Mental Arithmetic: Solve 314*159 & Discover Speciality

[Edgardo]

(a) Calculate this in your head: 314*159
(b) What's so special about these digits?

I would like to see how people solve this in different ways.

 

[Paul Wilson]

Answer is = 2226.

2^3 = 6

2^3 is equal to 2+2+2 as it happens the initial answer is 2226 which is the sum itself. 2^3=6 all the two's being the question and the last digit, 6, being the answer.

?

 

[K.J.Healey]

its als the first dgits of pi

 

[Edgardo]

Haley got the right answer to (b).
@Paul Wilson: I am not sure what you mean.

With respect to (a): I am asking you to write down how you
calculated 314*159 in your head.

Here's how I did it:

Step 1:
314*159 = 314*(160-1)


Step 2:
What's 314 * 160 ?
314*16
= 314 * (10 + 6)
= 3140 + 1884
= 3140 + 2000 - 116
= 5024

Therefore, 314*160 = 50240


Step 3 (Final Step):
We go back to Step 1, which was:
314*159 = 314*(160-1)
= 314*160 - 314
= 50240 - 314
=50000 - (314 - 240)
=50000 - 74
=49926

 

[RandallB]

Answer is = 2226.
2^3 = 6 ?

I thought 2 to the 3rd was 8 ??
and what is 2226 the answer to ?

Edgardo: your keeping those steps all seperat in your head - good for you!

How about 314*159 =
314 *100
plus 1/2
plus 3140
minus 314
===============
31400
add 1/2 one diget at a time working right to left
31600
32100
47100
now add the 3140
50240
and subtract the 314 working right to left
50236
50226
49926

 

[Gokul43201]

Answer is = 2226.

2^3 = 6

2^3 is equal to 2+2+2 as it happens the initial answer is 2226 which is the sum itself. 2^3=6 all the two's being the question and the last digit, 6, being the answer.

What ??

?

My sentiments exactly !



I'd do it Randall's way : 159 = 100 * ( 1 + 0.5 + 0.1 - 0.01 )

 

[rachmaninoff]

Same way, but I subtract things differently:

314 00 (314/2 = 150 + 7)
+157 00 (471 00)
+031 40 (471 + 31)~40
=502 40
240 + 74 = 314
50240 - 314
= 50000 - 74
= 499 26 (it's faster mentally for me to subtract from zeroes)

Answer is = 2226.

2^3 = 6

2^3 is equal to 2+2+2 as it happens the initial answer is 2226 which is the sum itself. 2^3=6 all the two's being the question and the last digit, 6, being the answer.

?

don't get it...

 

[gerben]

I followed these steps:

314*6 = 1884
1884*10 = 18840
18840 - 314 = 18526
18526 + 31400 = 49926

 

[vikasj007]

well, believe it or not, i solved it in my head in 20-25 seconds.

actually, a few days back i was reading a book which deals with a special branch of mathematics called VEDIC MATHEMATICS. actually it is an ancient Indian way of solving complex mathematical problems orally, and believe me they are very easy to learn. till now i could only read the part about multiplication, as my exams are just around the corner, but it deals with division, fractions(the vulgur ones), linear equations, etc. and all those can be solved without much difficulty and orally. you don't need to form equations or anything like that.


i solved this one using the criss cross method, which is very easy to learn, but i think i'll post it sometime later.




i hope some you must have heard about this, specially those who belong to india, as it is fairly popular here.

 

[T@P]

maybe I am not that sophisticated but i would just do
(3*100 + 10 + 4)* 159... really the biggest problem to me was remembering what the previous sum was...

 

[Edgardo]

Hello vikasj007,

I believe you. I once read about this criss-cross method. But I was too lazy to learn it.

@ T@P
I don't know how you do it, but if I try to remember all the numbers from each step to sum them up, I repeat them loudly. But sometimes, if the number is very long, I use some sort of coding (Major/Master system) to encode them.

--------------------------------------------------------------

NEW TASK:
Calculate 111*111 ( )

 

[ToxicBug]

12321

Its just to take the number of digits and write them front and back. Works only upto 9 though.

 

[vikasj007]

Hello vikasj007,

I believe you. I once read about this criss-cross method. But I was too lazy to learn it.

well, believe me it is very easy to learn, and if you find it interesting as i did, then i am sure you will not be able to leave it till you are done with it.

 

[K.J.Healey]

I did it in my head, standard multiplication way.
314
159
so 9*314 + 50*314 + 100*314
it took about 2 minutes, and a lot of talking to myself to keep the numbers from fading
I basically did
9*314,
then added 50*4, then 50*10, then 50*300
then added 100*4, 100*10, and 100*300
I had to talk outloud to myself though. Makes it easier.

