. Word problems often use words and phrases which indicate that performing calculations is needed to find a solution. Additionally, word problems will often include specific information such as numbers, measurements, and units that needed to be used to solve the problem.
• Is there a calculator that can solve word problems?
• Symbolab is the best calculator for solving a wide range of word problems, including age problems, distance problems, cost problems, investments problems, number problems, and percent problems.
• What is an age problem?
• An age problem is a type of word problem in math that involves calculating the age of one or more people at a specific point in time. These problems often use phrases such as 'x years ago,' 'in y years,' or 'y years later,' which indicate that the problem is related to time and age.

word-problems-calculator

• High School Math Solutions – Systems of Equations Calculator, Elimination A system of equations is a collection of two or more equations with the same set of variables. In this blog post,...

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