**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:

**1**42857 142857 142857… (and repeat)
If R = 2; the pattern starts at 2:

**2**857 142857 142857…
If R = 3; the pattern starts at 4:

**4**2857 142857 142857…
If R = 4; the pattern starts at 5:

**5**7 142857 142857…
If R = 5; the pattern starts at 7:

**7**142857 142857…
If R = 6; the pattern starts at 8:

**8**57 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.

**7**142857142857…**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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2011-2018. Edward Shore. Unauthorized use and/or unauthorized
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