Introduction
Mental math refers to the calculation of simple arithmetic expressions in mind, without using any paper, pencils and calculators. We often encounter some simple arithmetic problems in our daily life — to calculate the bus fare, to do the changes, to split the bill among a group of friends etc.. This is a skill always good to acquire in order to make your life more convenient, or more importantly, to impress your friends by giving out an arithmetic answer before they switch to the calculator app in their phones.
Quick Recap
In Part 1, I have introduced a way to calculate multiplication of two-digit numbers. In this article, we will look at some quicker ways to do multiplication for some special numbers. Let’s begin!
Multiplication by 11 and 111
54 * 11 = ?
There is a very convenient way to do multiplication of 11 (and 111, 1111, …). To do multiplication of 11, what you have to do is to “create a window of digit-length 2”, and sweep it from left to right. Perhaps a diagram would be better to visualize this idea:
In easier terms, the result of 54*11 is give by just simply inserting the answer of 5+4 into the middle of the digits.
Indeed, this idea can be generalized for multiplication of 11 to numbers with more digits:
32768 * 11 = ?
Note that in this example, the addition of some digits exceeded 9, eg. 7+6 and 6+8. In that case, the ten is added to the left-adjacent digit, similar to what to do in vertical addition form.
To further generalize this idea, we can also use this way to do multiplication of 111.
710935 * 111 = ?
Instead of looking at a window of 2, we now look at a window of 3. Similar to the previous example, first we have 7, then 7+1=8, then 7+1+0=8, then 1+0+9=10, then 0+9+3=12, then 9+3+5=17, then 3+5=8, then 5. In other words, we should come up with {7,8,8,10,12,17,8,5} for the digits in the final result, which is 78913785.
Although this method is applicable to any multiplication of 111, 1111, 11111 and so on, we see that the memory slot required grows remarkably larger and is not feasible for everyone. Actually, multiplication of 11 is the only and most frequent use case of this method. Imagine calculating the bill charge after 10% service fee, which means a multiplication of 1.1!
Multiplication with special fractions in mind
We often come across with multiplication of some numbers that are actually feasible to turn it into a problem of simpler division.
- Multiplication of 5 means divide by 2 and times 10
- Multiplication of 25 means divide by 4 and times 100
- Multiplication of 125 means divide by 8 and times 1000
- Multiplication of 625 means divide by 8, then 2, and times 10000
- Multiplication of 14 is approximately divide by 7 times 100
- Multiplication of 18 is approximately divide by 6 times 100
The idea behind is to convert multiplication of a certain number into a fraction with multiplication of 10 at the denominator, so that the problem would be easily converted into a division problem. This method is very handy to calculate exchange rates, from my personal experience.
Conclusion
This is a little trick that helped me do the mental math slightly faster than an average person. Remember: practice makes perfect. Less reliance on calculator in daily life can significantly reduce your computational time when you encounter them in the midst of a computational-heavy math problem. Next, we will look at some important constants to remember and estimation of value of square roots!
Other Parts of the Mental Math Series
Part 1: Two-digit Numbers Multiplication
Part 2: Special Multiplications [This Article]
