Mental Math Techniques

Question 1

Squaring a 2 digit number

Answer 1

1. Square each digit in the number and put them next to each other.
2. Multiply the two digits and then double them.
3. Put the answer of #2 as the middle two digits.
4. Add to get the squared value.

For example:

84^2

6416 + 0640

7056

Question 2

Squaring a 3 digit number

Answer 2

1. Square each digit in the number and put them next to each other to get a 6-digit number.
2. Multiply and double the first and second number to get the 2nd and 3rd values.
3. Multiply and double the second and third number to get the 4th and 5th values.
4. Multiply and double the first and third digits to get the 3rd and 4th digits on a 3rd layer.
5. Add.

For example:

367^2
093649 + 036840 + 004200
134689

https://youtu.be/HWF1y5FfGXo

Question 3

Multiplying any number by another number

Answer 3

1. Make it such that both numbers have the same number of digits: ex. if multiplying 356 by 28, make it 365 by 028
2. Multiply the ones place by ones place values
3. For the number of digits it is, keep going and carry over multidigit numbers if necessary for all the possible combinations of digits that can result in a value in a certain place value. For example, in 365 by 028, for the tens values, add 5 * 2 and 6 * 8 and put that number in the tens place. Carry over any multidigit values. For the hundreds place add 5 * 0 + 6 * 2 + 3 * 8 because ones by hundreds and tens by tends can result in values in the hundreds.
4. Keep going for all the place values until the last digits of the number has been reached (which you would get from multiplying the leftmost digits of each number to get the highest place value).

Example:

436 * 589

1. 6 * 9 = 54. 4 in ones place and carry over 5.
2. 3 * 9 + 6 * 8 + 5 = 80. 0 in tens place and carry over 8.
3. 4 * 9 + 3 * 8 + 6 * 5 + 8 = 98. 8 in hundreds place and carry over 9.
4. 4 * 8 + 5 * 3 + 9 = 56. 6 in thousands place and carry over 5.
5. 4 * 5 + 5 = 25. 5 in ten thousands place and 2 in ten thousands place.
6. Final answer: 256804

Question 4

Adding any fractions

Answer 4

1. Multiply the denominators of both fractions- this is the denominator of the sum fraction.
2. Multiply the numerator of one fraction with the denominator of the other fraction. Do this for both diagonals.
3. Add the values from Step 2, which will be the numerator of the sum fraction!

Example:

5/30 + 9/20

Denominator: 30 * 20 = 600

Numerator: 5 * 20 + 9 * 30 = 370

Final sum: 370/600

Question 5

Factoring A Quadratic

Answer 5

1. Find the two factors that multiply to A * C and add to B values.
2. Negate each of the two factors and divide them by the A value.
3. Those values are the roots to the equation!

Example:

2x^2 + 9x + 10

Factors: 5, 4

Final roots: -5/2, -4/2 or -2.5, -2

Question 6

x% of y is equal to…

Answer 6

y% of x

For example:

32% of 50 = 50% 32 = 16

Question 7

Find any percentage of a value

Answer 7

1. Multiply the nonzero leading numbers of both the percentage and the value being changed by the percentage.
2. This is the new value.

• BUT: It assumes you have two place values of 0 because it bypasses the dividing by 100.

For example:

WORKS:
20% of 130

2*13 = 26

DOESN’T WORK:

1. 65% of 30 (only 1 zero place value)

So here you do 65 * 3 and then divide by 10 to account for the missing place value. Final answer is 19.5.

2. 20% of 13 (only 1 zero place value)

So here you do 2 * 13 and then divide by 10 to account for the missing place value. Final answer is 2.6.

Question 8

Dividing by 9 (without Remainders)

Answer 8

1. Take the first digit of the dividend as the first digit of the quotient.
2. For each next digit of the quotient, calculate it by adding the sum of all previous digits of the dividend in the number (do for all digits except for the last digit of the dividend).
3. For the last digit of the dividend, divide the sum of all digits by the divisor and add it in the ones place column (not a new place value) to get the final answer.

Example:

1321101 / 9

146788 + 000001

146789

Question 9

Squaring ANY number ending in 5

Answer 9

1. Multiply the digits before the last digit (5) by 1 + the number that the previous digits form. That number is the first few digits of the answer.
2. The next digits of the answer are the squared value of the second digit.

