Mental Math Journey
In my quest to get a job, I have been practicing my mental math. I make no claim that these are the best ways to do mental math, and I don’t claim to be an expert, but here are some tricks I’ve been using to improve. All of these tips have come from practicing on Zetamac’s arithmetic trainer using the basic settings. https://arithmetic.zetamac.com/
Here is my progress Google Sheet (I highly recommend keeping track of your progress): https://docs.google.com/spreadsheets/d/1oRcaKdlU0qXqvuRJ7qZQqQ7sFfcHYC9uWN_vXFHEl5k/edit?gid=0#gid=0
(This blog assumes proficiency in the 12 times multiplication table.)
Make 11 the Easiest Number:
Multiplying and dividing by 11 becomes one of the easiest types of problems in Zetamac once you learn these simple tricks. Suppose we get 275 ÷ 11. First, look at the first two digits. We’re looking for the closest multiple of 11 to 27 that is also less than or equal to 27. In this case, it’s 22, which gives us 2 as the first digit of our answer. Next, because we know that each multiple of 11 up to 99 is a repeated digit (like 33, 44, 55), we can determine the next digit must be 5. This gives us 25 as the answer to 275 ÷ 11.
Another example is 407 ÷ 11. We see that 33 is less than 40 and 44 is greater than 40, so the first digit is 3. The last digit of 407 is 7, which we keep. This gives us an answer of 37. Now for multiplication, it’s even easier. Take 53 × 11. The first digit of the answer is 5, the middle digit is 5 + 3 = 8, and the last digit is 3. So the answer is 583. In general, if we have a two-digit number xy, then xy × 11 = x (x + y) y. If there is a carry, just account for it. For example, 56 × 11 becomes 5 + 6 = 11. We carry the 1 to get 616.
Dividing by 12:
In my opinion, dividing and multiplying by 12 is the hardest type of problem in Zetamac. Besides becoming familiar with the 12 times table, I haven’t found many tricks that really speed things up. I do have one tip for division.
Take 1188 ÷ 12. To find the first digit, estimate the usual way. Since 118 is greater than 108 and less than 120, we know the first digit is 9. Then look at the last digit of 1188, which is 8. The only multiples of 12 that end in 8 are ones involving 4 or 9. So how do we choose? Notice that 1188 is much closer to 1200 than to 1080, which suggests the correct second digit is 9. So the full answer is 99
Multiplying by 12:
I find this trick most useful when the number being multiplied by 12 ends in 9. For example, 12 × 29. First, calculate 12 × 20 = 240. Then, since 12 × 9 = 108, we add that to get 348. Another way to do this is 12 × 30 - 12 = 360 - 12 = 348, but I personally find subtraction and borrowing harder, so I prefer the first method. You can choose whichever approach works best for you.
Dividing: Looking for 5:
When dividing, if the number ends in 5, the answer often ends in 5 as well. For example, 245 ÷ 7. Since 5 appears at the end, we suspect the answer ends in 5. Then we find the first digit the regular way, which is 3. So the answer is 35. This also works with 270 ÷ 6. The number ends in 0. The only number besides 10 that multiplies by 6 to get a 0 in the last slot is 5. We find the first digit is 4, giving us 45 as the answer. I recommend always starting with the first digit because it makes typing the answer into Zetamac easier and keeps things clearer in my mind (Benjamin and Shermer 12).
Subtraction and Borrowing the One:
This is one of the more challenging problems to do, and here is my approach.
Suppose we have 153 - 65 = 88. Start with the leftmost digits: 15 - 6 = 9. Then look at the last digits: 3 - 5 requires borrowing. Using a triple (explained below), we know 13 - 5 = 8. So we subtract 1 from 9 to get 8, giving us the final answer of 88.
This is my current method, but I’m transitioning to one of two other approaches depending on how they work out. The second method is finding the “100 pair” for the smaller number. In this case, for 65, the pair is 35 because 65 + 35 = 100. Then add 35 to the remaining 53, which gives 88. This method turns the subtraction into a simpler two-digit addition.
The third method is to round 65 up to 70 and subtract: 153 - 70 = 83. Then add back the 5 we added to make 70, giving us 88. Sometimes I forget how much I added, which makes this method harder for me personally (Benjamin and Shermer 23).
Adding Two Digit Numbers:
This trick is one I developed through practice and later found echoed by Benjamin and Shermer in their book, which is cited below. When we have a problem like 54 + 34, we can break it into parts: first do 50 + 30 = 80, then add 4 + 4 = 8, which gives us a total of 88. This method becomes especially helpful when there’s a carry involved. For example, take 44 + 57. Start with 50 + 40 = 90, then add the remaining digits: 7 + 4 = 11, and 90 + 11 = 101. This process keeps things very clear in your head by handling the tens and ones separately.
I discovered this trick while practicing addition with numbers in the 90s. These can be easier than expected because of how the 9 behaves in mental math. For example, with 92 + 52, think of 9 + 5 = 14, and 2 + 2 = 4. So the total is 144. This works because adding 9 to a number is like adding 10 and subtracting 1. In this case, 90 + 50 = 140, then 2 + 2 = 4, giving us 144.
Starting with the first two digits and then looking at the second digits gives you a quick mental picture of where the answer will land. It makes the whole process feel more intuitive and less error-prone.
(WIP) Adding and Subtracting Triples:
I often look at certain additions and subtractions as triples. For example, 5 + 7 = 12. So I think of 5, 7, and 2 as a triple. This means if we have 27 - 5 = 22 or 27 + 5 = 32, I use the relationships between these numbers to solve quickly.
This doesn’t always work, for instance 25 - 7 = 18, but here 7 + 8 = 15, so 5, 7, and 8 could be another triple. I’m still figuring out how this trick works in general, but it has helped me especially when I struggle with sums like 8 + 6 or 7 + 6.
Works Cited:
Benjamin, Arthur, and Michael Shermer. Secrets of Mental Math: The Mathemagician’s Guide to Lightning Calculation and Amazing Math Tricks. 1st ed., Three Rivers Press, 2006.