Secrets of Mental Math

1. Mental Math is a Joyful and Empowering Skill

But there is joy and lifelong value in being able to do mathematics in your head.

Empowerment and lifelong value. Mental math is not just an academic exercise; it's a practical skill that empowers individuals in various aspects of life. From calculating expenses to estimating distances, the ability to perform math in your head offers a sense of independence and efficiency. Moreover, research suggests that engaging in mental math activities can keep the mind sharp and active as we age, making it a valuable tool for lifelong cognitive health.

Beyond the classroom. While most math education focuses on paper-based problem-solving, mental math provides a unique advantage in real-world scenarios where calculators are not readily available. Whether it's splitting a bill at a restaurant, comparing prices while shopping, or quickly assessing a financial situation, mental math skills enable individuals to make informed decisions on the spot. This practical application of math can boost confidence and improve overall decision-making abilities.

Number sense and standardized tests. Mental math reinforces number sense, which is helpful in all mathematics courses. Success at doing mental calculation and estimation can also lead to improvement on several standardized tests.

2. Master Addition and Subtraction by Thinking Left to Right

The “right” way to do mental math.

Left-to-right approach. Unlike traditional paper-based methods that start from the rightmost digit, mental addition and subtraction are more intuitive and efficient when performed from left to right. This approach aligns with how we naturally read and process numbers, allowing us to keep track of the running total more easily. By breaking down problems into smaller, manageable steps, we can minimize the cognitive load and improve accuracy.

Addition strategies. When adding numbers mentally, start by adding the largest place values first (hundreds, then tens, then ones). This allows you to quickly establish an estimate of the answer and adjust as you move to smaller place values. For example, to add 314 + 159, start by adding 300 + 100 = 400, then add 10 + 50 = 60, and finally add 4 + 9 = 13. Combining these results gives you 400 + 60 + 13 = 473.

Subtraction strategies. Similar to addition, mental subtraction involves working from left to right, subtracting the largest place values first. However, it's often helpful to use complements to simplify the process. For example, to subtract 835 – 497, treat 497 as 500 – 3. Subtract 835 – 500 = 335, then add back 3: 335 + 3 = 338.

3. Unlock Multiplication Through Simplification and Factoring

We calculate from left to right, and add numbers as we go.

Building blocks of multiplication. Mastering 2-digit by 1-digit and 3-digit by 1-digit multiplication is crucial for tackling more complex problems. These smaller calculations serve as the foundation for mental multiplication, allowing you to break down larger problems into manageable steps. By practicing these basic skills, you can develop fluency and confidence in your ability to multiply numbers mentally.

Addition method. The addition method involves breaking down one of the numbers into its place values and multiplying each part separately. For example, to multiply 432 × 3, calculate 3 × 400 = 1200, 3 × 30 = 90, and 3 × 2 = 6. Then, add the results: 1200 + 90 + 6 = 1296. This method is particularly useful when one of the numbers is relatively small.

Factoring method. The factoring method involves breaking down one of the numbers into its factors and multiplying in stages. For example, to multiply 23 × 16, recognize that 16 = 8 × 2. Then, calculate 23 × 8 = 184, and finally multiply 184 × 2 = 368. This method is effective when one of the numbers has easily identifiable factors.

4. Divide and Conquer with Strategic Simplification

When I was a kid, I remember doing lots of 1-digit division problems on a bowling league.

Division as repeated subtraction. Mental division can be approached as a process of repeated subtraction. Start by estimating how many times the divisor goes into the dividend, then subtract that multiple from the dividend. Repeat this process until you reach a remainder that is smaller than the divisor. The number of times you subtracted is the quotient.

Simplifying the problem. Look for opportunities to simplify the division problem before you begin. This might involve dividing both the dividend and divisor by a common factor or doubling both numbers to eliminate decimals. For example, to divide 1234 ÷ 5, double both numbers to get 2468 ÷ 10, which is simply 246.8.

Overshooting. With the problem 770 ÷ 79, we know that 79 × 10 = 790, which is too big by 20. Our ¿ rst answer is 10 with a remainder of –20, but the ¿ nal answer is 9 with a remainder of 79 – 20, which is 59.

5. Guesstimation Provides Practical Everyday Answers

Your body is like a walking yardstick, and it’s worth knowing things like the width of your hand from pinkie to thumb, or the size of your footsteps, or parts of your hand that measure to almost exactly one or two inches or one or two centimeters.

Rounding for quick estimates. Guesstimation is a valuable skill for quickly approximating answers in everyday situations. One common technique is rounding numbers to the nearest whole number, ten, hundred, or thousand, depending on the level of accuracy required. This simplifies the calculations and allows you to arrive at a reasonable estimate without precise measurements.

