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

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

You can do a similar trick with adding 18 and 27.  To add 18:  add 20 then subtract 2.  To add 27:  add 30 then subtract 3. 

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


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