Mental math for business – part 02 – fast multiplication and division

This is the second piece of writing in my series of articles on how to use mental math to improve business acumen. In my first article Mental math for business – part 01 – fast addition and multiplication, I have discussed why it is important for business professionals to acquire mental math skills in order to achieve their business goals. It also covers some of the basic techniques such as 2 digit & 3 digit addition, and 2×2 (2 digit by 2 digit) multiplication. Hence I would recommend you to have a look at that article before you continue reading here.

In this article, we will look at some more advanced techniques that would help you take your mental math skills one level higher. These techniques would require you to have a basic understanding of the techniques discussed in part 01.

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3×2 (3 digit by 2 digit multiplication)

Once you are comfortable with your 2×2 (2 digit by 2 digit) multiplication, you are prepared to try 3×2. Let us look at an example.

Take 184 x 27

We would apply the technique of breaking down here as well.

We could write 184 x 27 as (180 + 4) x 27 = (180 x 27) + (4 x 27)

Treat 180 x 27 as a separate 2×2 multiplication problem

180 x 27 = 18 x 27 x 10 = 18 x (20 + 7) x 10 = {(18 x 20) + (18 x 7)} x 10 = 4860

Similarly 4 x 27 = 108

So 184 x 27 = 4860 + 108 = 4968

Let us look at one more example

487 x 37 = (480 + 7) x 37 = (480 x 37) + (7 x 37)

480 x 37 = 48 x 37 x 10

Now try doing 48 x 37 in your head

48 x 37 = 1776, so 480 x 37 = 17760

Similarly 7 x 37 = 259

So 487 x 37 = 17760 + 259 = 18019

Practice a few more examples till you get comfortable with this. Practicing 3×2 multiplication will also make your 2×2 multiplication faster.

Calculating percentages of 3 digit numbers

In part 01, we had discussed how to extend the technique for 2×2 multiplication to find percentages of 2 digit numbers. Now that you have learnt how to do 3×2 multiplication in your head, let us apply it to finding percentages.

Let us try calculating 67% of 276

This is equal to 276 x (67/100) = (276 x 67)/100

Apply the break down method to find the product of 276 and 67

276 x 67 = (270 + 6) x 67 = (27 x 67 x 10) + (6 x 67)

Now 27 x 67 x 10 = (20 +7) x 67 x 10= (1340 + 469) x 10 = 18090

276 x 67 = 18090 + (6 x 67) = 18090 + 402 = 18492

So 67% of 276 = 18492/100 = 184.92

You could practice more examples. With a calculator always handy with us in our mobile phones, you could easily take a random example any time and validate your answer.

Division by one digit number

Being able to multiply numbers fast is really cool. But I feel division is even cooler, as you will be able to find your answers accurate to decimal points.

Here I am assuming that all of you are comfortable with division that does not leave a reminder. Like you would easily know 63 divided by 7 is 9. So I am not going to cover that here. Rather we will look at how to find the output of something like 65 divided by 7.

In this example, you would be left with a reminder of 2. And you would know the answer is between 9 and 10, or 9 point something. The next step is to find the numbers that come after the decimal.

In most of the scenarios, finding the answer accurate to 2 decimal points would serve your purpose. To do that add 2 zeroes to the right of the reminder, in this case 2. 2 becomes 200.

Now divide 200 by 7. You would know that the answer would be between 28 and 29. Here since we are trying to find only 2 digits after the decimal point, we are interested in identifying the first two digits, which is 28 in this case.

So 65 divided by 7 equals 9.28.

Let us look at another example.

Take 84 divided by 9.

The answer is 9 with a reminder of 3

Now to find 3 divided by 9, take 300/9. The answer would be 33 with reminder 3. But we are interested only in the first two digits of the answer, which is 33 in this case.

So 84 divided by 9 would give you 9.33.

You could look at more such examples by yourself.

Division by 2 digit number

In many scenarios, business professionals might have to estimate – if not find the accurate answer – values by dividing 2 or 3 digit numbers by a 2 digit number. Instead of opening a calculator, wouldn’t it be faster and more convenient to do it in your head? This would also help increase your confidence with numbers.

Let us dive straight away into an example.

Take 124 divided by 34. You get 3 as the quotient and 22 as the reminder (34 x 3 = 102, 124 – 102 = 22)

Now you need to find 22/34. Unlike the last case, instead of adding 2 zeroes to 22, we would deal with this in two separates steps.

Step 1 would be to add one zero to 22.