Order of magnitude. Before performing any calculations, it's essential to determine the order of magnitude of the answer. This involves estimating the number of digits in the result, which can be done by considering the number of digits in the original numbers. For multiplication, the product will have either x + y or x + y – 1 digits, where x and y are the number of digits in the original numbers. For division, the quotient will have either x – y or x – y + 1 digits.

Rule of 70. Suppose a bank offers an interest rate of 3% per year on its savings accounts. You can ¿ nd out how long it will take to double your money using the “Rule of 70”; this calculation is 70 divided by the interest rate.

6. Paper and Pencil Techniques Enhance Mental Math

Even if you haven’t been balancing your checkbook, you might now want to start.

Right-to-left calculations. When performing calculations on paper, it's generally best to follow the traditional right-to-left approach. This allows you to carry digits and manage place values more effectively. However, it's still helpful to maintain a mental estimate of the answer to ensure that your calculations are reasonable.

Complements in subtraction. When doing subtraction on paper, we can make use of complements. Imagine balancing your checkbook; you start with a balance of $1235.79, from which you need to subtract $271.82. First, subtract the cents: 79 – 82. If the problem were 82 – 79, the answer would be 3 cents, but since it’s 79 – 82, we take the complement of 3 cents, which is 97 cents.

Digit-sum check. The digit-sum check can be used to check the answer to a multiplication problem. Let’s try the problem 314 × 159 = 49,926. We ¿ rst sum the digits of the answer: 4 + 9 + 9 + 2 + 6 = 30. We reduce 30 to a 2-digit number by adding its digits: 3 + 0 = 3. Thus, the answer reduces to the number 3. Now, we reduce the original numbers: 314 ĺ 3 + 1 + 4 = 8 and 159 ĺ 1 + 5 + 9 = 15, which reduces to 1 + 5 = 6. Multiply the reduced numbers, 8 × 6 = 48, then reduce that number: 4 + 8 = 12, which reduces to 1 + 2 = 3. The reduced numbers for both the answer and the problem match.

7. Intermediate Multiplication Methods Boost Speed and Accuracy

The reason I like the factoring method is that it’s easier on your memory, much easier than the addition or the subtraction method, because once you compute a number, … you immediately put it to use.

Addition method. With this method, we round that number down to the nearest easy number. With 41 × 17, we treat 41 as 40 + 1 and calculate (40 × 17) + (1 × 17) = 680 + 17 = 697.

Subtraction method. With this method, we treat 89 as 90 – 1 and calculate (53 × 90) – (53 × 1) = 4770 – 53 = 4717. The subtraction method is especially handy when one of the numbers ends in a large digit, such as 7, 8, or 9. Here, we round up to the nearest easy number.

Factoring method. With this method, we factor 22 as 11 × 2. We now have 97 × 11 × 2, and we can use the 11s trick from Lecture 1. The result for 97 × 11 is 1067; we multiply that by 2 to get 2134.

8. Vedic Division Offers a Unique Approach to Complex Problems

There is more to Vedic mathematics than division, although we’ve seen much of it in this course already.

Vedic division. Vedic division is sort of like the subtraction method for multiplication applied to division. Let’s start with the problem 13,571 ÷ 39. The Vedic approach makes use of the fact that it’s much easier to divide by 40 than 39. Dividing by 40 is essentially as easy as dividing by 4. If we divide 13,571 by 4, we get the 4-digit answer 3392.75. (We consider this a 4-digit answer because it has 4 digits before the decimal point.) If we’re dividing 13,571 by 40, we simply shift the decimal point to get a 3-digit answer: 339.275.

Vedic approach. With Vedic division, we start off as follows: 4 goes into 13, 3 times with a remainder of 1. The 3 goes above the line and the 1 goes next to the 5 below the line. Now, here’s the twist: Instead of dividing 4 into 15, we divide 4 into 15 + 3; the 3 comes from the quotient digit above the line. We now have 15 + 3 = 18, and 4 goes into 18, 4 times with a remainder of 2. The 4 goes above the line, and the 2 goes next to the 7 below the line. Again, instead of dividing 4 into 27, we divide 4 into 27 + 4 (the quotient digit above the line), which is 31. We continue this process to get an answer to the original problem of 347 with a remainder of 38.

Why this works. To see why this method works, let’s look at the problem 246,810 ÷ 79. Essentially, when we divide by 79, we’re dividing by 80 – 1, but if the process subtracts off three multiples of 80, it needs to add back three to compensate, just as we saw in the subtraction method for multiplication. The idea behind the Vedic method is that it’s easier to divide by 80 than 79. For this problem, 80 goes into 246, 3 times, so we subtract 240, but we were supposed to subtract 3 × 79, not 3 × 80, so we have to add back 3 before taking the next step.