 

[Edgardo]

Here's how I calculated 111*111:

I used the formula $(a+b)^2 = a^2 + 2ab + b^2$.

111*111 = $(111)^2$ = $(110 + 1)^2$
= $110^2 + 2 \cdot 110 + 1^2=$
= $12100 + 220 + 1 = 12321$

 

[Gza]

Answer is = 2226.

2^3 = 6

2^3 is equal to 2+2+2 as it happens the initial answer is 2226 which is the sum itself. 2^3=6 all the two's being the question and the last digit, 6, being the answer.

Sorry, I just had to add my WTF? to this post as well.

 

[The Bob]

I did basic long multiplication and got 50556. I don't really know any other way to work it out.

The Bob (2004 ©)

 

[ToxicBug]

111² is too easy. Calculate 11^4 and 111^3. Show how you did it. Tools allowed: pen and paper.

 

[Edgardo]

Toxic, the question is whether one can calculate things in his/her head.
Otherwise with paper and pencil it's nothing special calculating 11^4 and 111^3

I calculated 11^4 in my head the following way:
11^4 = (121)*(121) = (120 +1 ) ^2
= 14400 + 2*120 + 1 = 14641

As for the second problem 111^3 I am still searching for an easy way to calculate this one in my head.

 

[Edgardo]

Ok, here's how I calculated 111^3:

111^3 = (111^2)*111 = 12321*111

Calculate 12321*111:
12321*111 = 12321*(100+10+1) = 1232100 + 123210 + 12321
= 13 676 31

I found calculating 12321*111 in my head difficult.

 

[gafoon]

here is how i did it,
314 x 100 = 31400
314 x 50 = 31400 / 2 = 15700
314 x 9 = 2826
31400 + 15700 + 2826 = 49926
but the shorter version is,
314 +(314/2) = 471
314 x 9 = 2826
471
+2826
49926

 

[joeboo]

314*159,
314*318/2,
(316-2)(316+2)/2,
( 316^2 - 2^2 )/2,
( (316+16)(316-16) +16^2 - 2^2 )/2,
( 332*300 + 16^2 - 2^2 )/2,
( 99600 + 256 - 4 ) / 2,
99852/2 =
49926

( I realize it looks like a lot of headwork, but some of the steps are realized quickly )

 

[Cheman]

About vedic maths - I've read the technique for how you multiply numbers who are of the same number of digets in length. eg - 123*524 or 678767*457098 using the vertically and crossing method. But is there a vedic method to multiply numbers of differing lengths? eg - 365*34543 or 97923*607050434?

Thanks in advance.

 

[Edgardo]

Hello Cheman,

I think you just have to put a zero in front of the number with less digits, for example:

345*45 = 345*045

 

[Felix83]

My thought processes:

314*159

314*100=31400
314*50=31000/2=15700
31400+15700=47100
300*9=2700
47100+2700=49800
10*9=90
49800+90=49890
9*4=36
49890+36=49926

But I did it much quicker the other day when I wasn't as tired
I just thought:
Hey, 314*159 = 49926 :)

If you think about it, the human brain performs operations much faster than a computer, yet a computer can do math much faster than us. Shouldn't the human brain be able to do math just as fast as a calculator?

 

[Moo Of Doom]

The reason we can't crunch numbers as fast as calculators is that, though we process more bits of information per second than calculators, we do a lot more with those numbers as well. Calculators just purely put the number in memory and run it through a process without ever knowing what number it is, what they are doing with it, and that they are spitting out an answer. The brain, on the other hand, deals macroscopically with the concept of each number and puts a conscious effort into manipulating it. This involves far more operations than calculators go through to add things.

 

[bomba923]

Hmm--Also I believe the concept of working memory (i.e., memory enabling us to store information and manipulate it at the same time) might apply here. Like Moo of Doom indicated, we do far more than just calculators to solve these problems; however as humans, our working memory "capacity" is simply not enough to, for example, calculate 352*765 in less than a second or two (well, some people in the world can...). Given the additional mental "stuff" we do, our working memory is further strained.
Without a calculator, we would do long multiplication (or another method) on paper, for example, to find 765*352. Not that we can't perform multiplication in our minds, but we cannot store the values of (2*765), (50*765), (300*765) while we simultaneously perform the multiplicative operations, carrying over the extra digits, adding, then repeating (multiplication)...etc. (for long multiplication). That's why we write them on paper, and then add them.

