approximate integer value?

Question

Hi,

whats the approximate value of 30Sqrt(2)= ? (approximate interger value).

sqrt(2)=1.414..

similarly,whats the approximate integer value of 40sqrt(3)=?


i need the shortest and quick method to find it.

----------------
so u r mutiplying 30*1.4 ?
i.e u r taking the two digits of the number and multiplying mentally and produing the approx result ?


I thought there might be some thumb rule kind of thing .

like, any_number * Sqrt(any_number)=Apply the thum rule=get the result

Answer 1

Doing the multiplication mentally is actually very fast in both of these cases.

Think about the fact that the final answer is going to have two digits. This means that, in order to keep two-digit accuracy, all we need is two-digit accuracy in the sqrt(2), or 1.4 (if it were going to be a three-digit number, we would want to use three digits of sqrt(2)).

Now, 30*1.4 is really easy. Just split it up into easier to understand numbers (you'll usually do this part mentally). 30 is the same as 3 * 10, which means that I can just multiply 3 by 1.4 and then multiply that result by 10. 1.4 is just 14 divided by ten, which means that I can use 14 / 10 instead:

3 * 14 * 10 / 10

You can see that the "tens" cancel, so it is just 3 * 14 or 42.

This is a very quick way to do the mental math that can be applied in the second problem as well:

40 * sqrt(2)

We need two-digit accuracy because our final answer is going to be two digits long:

40 * 1.4

(Mental math portion) 40 = 4 * 10 and 1.4 = 14/10, so I can use those numbers instead to help simplify the problem. The *10 and the /10 cancel out, so the answer is simply 4 * 14, or 56.

If the mental math part is confusing, just focus on the basic approximation concept: The numbers that you are multiplying need to have the same accuracy or greater as the final product. In this case, we always assumed that sqrt(2) = 1.4 because we only needed two-digit accuracy.

***EDIT***

By the way, I noticed DutchProf's mentioning of the need for three-digits in the extreme case of 99. This makes sense because the result is a three-digit integer, and we thus need three digits of sqrt(2) to maintain accuracy.

Answer 2

CALCULATOR!

30sqrt 2 = 30 * 1.414... = 42.4..., so it is close to 42.
40sqrt 3 = 40 * =1.732... = 69.2..., so it is close to 69.

Okay, now without the calculator :)

30 sqrt 2 = sqrt(30 * 30 * 2) = sqrt(1800)
40 sqrt 3 = sqrt(40 * 40 * 3) = sqrt(4800)

Note, first, that 1800 lies between 1600 (40^2) and 2500 (50^2), and is much closer to 1600. Therefore, sqrt(1800) will be in the lower 40s. You can calculate a few and compare:
40^2 = 1600
41^2 = 1681
42^2 = 1764 * so the sqrt of 1800 must lie between
43^2 = 1849 * these two numbers

Similarly, 4800 lies between 3600 (6^2) and 4900 (7^2), and is much closer to 4900. Therefore, sqrt(4800) will be in the upper 60s.
70^2 = 4900 * so the sqrt of 4800 must lie between
69^2 = 4761 * these two numbers, closest to 69

If you want to be precise, you have to decide whether sqrt(1800) is closer to 42 or to 43. The trick is to compare with 42.5^2. (Trick: 42.5^2 = 42*43 + 1/4 = 1806 1/4.) Because 1800 is slightly less than this, we conclude sqrt(1800) < 42.5 and round down to find 42.



===== EDIT =====

I see what you mean... You cannot just work with the first two digits, because the next digit me create a fair large carry over.

In general, if you multiply the sqrt with a number of n digits, you should take in account the n'th decimal of the decimal sqrt. In your case, 30 and 40 have two digits, so work with 1.41 and 1.73. Then

30 * 1.41 = 42.3, round down
40 * 1.73 = 69.2, round down

Worst case scenario:

99 * 1.41 = 139.59, round up gives 140
The correct answer is
99 sqrt(2) = 140.007..

so our method is approx. 1/2 off. This can occasionally cause a wrong rounding; if you want to be sure, move on to the next decimal.

Answer 3

Hmm...
I have put a lot of work into this but...

I Think:
1+1=3 ...... i mean 2!

Hope this helps!!

lol

Jordan 14, CA