How is your mental math?

B

bazookaworm

New member
Joined
Nov 15, 2006
Messages
13
Ok from what I've seen, most of you have really good math skills. How about doing math mentally? Are you slow or fast? When you're buying groceries how quickly can you calculate the price? Can you do tax in your head? Just curious :D
 
I'm not very good at mental math - I bought the book "secrets of mental math" by Dr. Arthur Benjamin who visited my university doing a mental math performance...there are a lot of good tricks - it takes practice like anything. I find my mental math is worse when I am under pressure to get the correct answer within a group of people...that's when my performance anxiety kicks in and interrupts the thought process.

One of the tricks I found useful is squaring any two digit number in my head, fast... lets say 25 * 25

I do (25+5) * (25-5) = (30 * 20) = 600 and then simply add the square of what ever number I subtracted and added.. so 25*25 = 625

John.
 
well, tax is easily approximated in texas we have 8.25% sales tax, which means its always less than 10%, and therefore if you move the decimal over, there ya go Or a 15% tip on a check is 10% + half of 10%, in reality my mental math ability is limited as shown by my number sense scores , on the other hand, hiding the calculator seems to help!
 
ilaggoodly said:
well, tax is easily approximated in texas we have 8.25% sales tax, which means its always less than 10%, and therefore if you move the decimal over, there ya go Or a 15% tip on a check is 10% + half of 10%, in reality my mental math ability is limited as shown by my number sense scores , on the other hand, hiding the calculator seems to help!
Click to expand...

8.25% sales tax? Ouch! We only have 5% here in Maine :D
 
I learned lots of tricks. My favorite is squaring 2-digit numbers that end in a five. Everyone's always like... How'd to do that!

\(\displaystyle \(\{d\}5\)^2\) = \(\displaystyle \{d (d+1)\}25\)

ex. (25*25). d=2. Answer = {2*3}25 = 625. I hope my notation makes sense to everyone.

The same works for bigger numbers. I.e. try \(\displaystyle 135^2\) with d=13.

I usually try to break down more difficult (but not THAT difficult) multiplications as follows: 121*13 = (120+1)*13 = 120*10 + 120*3 + 13 = 1573.

Or numbers close to a nice "round" number: 98*215 = 100*215 - 430 = 21500-430=21070.

Tax for restaurants is simple enough. I always tip 20%-ish so I just divide the check by 5 and round up. Or better, move the decimal once and multiply by two.
 
bazookaworm said:
14% Sales tax here :cry:
Click to expand...
Ouch! Its 8 and a quarter here too... If I had to pay 14% for my electronics I'd certainly cry too...
 
My parents had a gift shop in Lake George, New York.

One day, I was looking at an invoice. .One item was $30 per gross (144).

I was reaching for my calculator when my father, who was looking over my shoulder,
. . said, "About 21 cents."

Of course, I checked it . . . The calcuator said: 20.83333...

I was a math major at the time . . . "Okay, Dad, how'ja do that?"

And he told me: Multiply by 7 and point off 3 decimal places (to the left).

. . So: \(\displaystyle \;7\,\times\,30 \:=\:210\;\;\Rightarrow\;\;0.21\;=\;21\)¢


Of course, this works because: .\(\displaystyle \:\frac{7}{1000} \:\approx\:\frac{1}{144}\)

. . \(\displaystyle \frac{7}{1000}\:=\:0.007\;\;\;\;\frac{1}{144}\: = \:0.0069444...\)


My father was always surprising me with tricks like these.
I truly believe that I got my "math gene" from him.

 
Teacher to Jack (grade 1): what's 7 minus 2 3/4 ?
Jack: gee, dunno
Teacher: your father gave you 2 dollars and 75 cents as part of your 7 dollars allowance; how much does he owe you?
Jack: 4 bucks and a quarter, of course!
 
Similarly, I find some beginning students (in college) don't understand positive/negative arithmetic until I relate it to their bank accounts.
 
Here's a trick that may impress your friends
. . . or it may sending them running from the room.

Have them cube any two-digit number on a calculator.
When they give you the answer,
. . you instantly extract the cube root.


It takes a bit of "homework", but it's worth the effort.