220/34 would give a quotient 6 with a reminder of 16.

Now to find the second number in the answer, take the reminder 16, add a zero and then divide it by 34.

160/34 would give a quotient of 4

So the output of 124/34 is 3.64

Take 748/43 as the second example.

Break down 748 into 430 (43 x 10) and 318 (748 – 430 = 318)

318/43 would give 7 as quotient and 17 as reminder. So 748/43 would give 17 (10 + 7) as quotient and 17 as reminder.

Now divide 17 by 43

170/43 would give 3 as quotient and 41 as reminder (43 x 3 = 129, 170 – 129 = 41)

Now 410/43 would give 9 as quotient.

17/43 gives 0.39 as quotient

So 748/43 = 17.39

Calculating percentages using division

Percentages can also be calculated using the method of division. Let us see how through an example.

Let us say you want to calculate 45% of 147.

You could break down 45 as 4 x 10 and 5 x 1

45% would be (4 x 10%) + (5 x 1%)

10% of a number would that number divided by 10. So 10% of 147 is 14.7.

Similarly 1% of a number is that number divided by 100. So 1% of 147 is 1.47.

45% of 147 would be (4 times 10%) + (5 times 1%)

= (14.7 x 4) + (5 x 1.47) = 58.8 + 7.35 = 66.15

Similarly you could break down any percentages into 10s and 1s using the break down method, and apply it to find percentage of any number.

Multiplication – special cases

So far we have learnt how to use the break down technique to multiply any 3 digit number by a 2 digit number. Now let us look at some special cases where we can apply other methods to find the answer even faster.

Multiplication by 25

25 is a very interesting number. It is equal to 100/4

So multiplying any number by 25 is same as multiplying it first by 100, and then dividing it by 4.  Most of the times, it is easier to divide the number by 4, and then multiply it by 100

Let us look at an example to understand this better.

Take 369 x 25

Divide 369 by 4 first.

Division by 4 could give you only four types of answers – a whole number (when reminder is 0), a number that ends in 0.25 (when reminder is 1), a number that ends in 0.5 (when reminder is 2), and a number that ends in 0.75 (when reminder is 3).

So 369 by 4 = 92.25 (92 x 4 = 368)

Now multiply the answer with 100.

92.25 x 100 = 9225. So 369 x 25 = 9225

Similarly 827 x 25 = 206.75 x 100 = 20675

Multiplication by 50

Multiplication by 50 is even easier than 25. Multiplying by 50 is same as dividing by 2 and then multiplying by 100 (50 = 100/2)

Take 657 x 50

Divide 657 by 2 first. 657/2 = 328.5

328.5 x 100 = 32850

Multiplication by 75

75 = (3/4) x 100

Multiplying by 75 is same as multiplying by 3/4 and then by 100.

For multiplying a number by 3/4, first divide it by 4, and then multiply by it by 3

Let us look at an example now.

Take 486 x 75. This is equal to 486 x (3/4) x 100

486 x (3/4) = (486/4) x 3 = 121.5 x 3 = 364.5

So 486 x 75 = 364.5 x 100 = 36450

Multiplication by 11

Multiplication by 11 is also a special case.

Take 34 x 11

There are two ways in which you could do this.

34 x 11 = 34 x (10 + 1) =  (34 x 10) + (34 x 1) = 340 + 34 = 374

As a shortcut, you could first write the 2 digits of 34 as the first and last digit of the answer.

This means the answer would start in 3 and end in 4 with a number in the middle. And this number would be the sum of the two digits 3 and 4, that is 7 in this case.

So 34 x 11 = 374, and you could write this in one step.

Similarly 45 x 11 = 495

Now take 78 x 11

Here the sum of 7 and 8 is 15, which is more than 10. In such cases, carry the 1 to the first digit of the answer. So the first digit of the answer would be 7 + 1 = 8.

And 78 x 11 = 858.

The takeaway

In this article, we have discussed advanced multiplication and division. And when you apply these techniques to finding percentages, the horizon of application expands. I will be covering different business cases and scenarios where you can use these techniques in a separate article. However, feel free to find your own situations to apply these depending own your business context.

One point to note is that, making good use of mental math requires constant practice. You need not even find a dedicated time for this from your busy life. Try adding, multiplying or dividing numbers that you come across everyday, like numbers on a vehicle number plate, or numbers in your watch or clock, or even your own mobile number. As time passes, with more practice you will see your brain responding faster to arithmetic problems.

Please feel free to leave your comments and feedback. Happy ‘mental mathing’.

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