Mental Math Techniques. How to Look like a Math Whizz
By: Sofien Kaabar
Tags: Math, Writing, Numbers, Mathematics, Education

Squaring Numbers Ending in 5.

When squaring two-digit numbers that end in 5, the result will have 25 as the last number while the left part will be n x (n+1). Hence, 85 x 85 = (8 x 9) and 25 giving us a result of 7225. Let’s try out some more examples:

15 x 15 = (1 x 2) and 25 = 225.

45 x 45 = (4 x 5) and 25 = 2025.

95 x 95 = (9 x 10) and 25 = 9025.

The above trick only works with two-digit squares that end in five.

Multiplication when Digits add up to 10 with a Condition

The only conditions for the trick to work is to have a double-digit number that have both the right numbers equal to 10 and the left number is the same. The way to find the answer is to multiply the two right number and to do the same thing to the left numbers as we did in the first calculation, . Let’s check the below examples:

73 x 77 = (7 x 8) and (3 x 7) = 5621.

81 x 89 = (8 x 9) and (9 x 1) = 7209.

32 x 38 = (3 x 4) and (8 x 2) = 1216.

Multiplication by 11.

Multiplying by 11 is really simple. Here’s how. Consider 81 x 11. All we have to do is to add the two numbers composing the two-digit number, meaning 8 + 1 = 9. Then put it between the original numbers so that the result is 891. How about 42 x 11? In lightning speed, you’ll answer 462, which is correct. Let’s now see a slightly more complex example. What is 87 x 11? Well, 8 + 7 = 15, does that give us 8157? Not quite, we have to simply carry it so that we have 957. More examples:

56 x 11 = 616.

63 x 11 = 693.

25 x 11 = 275.

Multiplying Two-Digit Numbers Close to 100.

Here’s the trick, subtract the two numbers from 100, giving us a 3 and a 5. Add them together to get 8 and then multiply them together to get 15. For the final step, subtract the added number 8 from 100 giving you 92. The answer is 9215. Trying out more examples:

91 x 96 = (100–9–4) and (4 x 9) = 8736.

88 x 94 = (100–12–6) and (12 x 6) = 8272.

81 x 93 = (100–19–7) and (19 x 7) = 74 and 133, the key here is to add the 1 to 74, giving us 75 and 33. The answer is 7533.

It doesn’t get any simpler than this.

Multiplying by 101 and by 111.

This quick and simple trick is used when you have a two-digit number multiplied by 101. The answer will be that same two-digit number written twice. Here are some examples:

28 x 101 = 2828.

74 x 101 = 7474.

36 x 101 = 3636.

To find the result of a two-digit number multiplied by 111, we simply follow this example. Consider 63 x 111. To find the answer, we sum the 6 and the 3 which gives us 9 then we write it two times between them. Therefore, the answer is 6993. If the two digits add up to more than 10, we carry the two numbers twice. Here’s an example to understand it more:

78 x 111 = 8658. Why? First, we add 7 + 8 giving us 15. Therefore, we have to carry the first time (65 instead of the usual 55 intuition like the above). Next we carry again, so, instead of starting with 7, we start with 8 which gives us 8658.

49 x 111 = 5439. Using the same intuition, we add 4 + 9 for a total of 13 which makes the middle numbers 43. As we carry the second time, we will start with 5. The end result is 5439.

Squaring Two-Digit Numbers

Here’s a very simple technique for squaring two-digit numbers. Suppose we want to find out 34 squared. How should we approach it? Well, first of all, we try to approach an easy number by subtracting or adding. If we want to approach 30, then we subtract 4. Now, we have to also add 4 on another spectrum. This gives us two numbers to keep in mind: 30 and 38. We multiply them together which is 38 x 10 x 3 = 1140. As a final step, we add to this final number the square of the subtracted/added number (which is 4), therefore 16 + 1140 = 1156. Let’s see some more examples:

18 x 18 = 20 x 16 = 320 + 4 = 3240

88 x 88 =90 x 86 = 7740 + 4 = 7744.

46 x 46 = 50 x 42 = 42 x (100/2) = 2100 + 16 = 2116.

199 x 199 = 200 x 198 = 39600 + 1 = 39601.

With a quick method for two-digit numbers between 50 and 59

Consider squaring the number 57. We will square the right digit 7. This gives us 49. Now, add 25 to the 7, thus, giving us 32. The result is 3249. Let’s look at more examples:

Squaring 51 = 1 x 1 = 01 (We have to add the zero to the left when it’s a one-digit number. Now, we add 25 to 1, giving us 26. The result is therefore 2601.

Squaring 59 = 9 x 9 = 81. Now, we add 25 to 9, giving us 34. The result is therefore 3481.

Finding Cube Roots.

