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655-705 (Hard)|   Fractions and Ratios|            
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Username1512
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If t = 1/(2^9*5^3) is expressed as a terminating decimal, ho 09 Jul 2026, 23:28
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Bunuel can you please explain Or another way:



Now,
[ltr]164,000164,000[/ltr]
is greater than
[ltr]1100,000=0.000011100,000=0.00001[/ltr]
and less than
[ltr]110,000=0.0001110,000=0.0001[/ltr]
, so
[ltr]164,000164,000[/ltr]
is something like
[ltr]0.0000xxxx0.0000xxxx[/ltr]

I'm with you till 1/64000 but I'm a little lost after.



Bunuel
If \(t = \frac{1}{(2^9*5^3)}\) is expressed as a terminating decimal, how many zeros will t have between the decimal point and the fist nonzero digit to the right of the decimal point?

A. Three
B. Four
C. Five
D. Six
E. Nine

Given: \(t=\frac{1}{2^9*5^3}\).

Multiply by \(\frac{5^6}{5^6}\):


\(t=\frac{5^6}{(2^9*5^3)*5^6}=\frac{25*625}{10^9}=\frac{15625}{10^9}=0.000015625\).

Hence \(t\) will have 4 zeroes between the decimal point and the fist nonzero digit.

Answer: B.

Or another way:


\(t=\frac{1}{2^9*5^3}=\frac{1}{(2^3*5^3)*2^6}=\frac{1}{10^3*64}=\frac{1}{64000}\).


Now, \(\frac{1}{64,000}\) is greater than \(\frac{1}{100,000}=0.00001\) and less than \(\frac{1}{10,000}=0.0001\), so \(\frac{1}{64,000}\) is something like \(0.0000xxxx\).

Answer: B.
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If t = 1/(2^9*5^3) is expressed as a terminating decimal, ho 10 Jul 2026, 02:41
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Username1512
Bunuel can you please explain Or another way:



Now,
[ltr]164,000164,000[/ltr]
is greater than
[ltr]1100,000=0.000011100,000=0.00001[/ltr]
and less than
[ltr]110,000=0.0001110,000=0.0001[/ltr]
, so
[ltr]164,000164,000[/ltr]
is something like
[ltr]0.0000xxxx0.0000xxxx[/ltr]

I'm with you till 1/64000 but I'm a little lost after.




Show more

The key is to connect the size of the decimal with the number of zeros.

After we get \(\frac{1}{64,000}\), compare it with nearby powers of 10:

\(\frac{1}{10,000}=0.0001\) has 3 zeros after the decimal point before the first nonzero digit.
\(\frac{1}{100,000}=0.00001\) has 4 zeros after the decimal point before the first nonzero digit.

Since 64,000 is between 10,000 and 100,000, \(\frac{1}{64,000}\) is between \(0.0001\) and \(0.00001\).

So the decimal must be smaller than \(0.0001\) but bigger than \(0.00001\). That means it must look like:

\(0.0000...\)

Therefore, there are 4 zeros between the decimal point and the first nonzero digit.
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Bunuel
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