First, memorize the first 9 cubes, and note their "endings".
. . \(\displaystyle \begin{array}{ccccc} 1^3 & = & 1 & \;\Rightarrow\; & 1 \\ 2^3 & = & 8 & \Rightarrow & 8 \\ 3^3 & = & 27 & \Rightarrow & 7 \\ 4^4 & = & 64 & \Rightarrow & 4 \\ 5^5 & = & 125 & \Rightarrow & 5 \\ 6^3 & = & 216 & \Rightarrow & 6 \\ 7^3 & = & 343 & \Rightarrow & 3 \\ 8^3 & = & 512 & \Rightarrow & 2 \\ 9^3 & = & 729 & \Rightarrow & 9 \end{array}\)


The endings can be easily memorized.

Some are "self-enders": 1, 4, 5, 6, 9
. . \(\displaystyle \begin{array}{c} 1^3\text{ ends in 1} \\ 4^3\text{ ends in 4} \\ 5^3\text{ ends in 5} \\ 6^3\text{ ends in 6} \\ 9^3\text{ ends in 9}\end{array}\)
One "on each end", and the three "in the middle".

The others are "10-complements".
. . \(\displaystyle \begin{array}{c}2^3\text{ ends in 8} \\ 3^3\text{ ends in 7} \\ 7^3\text{ ends in 3} \\ 8^3\text{ ends in 2}\end{array}\)


Example: You are given \(\displaystyle 438,976\)

Disregard the rightmost three digits; consider the leading three-digit number: \(\displaystyle 438\)

Find the largest cube that "fits" in 438.
. . It is: .7³ = 343.
. . Hence, the tens-digit is 7.

The cube ends in 6, a self-ender.
. . Hence, the cube root ends in 6.

Therefore: \(\displaystyle \:\sqrt[3]{438,976} \;=\;76\)



Example: You are given 54,872

The leading three-digits number is; \(\displaystyle \,054\)

The largest cube is: .3³ = 27
. . The tens-digit is: 3.

The cube ends in 2, a 10-complement.
. . Hence, the cube root ends in 8.

Therefore: \(\displaystyle \:\sqrt[3]{54,872} \;=\;38\)

 
How about Instant Square Roots?

Have someone square any two-digit numbers.
. . You instantly extract the square root.

There is a trick to it, a bit more elaborate than cube roots.


First, memorize the first nine squares and note their endings.

. . \(\displaystyle \begin{array}{ccc}1^2 & \Rightarrow & 1 \\ 2^2 & \Rightarrow & 4 \\ 3^2 & \Rightarrow & 9 \\ 4^2 & \Rightarrow & 6 \\ 5^2 & \Rightarrow & 5 \\ 6^2 & \Rightarrow & 6 \\ 7^2 & \Rightarrow & 9 \\ 8^2 & \Rightarrow & 4 \\ 9^2 & \Rightarrow & 1\end{array}\)

If \(\displaystyle n^2\) ends in 5, then \(\displaystyle n\) ends in 5 ... the only "self-ender".

If \(\displaystyle n^2\) ends in 1, then \(\displaystyle n\) ends in 1 or 9.
If \(\displaystyle n^2\) ends in 4, then \(\displaystyle n\) ends in 2 or 8.
If \(\displaystyle n^2\) ends in 9, then \(\displaystyle n\) ends in 3 or 7.
If \(\displaystyle n^2\) ends in 6, then \(\displaystyle n\) ends in 4 or 6.

These four endings have two choices; note that they are 10-complements.


Example: 2304

[1] Determine the tens-digit.

Consider the first (leftmost) two digits, 23.

The largest square in 23 is 4².
. . The tens-digit is 4.


[2] Determine the unit-digit.

The square ends in 4.
. . The square root ends in either 2 or 8.
Which one?

Multiply the rightmost digit by the next integer: \(\displaystyle \,4\,\times\,5\:=\:20\)
Compare the leftmost two-digit number (23) to this product (20).
Since 23 > 20, we use the larger choice.
. . The unit-digit in 8.

Therefore: \(\displaystyle \:\sqrt{2304} \:=\:48\)



Example: 6889

[1] Determine the tens-digit.

