## Saturday, August 4, 2018

### Mental Math and Some Numerical Musings

Mental Math and Some Numerical Musings

Working with mathematics for a real long time, everything from doing this blog, programming, watching a lot of game shows, and being asked about prices after discounts, I picked up a few mental math pointers.  I don’t claim to be a mental math prodigy (those kids that get featured who are, are amazing).

Mentally adding 9 involves one of two cases:

If the last digit is 0, change it to a 9.

Example:  1820 + 9.  The last digit now becomes a 9.  Hence: 1820 + 9 = 1829.

If the last digit isn’t a 0, then add 10, then subtract 1.

Example:  1821 + 9 = 1821 + 10 – 1 = 1831 – 1 = 1830

Example:  1827 + 9 = 1827 + 10 – 1 = 1837 – 1 = 1836

Example: Add 18 to 1827, then 27 to the resulting sum.

(mentally) 1827 + 18 + 27 = 1827 + 20 – 2 + 30 – 3 = 1847 – 2 + 30 – 3 = 1845 + 30 – 3 = 1875 – 3 = 1872

It’s like give to the tens digit, take from the ones digit.

Mentally Multiplying and Dividing by 10

Mentally, it’s a matter of moving the decimal point.  When multiplying by 10, move the decimal right (and fill in a zero if necessary).  Dividing by 10 will cause the decimal point to move to the left.

Example:  58.238 * 10

Move the decimal point to the right and get the answer:  582.38

Example:  58.238 ÷ 10

Move the decimal point to the left and get the answer:  5.8238

The 10% Discount

Need to find out the discount when something is 10% off?  Fairly simple, just recognize that 10% is multiplying by 0.1, which is dividing by 10.  Mentally, move one decimal point to the left.

Example:  What is 10% of \$38.99?   Multiplying by 10% is the same as dividing by 10, hence move the decimal point to the left, and we get the answer:  \$3.899 or rounding to the nearest cent, \$3.90.

The 10% Tip

How about if we add 10% to an amount?  Note that adding 10% is equivalent to multiplying the number by 1.10.  Let n be the number, and:

n + 10% = n * 1.1 = n * 1 + n * 0.1 = n + n ÷ 10

Example:  If a restaurant bill is \$32.00 and we needed to find the total cost after adding 10% tip:

32.00 + 10% = 32.00 + 32.00 ÷ 10 = 32.00 + 3.20 = 35.20

Dividing a Number by 5

To mentally divide a number by 5, double the number and divide the result by 10.  Why does this work?

n ÷ 5 = n * (2 ÷ 10) because the fraction 2/10 is equal to 1/5.

Example:  Divide 64 by 5.

Step 1:  Double 64.  Now we have 128.
Step 2:  Divide by 10.  Move the decimal point left.  (think that we starting with 128.0).  The result is 12.8.

Multiplying a Number by 5

To mentally multiply a number by 5, multiply the number by 10 and then half the result.  Observe that for any number n:

n * 5 = n * (10 ÷ 2) = (n * 10) ÷ 2

Example:  Multiply 753 by 5.

(mentally) 753 * 5 =  753 * 10 ÷ 2 = 7530 ÷ 2 = 3765

Dividing Whole Numbers by 3, Will It Divide Evenly?

The way we can tell if a whole number divides by 3 evenly (no remainder, the quotient is also a whole number) is that if the sum of all its digits is also divisible by 3.

Example:  780, 1959, 4839, and 55101 are all divisible by 3.  Why?

780:  7 + 8 + 0 = 15; 15 is divisible by 3.  Also, 1 + 5 = 6.  (780 ÷ 3 = 360)

1959: 1 + 9 + 5 + 9 = 24; 2 + 4 = 6.  Divisible by 3.  (1959 ÷ 3 = 653)

4839: 4 + 8 + 3 + 9 = 24.   Divisible by 3.  (4839 ÷ 3 = 1613)

55101: 5 + 5 + 1 + 0 + 1 = 12  (1 + 2 = 3).  Divisible by 3.  (55101 ÷ 3 = 18367)

Dividing by 7

I have not memorized this.  However, something interesting when you divide numbers that are not multiples of 7 happens:

1/7 = 0.142857142…
2/7 = 0.285714285…
3/7 = 0.428571428…
4/7 = 0.571428571…
5/7 = 0.714285714…
6/7 = 0.857142857…

8/7 = 1.142857142…
9/7 = 1.285714285…
10/7 = 1.428571428…
11/7 = 1.571428571…
12/7 = 1.714285714…
13/7 = 1.857142857…

The decimal portion always follows the pattern 1, 4, 2, 8, 5, 7.  So the next time you divide a whole number by 7 and figure the remainder, you can figure out which part of the pattern to attach if your answer is required as a decimal answer:

If R* = 1; the pattern starts at 1:  142857 142857 142857… (and repeat)
If R = 2; the pattern starts at 2:  2857 142857 142857…
If R = 3; the pattern starts at 4:  42857 142857 142857…
If R = 4; the pattern starts at 5:  57 142857 142857…
If R = 5; the pattern starts at 7:  7 142857 142857…
If R = 6; the pattern starts at 8:  857 142857 142857…

* R: remainder

Example:  1720 ÷ 7.  The division results as 245 with the remainder of 5.  The decimal patter starts at 7, hence 1720 ÷ 7 = 245.7142857142857…

Squaring Any Integer That Ends in 5

Why does squaring every whole number ending in 5 results in the square ending with 25?

Check it out:

5^2 = 25
15^2 = 225
25^2 = 625
35^2 = 1225
45^2 = 2025
55^2 = 3025
65^2 = 4225
185^2 = 34225
(feel free to use a calculator to check for other numbers)

Let n be a whole number whose last digit is 5.  (n = {5, 15, 25, 35, 45, ... 155 … }).  Then:

n^2
= (n – 5 + 5)^2

Let ϕ = n – 5.   Observe that ϕ is multiple of 10.  (Example:  If n = 25, then ϕ = 25 – 5 = 20)

Then:
n^2
= (ϕ + 5)^2
= ϕ^2 + 10 * ϕ + 25

Note that ϕ^2 and 10*ϕ will be multiples of 100.

The mental trick given when squaring a whole number ending in 5 is:

Step 1: Spilt the number into two parts, separating the last digit 5 from the rest of the number.  Treat the detached as a separate number.

Step 2:  Square the detached number and the detached number to the result.

Step 3:   “Attach” a 25 to the right side of the result.

Example:  25^2.

Step 1: “Split and detach” the number:  2 | 5

Step 2:  Square the detached number and add the detached number to the result:
2^2 + 2 = 6

Step 3:  “Attach” a 25 to the right side of result:  625

Hence:  25^2 = 625

If we use the formula:  n = 25, ϕ = 25 – 5 = 20:

Then 25^2 = 20^2 + 10 * 20 + 25 = 400 + 200 + 25 = 625

Example:  215^2

Step 1:  “Detach”:  21 | 5

Step 2:  Square detached, add the detached to the result:  21^2 + 21 = 441 + 21 = 462

Step 3:  “Attach” a 25 to the right end:  46225

215^2 = 46225

If we use the formula:  n = 215, ϕ = 215 – 5 = 210

Then 215^2 = 210^2 + 10 * 210 + 25 = 44100 + 2100 + 25 = 46225

I hope you find this helpful.  This is some of the math I can do mentally (except I haven’t memorized the 142857 pattern when dividing numbers by 7), it comes with practice and patience.  Of course, it doesn’t hurt to check for accuracy.

Happy August,

Eddie

All original content copyright, © 2011-2018.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.  Please contact the author if you have questions.

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