Example:

1. 115^2
   11*12 = 132

Add 5^2 as last two digits

Final answer: 13225

2. 12305^2

1230*1231 = 1514130

Add 5^2 as last two digits

Final answer: 151413025

Question 10

Squaring ANY number that has only 1s as its digits

Answer 10

1. Count up the number of digits in the number. The sum will be the largest digit and the middle value of the number.
2. For your final answer, start from the digit 1 and make your way up to the summation of the digits, increasing by one each time. When that summation number is reached, go down consecutively by 1 until reaching 1 as the last digit. That is the final squared value.

For example:

11111111^2

12345678987654321

Question 11

Multiplying any number by 11

Answer 11

1. Take the last digit of the number being mutiplied by 11 as the last digit of the product.
2. Working your way from the last digit to the first digit take the sums of groups of two numbers, carrying over as necessary, and add the sums as preceding digits in reverse order.
3. For the first digit of the number, just take the first digit of the number being multiplied by 11 and add any carried over numbers.

Example:

738247 * 11

7 = last digit of product

7, digit before is 1, digit before is 6 + carried over 1 so 7, digit before is 0, digit before is 2, digit before is 1, first digit is 7 + carried over 1 which is 8

Final product: 8120717

Question 12

Sine Rule (finding angles and sides in NON-right triangles)

Answer 12

sin A / side a = sin B / side b = sin C / side C

Example:
You are given a non-right triangle with sides a, b, and c, and side a measures 7 mm. Angle A is the angle opposite side a, and Angle B is the angle opposite side b. Angle A measures 53 degrees, while Angle B measures 36 degrees. Find side b.
__
\/

sin A / side a = sin B / side b
sin(53)/7 = sin(36)/x
x = sin(36)/(sin(53)/7)

Side b measures 5.15190812 mm.

Question 13

Quickly multiplying two digit numbers

Answer 13

1. Add either number to the distance that the other number is from 10.
2. Multiply that value by 10.
3. Multiply the distance from 10 that the first number is by the distance from 10 that the second number is.
4. Add the results from Steps #2 and #3 together to get the product.

Example 1:

14 * 18

a) 14+8 = 22
22 * 10 = 220

b) 4*8 = 32

c) 220 + 032 = 252

Product: 252

Example 2:

32*41

a) 32 + 31 = 63
63 * 10 = 630

b) 22 * 31 = 682

c) 630 + 682 = 1312

Product: 1312

Question 14

Convert kg to pounds

Answer 14

1. Take the # of kgs multiplied by 2 as one of the addends.
2. Take the value from Step 1 divided by 10 as the other addend (since this will give the value of 0.2 * kg #).
3. The final answer in pounds is the sum of both addends.

Example:
53 kg to lbs

53*2 = 106
106/10 = 10.6
Final answer: 106 + 10.6 = 116.6 lbs

This is an approximate, not exact value, since 1 kg does not exactly equal 2.2 pounds.

Question 15

Converting pounds to kg

Answer 15

1. Take the number of pounds and divide by 2.
2. Multiply the value from Step 1 by 10.
3. Divide the value from Step 2 by 11 to get the final answer in kgs.

Example:
53 pounds to kg

53/2 = 26.5
26.5 * 10 = 265
265/11 = 24.09

24.09 kgs

This is an approximate, not exact value, since 2.2 pounds does not exactly equal 1 kg.

Question 16

Calculate the cube root of any number.

Answer 16

1. Find the nearest perfect cube number (i.e. 64) that is lower than the value you have to cube root, and take its cube root (i.e. 4) as the whole number of your mixed number answer.
2. For the fractional part of the mixed number, the numerator should be the different between the cubes being cube rooted.
3. The denominator should be the cube root of the closest lower perfect cube value you found, but squared and multiplied by 3. This will get you your final cube root answer.

Example:

Cube root of 523

Closest perfect cube is 512, and its cube root is 8 <– whole number part of answer.

523-512 = 11 <– numerator of fraction

(8 ^ 2) * 3 = 192

Final cube root: 8 + (11/192) or ~8.057

Very close to exact answer of 8.0568862

This method gets you approximate values that are generally within 1/10th of the precise answer.

Question 17

Find any fraction of a number

Answer 17

1. Depending on whether there is a remainder if you divide by the denominator of the fraction or not, either…

A) divide the number by the denominator and then multiply it by the numerator.

OR B) multiply by the numerator of the fraction and then divide by the denominator.

A and B are interchangeable.

Example:

A is applicable:
3/7 of 84
84/7 * 3 = 36

B is applicable:
3/7 of 83
83 * 3 / 7 = 249/7