# Mental arithmetic

1. Mar 17, 2005

### 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.

2. Mar 17, 2005

### 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.

?

3. Mar 17, 2005

### K.J.Healey

its als the first dgits of pi

4. Mar 17, 2005

### 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

5. Mar 17, 2005

### RandallB

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

6. Mar 17, 2005

### Gokul43201

Staff Emeritus
What ??

My sentiments exactly !

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

7. Mar 17, 2005

### 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)

don't get it...

8. Mar 17, 2005

### gerben

I followed these steps:

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

9. Mar 18, 2005

### 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 dont 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.

10. Mar 18, 2005

### T@P

maybe im 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...

11. Mar 19, 2005

### 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 ( )

12. Mar 19, 2005

### ToxicBug

12321

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

13. Mar 22, 2005

### vikasj007

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.

14. Mar 22, 2005

### 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.

15. Mar 22, 2005

### Edgardo

Here's how I calculated 111*111:

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

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

16. Mar 22, 2005

### Gza

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

17. Mar 22, 2005

### 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 ©)

18. Mar 22, 2005

### ToxicBug

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

19. Mar 23, 2005

### 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.

20. Mar 24, 2005

### 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.

21. Mar 27, 2005

### 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

Last edited: Mar 28, 2005
22. Mar 28, 2005

### 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 )

23. Mar 29, 2005

### Cheman

About vedic maths - ive 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.

24. Apr 11, 2005

### 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

25. Apr 11, 2005

### 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?

Last edited: Apr 11, 2005
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