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Determining zero and non zero digits in a terminating decim

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Determining zero and non zero digits in a terminating decim

by Amrabdelnaby » Thu Dec 31, 2015 2:46 am
Dear Pros,

I have a conceptual question.

How do we determine the zero digits and non zero digits in a terminating decimal?

for example, how do we determine the zero and non zero digits from the decimal 1/(2^17 x 5^11)

I thought about it in an intuitive way.

First to determine the number of zeros i realized that the denominator will look like this: 10^11 x 5^6

So this gives us 11 zeros.

Now to check for more zeros together with non zeros, I did the following

I evaluated 1/5 to be 0.2
1/25 to be 0.04
1.125 to be 0.008

I noticed a pattern here, the number is multiplied by 2 and the zeros are increasing by one zero in each operation

Hence I concluded that 1/5^6 will yield to 0.0000064

Hence the decimal will have 17 zero digits ( 11 from 10^11 and 6 from 1/5^6 ) and 2 non zero digits

Is this right? Did I miss something?

Please explain.

Thanks
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by MartyMurray » Thu Dec 31, 2015 6:05 am
Actually you reversed the positions of the exponent values. The denominator will be 10¹¹ x 2�.

Also, looking at the math you did, it looks as if you added an extra 0, because when you got to 16, 32 and 64 you didn't count the 1, 3 or 6 as decimal places.

For example, .002 x .2 = .0004, but .008 x .2 ≠ .00016. .008 x .2 = .0016.

Maybe work it again and come back.
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by DavidG@VeritasPrep » Thu Dec 31, 2015 6:46 am
One of our instructors offers a thorough solution here: https://www.veritasprep.com/blog/2013/12 ... -the-gmat/

(You can scroll down to the second problem, but you may find the first worthwhile to do as well.)
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by Amrabdelnaby » Fri Jan 01, 2016 1:46 pm
Ok then.

Correct me if I am wrong pls :)

1/ (2^17 x 5^11)

lets work out the denominator first.

10^11 (2^6) --> 10^11 has 11 zeros

now lets focus on 2^6 in the denominator

1/2 = 0.5

1/2^2 = 0.25

1/2^3 = 0.125

1/2^4 = 0.0625

1/2^5 = 0.03125

1/2^6 = 0.015625 --> here we have two zeroes and 5 non zero digits

hence the above expression will yield to 13 zeros ( 11 from 10^11 and the two zeros from 1/2^6) and 5 non zero digits.

Am I right? I hope I am :)


Marty Murray wrote:Actually you reversed the positions of the exponent values. The denominator will be 10¹¹ x 2�.

Also, looking at the math you did, it looks as if you added an extra 0, because when you got to 16, 32 and 64 you didn't count the 1, 3 or 6 as decimal places.

For example, .002 x .2 = .0004, but .008 x .2 ≠ .00016. .008 x .2 = .0016.

Maybe work it again and come back.
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by MartyMurray » Fri Jan 01, 2016 4:37 pm
Now I see what's going on. You are counting the zero to the left of the decimal point, in the ones place, as one of the zeros. You did get the right answer, if you include the zero to the left of the decimal, but the way you got to it seems a little off.

Maybe you should not count zeros to the left of the decimal, as counting them does not really work out, especially when you have two decimals to multiply and you want to know how many zeros the product will have.

With that in mind, how many zeros to the right of the decimal point does 1/10 have?

So, just to be clear about how this works, how many zeros to the right of the decimal point does 1/100 have and how many does 1/10¹¹ have?

Now how many zeros to the right of the decimal does 1/(10¹¹ x 2�) have?
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by Amrabdelnaby » Sat Jan 02, 2016 1:36 am
Ok, So when counting the number of zeros in a number, I shouldn't include the one on the left of the decimal point?

As for your question, 1/10 has no zeros to the right of the decimal point; it has one non zero digit, one.

1/100 has one zero to the right of the decimal point and one non zero digit, one.

since 1/10 is equivalent to 1/10^1 and it yields to no zeros on the right of the decimal point and 1/100, which is equivalent to 1/10^2 yields to one zero to the right of the decimal point, then 1/10^11 will yield to 10 points on the right of the decimal point, according to the above pattern.

As for 1/10^11 x 2^6

1/2 ->0.5

1/4 --> 0.25

1/8 --> o.125

1/16 --> 0.0625

1/32 --> 0.03125

1/64 --> 0.015625

hence the entire fraction will have 12 zero digits and 5 non zero digits.

right? is my thinking process correct? or am i missing something?

also is there a better way to approach these kinds of questions?
Marty Murray wrote:Now I see what's going on. You are counting the zero to the left of the decimal point, in the ones place, as one of the zeros. You did get the right answer, if you include the zero to the left of the decimal, but the way you got to it seems a little off.

Maybe you should not count zeros to the left of the decimal, as counting them does not really work out, especially when you have two decimals to multiply and you want to know how many zeros the product will have.

With that in mind, how many zeros to the right of the decimal point does 1/10 have?

So, just to be clear about how this works, how many zeros to the right of the decimal point does 1/100 have and how many does 1/10¹¹ have?

Now how many zeros to the right of the decimal does 1/(10¹¹ x 2�) have?
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by MartyMurray » Sat Jan 02, 2016 6:30 am
Amrabdelnaby wrote:Ok, So when counting the number of zeros in a number, I shouldn't include the one on the left of the decimal point?
When working with decimals this way, at least when you are doing the calculations, you are better off not counting a 0 to the left of the decimal point, as doing that does not provide the best information for your purposes.

When answering GMAT decimal questions, be sure to choose the answer choice that fits the exact question asked, which could refer to digits only to the left, only to the right or to the left and right of the decimal point.
As for your question, 1/10 has no zeros to the right of the decimal point; it has one non zero digit, one.

1/100 has one zero to the right of the decimal point and one non zero digit, one.

since 1/10 is equivalent to 1/10^1 and it yields to no zeros on the right of the decimal point and 1/100, which is equivalent to 1/10^2 yields to one zero to the right of the decimal point, then 1/10^11 will yield to 10 points on the right of the decimal point, according to the above pattern.

As for 1/10^11 x 2^6

1/2 ->0.5

1/4 --> 0.25

1/8 --> o.125

1/16 --> 0.0625

1/32 --> 0.03125

1/64 --> 0.015625

hence the entire fraction will have 12 zero digits and 5 non zero digits.

right? is my thinking process correct? or am i missing something?

also is there a better way to approach these kinds of questions?
That looks tight to me.
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