When presented with a number and tasked to find its cube, you should drop the right three digits, then find the remaining number’s closest cube (by approximating the ones you memorize). Finally, the last number of the original number is used to find the right digit number of the cube. Math is always better with examples, let’s review and memorize first the cubing of numbers between 1 and 9:

Ignore 704 and focus on 592 for now giving us the closest cube 8 (because 8 x 8 x 8=512), so the first number of the square cube is 8. Now, we focus on the whole number 592704 and we take the digit that has a cube ending with that number, in this case it is 4 (because 4 x 4 x 4 = 64). The answer is 84.

Ignore 625 and focus on 15 for now giving us the closest cube 2 (because 2 x 2 x 2 = 8), so the first number of the square cube is 2. Now, we focus on the whole number 15625 and we take the digit that has a cube ending with that number, in this case it is 5 (because 5 x 5 x 5 = 125). The answer is 25.

Ignore 375 and focus on 857 for now giving us the closest cube 9 (because 9 x 9 x 9 = 729), so the first number of the square cube is 9. Now, we focus on the whole number 857375 and we take the digit that has a cube ending with that number, in this case it is 5 (because 5 x 5 x 5 = 125). The answer is 95.

Finding Percentages

When asked what is 6% of 2400 or 4.5% of 1800, it may seem intimidating but there are shortcuts that really help us either find the exact answer or approximate it. Let’s approach the first calculation.

6% x 2400 = What is 1% of 2400? Remove two zeroes and you’ll get 24. Now, all we have to do is multiply 24 by 6 which is a much simpler task. Simplifying the calculation to (20 x 6) + (6 x 4) gives us 144. So, 6% of 2400 is 144.

4.5% x 1800 = What is 1% of 1800? Well, the same method applies here to find 18. Now, the trick is to find 0.5%, which is simply 1% divided by two, hence, it is 9. So, 4.5% is 18 multiplied by 4 and added to 9. (18 x 4) + 9 = 81.

3% x 5000 = This one is very easy, 1% of 5000 is 50. Let’s multiply it by 3 and find 150 our answer.

1.1% x 1200 = This one is also easy. 1% of 1200 is 12. If we divide that by 10 to find the 0.1%, we get 1.2. Thus, 1.1% of 1200 is 13.2.

87% x 230 = What is 10% of 230? 23, right? Now, multiply it by 8 to get 184. What is 1% of 230? The answer is simply 2.3 which we have to multiply by 7 to get 16.1. The answer is 184 + 16.1 = 200.1

Fractions

One of the most basic tasks in Math is memorizing fractions. Some are easy such as 1/4 and 1/2 while some are difficult such as 3/7 and 5/9. However, there are patterns and shortcuts to follow that will help us calculate the fraction in lightning speed. Let’s start with the most common and interesting ones.

Fractions of 3 are known. Examples:

1 / 3 = 0.33333333333.

2 / 3 = 0.66666666666.

1 / 3 + 2 / 3 = 3 / 3 = 1.00

Fractions of 6 are a little more complicated. Examples:

1 / 6 = 0.16666666666.

2 / 6 = 1 / 3 = 0.33333.

3 / 6 = 1 / 2 = 0.55555.

4 / 6 = 2 / 3 = 0.66666.

5 / 6 = 1 / 6 + 4 / 6 = 0.16666666 + 0.6666666 = 0.833333333.

Fractions of 7 follow a special pattern. Here’s how:

1 / 7 = 0.142857 (Which continues to infinity).

2 / 7 = 0.285714 (Which also continues to infinity).

3 / 7 = 0.428571 (Which also continues to infinity).

4 / 7 = 0.571428 (Which also continues to infinity).

5 / 7 = 0.714285 (Which also continues to infinity).

6 / 7 = 0.857142 (Which also continues to infinity).

Fractions of 9 are simple in that they are repeated to infinity. Examples:

1 / 9 = 0.11111111111.

3 / 9 = 1 / 3 = 0.33333.

7 / 9 = 0.77777777777.

The pattern is always the same but changes place. To memorize it, think about it this way; the higher you go, the higher the number to start with. Check the pattern of the first numbers after the zero (1, 2, 4, 5, 7, 8).

Bonus: Special Number

Also known as Kaprekar’s constant, this number has a special feature if you follow the below steps (Taken from various sources, but let’s just say Wikipedia):

Take any four-digit number (at least two digits should be different).

Arrange the digits in descending and ascending order to get two new four-digit numbers.

to get two new four-digit numbers. Now, subtract the smaller number from the bigger number.

Redo step 2.

If you do this for multiple steps, you will always end up with 6174 and that is the mysterious thing about. Why do we always end up with this number no matter which numbers you start with. Let’s take an example of 2714:

7421 -1247 = 6174

Another example of 3687:

8763 -3678 = 5085;

8550 -0558 = 7992;

9972 -2799 = 7173;

7731 -1377 = 6354;

6543 -3456 = 3087;

8730 -0378 = 8352;

8532 -2358 = 6174

Now, if we have 6174, we will always stay at 6174 because 7641 -1467 = 6174.

It is also a Harshad number, meaning it is divisible by the sum of its constituents: 6174 / (6 + 1 + 7 + 4) = 6174 / 18 = 343. So, that adds to its coolness.