Btw, what is generally classified as "long" addition/subtraction/multiplication/division ? Four-digit numbers, three digits? Operations of two-digit numbers with 5-digit numbers? What exactly classifies addition/subtraction/multiplication/division as being "long" ?

 

[Jimmy Snyder]

(well, some people in the world can...).

I suppose that if some can, then all can.

Btw, what is generally classified as "long" addition/subtraction/multiplication/division ? Four-digit numbers, three digits? Operations of two-digit numbers with 5-digit numbers? What exactly classifies addition/subtraction/multiplication/division as being "long" ?

When my daughter was 4, I used to tell people she could add 13 digit numbers in her head. After the initial reaction of oohs and aahs, I would ask her "How much is 2 trillion plus 3 trillion?"

 

[BobG]

(a) Calculate this in your head: 314*159
(b) What's so special about these digits?

I would like to see how people solve this in different ways.

I try to at least get close, then get closer, pushing anything hard to calculate as far back as possible.

300 * 160 =48 000

16*16 0= 256 0 , pushing me to 50560 ... (314 = 300 + 16 - 2)

minus 2*160 = 50240 ... (which is 314, too high)

minus 300 = 49940 ... (314 = 300 + 14)

minus 14 = 49926.

 

[megavolts61]

(a) Calculate this in your head: 314*159
(b) What's so special about these digits?

I would like to see how people solve this in different ways.

I know this post is way old, but here's how I'd solve this...using Vedic math:

314
x 159
--------------
starting from the right, multiply 4.x6 write down the six and remember the 3

then add 1x9 plus 5x4 plus that 3 you remembered. ot get 32. write the 2 and remember the 3 (note that you crisscross the last two digits of each number) last two digits are 26

next, you add 3x9 +4x1 +1x5 + 3 to get 39 (note that you criss cross, then multiply the two middle digits) write down the 9 and remember the 3 so your last three digit are 926

next you do the crisscross and add 3x5 + 1x1, then add that 3 to get 19 write down the 9 and remember the 1 so now your last four digits are 9926.

finally, you multiply the first two digit 3x1 and add that 1 to get 4

and there you have it. the result is 49926

someone already noted in part (b) that those numbers are the first six digits of pi

This crisscross method can be done in your head with just a little practice and can be extended to any number of digits to be multiplied. Two four-digit number are just a little more complicated in that you extend the concept to crisscross the first and last digits and then crisscross the two middle digits as in:

abcd
x efgh
-------------
(ah)+(de)+(bg)+(cf) + any remainder from the addition of the previous step which would be (bh)+(df)+(cg) and any remainder from the previous step.

I hope that is clear enough

the 111 x 111 isn't too hard eigherl. using abc x def, first multiply bc x ef, in this case, it would be 121. the last two digit of the result therefor are 21. Also in this case, the result was over 100, so remember that 1.

then you can simply add abc +ef or bc + def...will get the same result either way. In this case the result would be 111 + 11 or 122. then add that one that you remember to get 123.
those are the first three digits, so the final result is 12321.

 

[megavolts61]

Ok, here's how I calculated 111^3:

111^3 = (111^2)*111 = 12321*111

Calculate 12321*111:
12321*111 = 12321*(100+10+1) = 1232100 + 123210 + 12321
= 13 676 31

I found calculating 12321*111 in my head difficult.

as from my previous reply, you can extend the vedic crisscross method you simple multiply the 12321x 00111
stacking them makes is easier:
12321
00111
so
1x1 is the last digit
2x1 + 1x1 second digit is 3 (no remainder)
then 3x1 + 1x1 + 2x1 = 6 so last three digits are 631.
then add 2x1 + 1x0 + 3x1 + 2x1 = 7 so the last four digits are 7631
then add:
1x1 + 1x0 + 2x1 + 2x0 + 3x1 to get 6 so you have the number ending 67631
then add:
1x1 + 2x0 + 2x1 + 3x0 to get 3 are you seeing a pattern here? now the number ends with 367631
it's getting easier now haha
add:
1x1 + 3x0 + 2x0 = 1 giving you 1367631...if there were not two zeros in the bottom number, you would continue with the last two steps, but in this case there is no need...or you'd wind up with 001367631 which is still 1367631
cheers.
Mike

 

[tedbradly]

Answer is = 2226.

2^3 = 6

2^3 is equal to 2+2+2 as it happens the initial answer is 2226 which is the sum itself. 2^3=6 all the two's being the question and the last digit, 6, being the answer.

?

This post gave me a good laugh for about 40 seconds.