Question

Reduce the following expression ${{94}^{g}}23'87''$ to sexagesimal measure.

Answer 1

Hint:The number ${{94}^{g}}23'87''$ is in centesimal scale. First divide 87 by 100 then it will convert into minutes. After that add 23 and $\dfrac{87}{100}$ which will give you $\dfrac{2387}{100}$ then dividing this expression by 100 will give you an answer in grades i.e. 0.2387. So, the given number is reduced to ${{\left( 94.2387 \right)}^{g}}$.The sexagesimal scale of a number comprises degree part followed by minute’s part and followed by second’s part. Multiplying ${{\left( 94.2387 \right)}^{g}}$ with $\dfrac{90}{100}$ will get the answer in degrees i.e. \[{{\left( 84.81483 \right)}^{0}}\].On multiplication of 0.81483 by 60 will give the minute part i.e. $48.8898$. Similarly for the second’s part of the sexagesimal scale, multiply 0.8898 by 60.

Complete step-by-step answer:
The number ${{94}^{g}}23'87''$ given in the question is in centesimal scale.
First of all we are going to convert $23'87''$ to grades.
We know that in sexagesimal scale,
1 minute = 100seconds so 87 seconds = .87 minutes.
100 minutes = 1 grade so 1 minute = 0.01 grades.
Total minutes = 23.87. Converting minutes into grades we get,
$23.87\left( 0.01 \right)=.2387grades$
Or we can write $23'87''$ in terms of grades as ${{\left( .2387 \right)}^{g}}$.
So, we can write the number ${{94}^{g}}23'87''$ in terms of grades as ${{\left( 94.2387 \right)}^{g}}$.
We know that in sexagesimal scale,
100grades = 90°
$\Rightarrow 1grade=\dfrac{90}{100}\deg rees$
Now, converting ${{\left( 94.2387 \right)}^{g}}$ into degrees we get,
$\begin{align}
  & 94.2387\left( 0.9 \right) \\
 & =84.81483\deg rees \\
\end{align}$
Sexagesimal scale in which a number is written in degrees, minutes and seconds so the degree part of ${{94}^{g}}23'87''$ is:
Degrees = 84°
As the sexagesimal scale contains degree followed by minutes which is followed by seconds so we can write the minute’s part as:
Multiplying 0.81483 by 60 minutes we get the minutes part.
$\begin{align}
  & 0.81483\times 60 \\
 & =48.8898 \\
\end{align}$
From the above, the minute’s part of the given number in the sexagesimal scale is 48’.
Multiplying .8898 by 60 seconds we get the second’s part of the sexagesimal scale of the given number.
$\begin{align}
  & 0.8898\times 60 \\
 & =53.388\sec onds \\
\end{align}$
From the above, the second’s part of the given number in the sexagesimal scale is 53.388”.
Hence, the conversion of ${{94}^{g}}23'87''$ into sexagesimal scale is ${{84}^{0}}48'53.388''$.

Note: You might be thinking that in this ${{94}^{g}}23'87''$ we just have to convert the grades part into degrees and we have the answer in the sexagesimal scale but this is wrong.
There is a difference between minutes and seconds in centesimal scale and sexagesimal scale.
In centesimal scale,
1minute = 100seconds
Whereas in sexagesimal scale,
1minute = 60seconds
This is the reason why we cannot write the minutes and seconds of centesimal scale as same as that of sexagesimal scale.