Question

(a) The number $10$ to the power $100$($10000$ sexdecillion) is called a ‘Googol’! If it takes $\dfrac{1}{5}$ second to write a zero and $\dfrac{1}{{10}}$ second to write a ‘one’, how long would it take to write the number $100$ ’Googols’ 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? You may assume that your pen has enough ink.

Answer 1

Hint: We are given a number with an exponential power. That number is given a particular name. We are given a time of fraction of seconds taken to write the digits to write a particular number. In this firstly we will obtain a number and then find the number of digits it contains. Then calculate the total time taken by each kind of total number of digits. Then simply add the total time taken. In the second case we are given a number with exponential power of a word. We have to find time taken in years to write that word in full. In this case we will first put the value of that word in power then find the total number of digits of each number. Then using the speed and time taken we will find the total time in seconds.
Formula used for conversions:
1 min = 60 seconds
1 hour= 60×60 seconds
1 day = 24×60×60 seconds
1 year = 365 days
1 year = 24×60×60×365 days

Complete step-by-step solution:
Step1: we are given a number $10$ on which it has power of $100$ and that number is called ‘Googol’ and time taken to write a zero is $\dfrac{1}{5}$ second and a one in $\dfrac{1}{{10}}$ second. We find time taken to write the $100$Googols
For this first we will obtain googol which is given ${10^{100}}$ , also $10000$ sexdecillion.
The number $10$ to the power $100$ written in full $ = 1$ Googol
We will find the value of $100$ ’Googols’ by multiplying $100$ to Googols:
 $100$ ’Googols’ $ = 100 \times {10^{100}}$
Converting into exponential form:
 $ = ({10^2})({10^{100}})$
Exponents having same base get their power added:
 $ = {10^{102}}$
Therefore, the number $100$ googols has one digit $1$ followed by $102$ zeros.
Step2: To write the first digit $1$ time taken will be:
$ \Rightarrow 1 \times \dfrac{1}{{10}} = 0.1$ seconds.
To write the following digits of $102$ zeroes @ the rate of $\dfrac{1}{5}$ seconds per zero, it takes:
$ \Rightarrow 102 \times \dfrac{1}{5} = 20.4$ seconds.
Therefore total time taken to write the number $100$ Googols in full is:
(Total time taken to write zero) + (total time taken to write one)
$ \Rightarrow 20.4 + 0.1 = 20.5$ seconds.
Step3: In the second case we are given a number ‘Googolplex’. Its value is equal to $10$ to the power of a ‘Googol’ speed of writing the digits is the same as ‘Googol’. As we have calculated the value of ‘Googol’ above it is ${10^{100}}$
Hence value of ‘Googolplex’ will be ${10^{Googol}}$
‘Googolplex’ $ = {10^{{{10}^{100}}}}$
This number has ${10^{100}}$ zeroes following $1$
Step4: Hence time taken to write $1$ is $\dfrac{1}{{10}}$ seconds here no of digits of $1$ is one
Total time taken to write $1$ is $\dfrac{1}{{10}}$ seconds
Time taken to write one zero is $\dfrac{1}{5}$ seconds. Total no zeroes are ${10^{100}}$ . Hence total time taken to write all zeroes are: (No of zeroes× time taken by one zero)
 $ = {10^{100}} \times \dfrac{1}{5}$ seconds
Total time taken to complete writing ‘Googolplex’ is:
(total time taken to write one + total time taken to write all zeros)
$ = \left( {\dfrac{1}{{10}}} \right) + \left( {\dfrac{{{{10}^{100}}}}{5}} \right)$ seconds.
Converting seconds to days
\[60\] sec =\[1\]min
\[60\] min =\[1\]hour
\[24\] hour =\[1\]day
$60 \times 60$ seconds $ = 1$hour
$60 \times 60 \times 24$ seconds $ = 1$day
Hence to convert seconds into days we divide the seconds with $60 \times 60 \times 24$
$ = \left( {1 + {{10}^{100}}} \right) \times \left( {\dfrac{1}{5}} \right) \div \left( {24 \times 60 \times 60} \right)$ days
Now converting days into year
$365.2425$ days $ = 1$year
$60 \times 60 \times 24 \times 365.2425$ seconds $ = 1$year
To convert seconds into year we will divide it by $60 \times 60 \times 24 \times 365.2425$
$ = \left( {1 + {{10}^{100}}} \right) \times \left( {\dfrac{1}{5}} \right) \div \left( {24 \times 60 \times 60 \times 365.2425} \right)$
On further solving we will get:
$\left[ {\dfrac{1}{{10}} + ({{10}^{100}}) \div 5} \right] \div \left[ {24 \times 60 \times 60 \times 365.2425} \right]$
On solving brackets:
$(3.17 \times {10^{ - 9}}) + (6.337747701 \times {10^{91}})$ years
$ = 6.337747701 \times {10^{91}}$ years approximately to write the complete number in full. We neglected the first term as compared to the magnitude second numerical expression which has${10^{91}}$, the first expression very very small
Step5: Final answer: (a)Time taken is $20.5$ seconds (b) Time taken is $6.337747701 \times {10^{91}}$ years

Note: In such questions students do mainly one mistake that is of calculation which is quite tedious
Solve step by step. We should use here exponential form as much as possible because writing such big numbers in a compact exponential form reduces the time of writing such a long number fully. The number of days counted in a year as $365.2425$. This is counted on the basis of the number of leap years in $400$ years. There are $100 - 3$ leap years in a period of $400$ years. So, the average days in a year is $(365 \times 400 + 100 - 3) \div 400 = 365.2425$ days. And students should keep in mind the following conversions: they mainly make mistakes in conversions; they get confused whether to divide or multiply to convert from one form to another.
Commit to memory:
\[60\] sec =\[1\] min
\[60\] min =\[1\] hour
\[24\] hour =\[1\] day
$60 \times 60$ seconds $ = 1$hour
$60 \times 60 \times 24$ seconds $ = 1$day
$365.2425$ days $ = 1$year