Secrets of Fast & Easy Mathematics #2

This is my second article on easy and fast mathematics. If you have not read my first article then please read that first so you can understand this one much better.

I repeat, fast mathematics is not a trick but a skill. If you really want to be good at it, you have to practice a lot.

So here are some new methods:

Multiply a number between 10 and 20 by a 1 digit number

If you want to multiply a number between 10 and 20 by a 1 digit number then multiply that 1 digit with 10 first, then multiply it by the second digit in the 2 digit number and add the products.

Example: 12 x 6 = (10 x 6) + (2 x 6) = 60 + 12 = 72

Example: 15 x 8 = (10 x 8) + (5 x 8) = 80 + 40 = 120

Multiply two numbers that are both between 10 and 20

If you want to multiply two numbers that are both between 10 and 20, then first add the first number and last digit of second number, multiply the result with 10 and after that, add that result to the product of last digits of both the numbers.

Suppose you want to multiply 12 x 16 then add the first number (12) and last digit of second number (6). It is 12+6=18. Then multiply the result (18) with 10, so it is 18 x 10 = 180. After that, add this result (180) to the product of last two digits (2 x 6 = 12) of both numbers, so it is now 180+12 = 192.

Example: 15 x 16 =
15 +6 = 21
21 x 10 = 210
210 + (5 x 6)
210 + 30
so answer is 240

Example: 18 x 13
18 + 3 = 21
21 x 10 = 210
210 + (8 x 3)
210 + 24
so answer is 234

The art of addition

Here we will talk about the methods related to addition problems.

If you want to add a 2 digit number with 1 digit number then add the 1s digits first. Suppose you want to add 46 + 3. Then add 6 & 3 first, which is 9. So 46 + 3 is 49.

Example 82 + 5: add 1s digits first, 2 +5 = 7, so 82+5 = 87

This is not a method but a thinking pattern. You have to adopt this thinking pattern to solve large calculations. So I am explaining this here.

If you are working addition with 3 digit numbers, it is easy when one or both numbers are multiple of 100

Example: 600 + 236 = 836

Or when both numbers are multiple of 10

Example: 250 + 160 = 410

While adding 3 digit numbers, first add the 100s, then 10s, then 1s.

Example: 124 + 265

124 = 100 (100s number) + 20 (10s number) + 4 (1s number)
265 = 200 (100s number) + 60 (10s number) + 5 (1s number)

So, while adding 124 and 265,
add 265 + 100 = 365 first (use large number first and small later)
then add 365 + 20 = 385 (keep 300 in mind and focus on 65 + 20)
then add 385 + 4 = 389

If you want to add 865 + 592 by adding the 100s, 10s, and 1s digits, but each
step would involve a carry. Addition problems that involve carrying can be turned into easy subtraction problems, as follow:

Example: 865 + 592

592 = 600 – 8

So we can add 865 + 600, then subtract 8 so answer is

865 + 600 = 1465
1465 – 8 = 1457

While doing mental subtraction, we also work one digit at a time from left to right.

Example: 54 – 39
54 – 30 = 24
24 – 9 = 15

A subtraction problem that would normally involve borrowing can usually
be turned into an easy addition problem with no carrying. For 181 – 87,
subtract 90, then add back 3: 181 – 90 = 91 and 91 + 3 = 94.

With 3 digits number, we can use the same method of subtract the 100s, then 10s, then the 1s. For 754 – 232, subtract 200, then 30 and then 2. So

Example: 754 – 232
754 – 200 = 554
554 – 30 = 524
524 – 2 = 522

Three-digit subtraction problems can often be turned into easy addition
problems. For 835 – 497, treat 497 as 500 – 3. Subtract 835 – 500, then add back 3: 835 – 500 = 335 and 335 + 3 = 338.

The concept of compliments

One must understand the concept of compliments to solve large and difficult calculations. The compliment of 75 is 25 because 75 + 25 = 100. To find the compliment of a 2 digit number, find the number that when added to the first digit will yield 9 and the number that when added to the second digit will yield 10.

Example: Find the compliment number for 65.

For 65, notice that 6 + 3 = 9 and 5 + 5 = 10. So it is 35.

If the number ends in 0, such as 60 then find the number, that when added to the first digit will yield 10 instead of 9.

Example: Find the compliment number for 60.

For 60, notice that 6 + 4 = 10. So it is 40.

Always remember that if the number ends in 0, such as 80, then the complement will also end in 0.

Knowing that, let’s try 835 – 467. We first subtract 500 (835 – 500 = 335), but then we need to add back something. How far is 467 from 500, or how far is 67 from 100? Find the complement of 67 (33) and add it to 335: 335 + 33 = 368.

To find 3-digit complements, find the numbers that will yield 9, 9, 10 when added to each of the digits. For example, the complement of 234 is 766. Exception: If the original number ends in 0, so will its complement, and the 0 will be preceded by the 2-digit complement. For example, the complement of 670 will end in 0, preceded by the complement of 67, which is 33; the complement of 670 is 330.

As you practice mental addition and subtraction, remember to work one digit at a time and look for opportunities to use complements that turn hard addition problems into easy subtraction problems and vice versa.

We will discuss more about these methods in next articles.

Thanks.

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