Question

The number $ 10 $ to the power $ 100 $ is called a ‘Googol’. If it take $ \dfrac{1}{5} $ second to write a zero and $ \dfrac{1}{{10}} $ second to write a ‘one’ how long would it take to write the number $ 100 $ ’Googol’ in full?
b) The number $ 10 $ to the power of a ‘Googol’ is called a ‘Googolplex’. Using the same speed of writing, how long in years would it take to write $ 1 $ ‘Googolplex’ in full?

Answer 1

Hint: For this we problem we first see calculate how many zero’s and one comes in expansion of $ {\left( {10} \right)^{100}} $ and using time given to write one ‘one’ and one ‘zero’ we find total time to write one Googol and then by unitary method we find total time to write $ 100 $ Googol.

Complete step-by-step answer:
For first problem, it is given one Googol = $ {\left( {10} \right)^{100}} $
As, we knows that for $ {\left( {10} \right)^2}\,\,there\,\,are\,\,two\,zeros,\,\,and\,\,for\,\,{\left( {10} \right)^3}\,\,there\,\,are\,\,three. $
Hence, using same fact we can say there for $ {\left( {10} \right)^{100}} $ there are $ 100 $ zero’s in the expansion of $ {\left( {10} \right)^{100}} $ .
Time taken to write a ‘one ’zero’ = $ \dfrac{1}{5} $ second
Time taken to write a ‘one’ = $ \dfrac{1}{{10}} $ second
As, there is only one $ 1 $ and $ 100 $ in one Googol. As it is given, one Googol is $ {\left( {10} \right)^{100}} $ .
Therefore, time taken to write one Googol is given as:
 $
  \dfrac{1}{{10}} \times (1) + \dfrac{1}{5} \times \left( {100} \right) \\
   \Rightarrow \dfrac{1}{{10}} + 20 \\
   = 20\dfrac{1}{{10}} \;
  $
Therefore, time taken to write on Googol is $ 20\dfrac{1}{{10}} $ seconds.
To calculate time taken to write $ 100 $ Googol we will multiply time taken to write one Googol by $ 100 $ .
\[
   \Rightarrow 100 \times \left( {20\dfrac{1}{{10}}} \right) \\
   \Rightarrow 100 \times \dfrac{{201}}{{10}} \\
   \Rightarrow 10 \times 201 \\
   = 2010 \;
 \]
Hence, time required to write $ 100 $ Googol is $ 2010\,\,\,seconds $ .
Also, it is given that a Googolplex is = $ {\left( {10} \right)^{Googol}} $ .
So, from above we see that in Googol and Googolplex there is a difference of zero’s present in both. Here both have one only once but the number of zeros in Googolplex are more than of Googol.
As, speed is given same so time taken to write one Googolplex time can be calculated as:
 $ \dfrac{1}{{10}} + \dfrac{{{{\left( {10} \right)}^{100}}}}{5} $ seconds.
Also, we know that there are $ 3153600\,\,\,seconds\,\,in\,\,a\,\,year. $
So, time to write one Googolplex in full in year = $ \dfrac{{\dfrac{1}{{10}} + \dfrac{{{{\left( {10} \right)}^{100}}}}{5}}}{{3153600}} $
Which is approximately equal to $ {\left( {10} \right)^{93}} $ years.

Note: One year is $ 365\,days,\,\,one\,\,day\,\,is\,\,24\,\,hours,\,\,one\,\,hour\,\,is\,\,60\,\,\,minutes\,\,and\,\,one\,\,\,minute,\,\,is\,\,60\,\,\,seconds. $
So $ 1 $ year = $ 365 \times 24 \times 60 \times 60 = 31536000\,\,\,seconds $ .