9. Memorization Techniques Enhance Calculation and Recall

I can tell you from experience that if you use a list a lot, like, say, the presidents or a particular credit card number, then eventually, the phonetic code fades away and the numbers are converted to long-term memory, or you remember the numbers using other contextual information.

Phonetic code. The Major system, which has been in the English language for nearly 200 years, is a phonetic code that assigns consonant sounds to digits. For example 1 gets the t or d sound, 2 gets the n sound, and so on. By inserting vowel sounds, numbers can be turned into words, which make them easier to remember.

Peg system. A way to remember lists of objects, especially when the items of the list are given a number, such as the list of presidents, elements, or constitutional amendments. Each number is turned into a word using a phonetic code, and that word is linked to the object to be remembered.

Computer security. You can use the phonetic code to provide more security to your computer password by adding extra digits. For instance, you might have one password that you like to use, such as BUNNY RABBIT, but you want to make slight alterations for each of your accounts. To adapt the password for your Visa account, you might attach the digits 80,741 (= VISA CARD).

10. Calendar Calculating Unlocks Historical Dates

Sometimes people ask me the days of the week of ancient history, like what day of the week was January 1 in the year 0?

Year codes. Starting with the year 2000, every year gets a code number. The code for 2000 is 0. The codes for Monday through Saturday are 1 through 6; the code for Sunday is 7 or 0. There are also codes for every month of the year: 6 (Jan.), 2 (Feb.), 2 (March), 5 (April), 0 (May), 3 (June), 5 (July), 1 (Aug.), 4 (Sept.), 6 (Oct.), 2 (Nov.), 4 (Dec.). In a leap year, January is 5 and February is 1.

Formula. To determine the day of the week for any year, we use this formula: month code + date + year code. For the date January 1, 2000, we go through these steps: The year 2000 was a leap year, so the month code for January is 5; add 1 for the date and 0 for the year. Those numbers sum to 6, which means that January 1, 2000, was a Saturday. If the sum of the codes and date is 7 or greater, we subtract the largest possible multiple of 7 to reduce it.

Leap years. The general rule for leap years is that they occur every 4 years, with the exception that years divisible by 100 are not leap years. An exception to this exception is that if the year is divisible by 400, then it is still a leap year.

11. Advanced Multiplication Techniques Tackle Enormous Problems

As I promised, these problems are de¿ nitely a challenge!

Squaring 3-digit numbers. To square 3-digit numbers quickly, you must be comfortable squaring 2-digit numbers. Let’s start with 1082. As we’ve seen before, we go down 8 to 100, up 8 to 116, then multiply 100 × 116 = 11,600; we then add 82 (= 64) to get 11,664. A problem like 1262 becomes tricky if you don’t know the 2-digit squares well, because you’ll forget the ¿ rst result (152 × 100 = 15,200) while you try to work out 262. In this case, it might be helpful to say the 15,000, then raise 2 ¿ ngers (to represent 200) while you square 26.

Squaring 4-digit numbers. One way to get better at 3-digit squares is to try 4-digit squares. In most cases, you’ll need to use mnemonics for these problems. Let’s try 23452. We go down 345 to 2000, up 345 to 2690. We then multiply 2000 × 2690, which is (2000 × 2600) + (2000 × 90) = 5,380,000. The answer will begin with 5,000,000, but the 380,000 is going to change.

3-digit-by-2-digit multiplication. The easiest 3-by-2 problems have numbers that end in 0, because the 0s can be ignored until the end. Also easy are problems in which the 2-digit number can be factored into small numbers, which occurs about half the time. To ¿ nd out how many hours are in a typical year, for example, we calculate 365 × 24, but 24 is 6 × 4, so we multiply 365 × 6, then multiply that result by 4.

12. Mental Math Mastery Benefits Everyone

You’ll notice that cube rooting of 2-digit cubes doesn’t really require much in the way of calculation.

Exploiting problem features. One of these lessons is that it pays to look at the numbers in a problem to see if they can somehow help to make the job of ¿ nding a solution easier. Can one of the numbers be broken into small factors; are the numbers close together; or can one of the numbers be rounded to give a good approximation of the answer?

Math of least resistance. We’ve also learned that dif¿ cult addition problems can often be made into easy subtraction problems and vice versa. In fact, if you want to become a “mental mathlete,” it’s useful to try to do problems in more than one way. We can approach a problem like 21 × 29, for example, using the addition, subtraction, factoring, or close-together methods.

Constant simplification. Finally, we’ve learned that mental calculation is a process of constant simpli¿ cation. Even very large problems can be broken down into simple steps. The problem 47,8932, for example, can be broken down into 47,0002 + 8932 + 47,000 × 893 × 2. As we go through this problem, we make use of the criss-cross method, squaring smaller numbers, complements, and phonetic code—essentially, this is the math of least resistance.

Last updated: April 13, 2025