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
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.