Consider the first (leftmost) two digits, 68.

The largest square in 68 is 8².
. . The tens-digit is 8.


[2] Determine the unit-digit.

The square ends in 9.
. . The square root ends in either 3 or 7.
Which one?

Multiply the rightmost digit by the next integer: \(\displaystyle \,9\,\times\,10\:=\:90\)
Compare the leftmost two-digit number (68) to this product (90).
Since 68 < 90, we use the smaller choice.
. . The unit-digit in 3.

Therefore: \(\displaystyle \:\sqrt{6889} \:=\:83\)


 
Not necessarily related. Some cool things I noticed about a calculator one day while I was supposed to be working. If you take any number of digitts say 78552, add to it its mirror image sequence of buttons on the number pad, 32558 you'll get the same as the sum of the reverses as these, 25587+85523 = 78552+32558. It works for any such combinations as long as you do it right. The reverse buttons will be the ones that are the relection along the diagonal, so 7-3,1-9.

Some easy examples (look at the NumbPad to undertsand what is going on):

741+369 = 147+963

751+359 = 953+157

999+111=777+333

123456+987654 = 654321+456789

It looks like just because the mirror image of k=10-k.
 
daon,

You do have too much free time~~~!!!

And me too - I suppose - since I have been reading all these and trying out......
 
Subhotosh Khan said:
daon,

You do have too much free time~~~!!!

And me too - I suppose - since I have been reading all these and trying out......
Click to expand...

Haha, yeah I do, but only because I put off the things that need to be done until the last minute. Not good, I know.
 
More on Squaring

You may noticed this long ago . . .

. . \(\displaystyle \begin{array}{cccc}\text{Square} & \text{Difference} \\ \\ 0^2\:=\:0 \\ & 1 \\ 1^2 \:=\:1 \\ & 3 \\ 2^2 \:=\:4 \\ & 5 \\ 3^2\:=\:9 \\ & 7 \\ 4^2\:=\:16 \\ & 9 \\ \vdots & \vdots \end{array}\)

Consecutive squares differ by consecutive odd numbers.


If we know a particular square, we can determine the next square
. . by adding the appropriate odd number.


We've already learned the trick for "squares ending in 5".
. . So we know, for example, that: .\(\displaystyle 35^2 \,=\,1225\)

Double the 35 and add one: .\(\displaystyle 2(35) + 1 \:=\:71\)

To determine \(\displaystyle 36^2\), we add 71: .\(\displaystyle 1225 + 71 \:=\:1296 \:=\:36^2\)


And now we can continue the list with mere addition . . .

. . \(\displaystyle \begin{array}{ccc}35^2 &=& 1225 \\ & & \;+71 \\ 36^2 & = & 1296 \\ & & \;+73 \\ 37^2 &=& 1369 \\& & \;+75 \\ 38^2 &=& 1444 \\ \vdots & & \vdots\end{array}\)

 
How about this?

Calculate the day of the week, for anyone's birthday!

Memorize this table: (day codes) Monday = 1, Tuesday = 2, Wednesday = 3, Thursday = 4, Friday = 5, Saturday = 6, Sunday = 7 or 0
Jan = 6*, Feb = 2*, March=2, April=5, May=0, June=3, July=5, August=1, Sep=4, oct=6, nov=2 dec=4
(* months on leap year, jan = 5, feb = 1)

Now we have to figure out a year code: for any year between 1990 and 1999, take the last two digits, divide by 4, and discard the remainder. Your year code will then be [1 + (last two digits) + (last two digits / 4) ] - (largest multiple of 7 that is less than the value contained within the brackets) The +1 is for any date in the 1900's.

The day code for the day of the week for which the person was born will be [Year code + date + month code]

I was born on July 26th, 1987
(87) + (21) + 1 = 109
Year code = 109 - 7(15) = 4

4 + 26 + 5 = 35
35 - 7(5) = 0

I was born on a Sunday.
 
OMG!

i have such bad mental math skills it's not even funny! i have to apologise to the cashieer for taking so long sometimes... :oops:

then i feel really bad for wasteing their time... :( :(
 
You must log in or register to reply here.
Top