Question

Divide $44{}^\circ 8'$ into two parts such that the number of sexagesimal seconds in one part may be equal to centesimal seconds in the other part.

Answer 1

Hint: In order to solve this question, we will assume a variable x as the equal part of seconds. And then we should know that in the sexagesimal system, $1{}^\circ =3600\text{ seconds}$, and in centesimal system, $1{}^\circ =11111.11\text{ seconds}$. So, we will add sexagesimal seconds and centesimal seconds which will be equal to $44{}^\circ 8'$ in seconds.

Complete step-by-step answer:
In this question, we are asked to divide $44{}^\circ 8'$ into two parts such that the sexagesimal seconds in one part will be equal to centesimal seconds in the other part. For that, we will first convert $44{}^\circ 8'$ into sexagesimal seconds by using the rule, $1{}^\circ =60\text{ minutes}$ and 1 minute = 60 seconds. So, we can write $44{}^\circ =44\times 60$ minutes, which is nothing but $2640'$ and so, we can write $44{}^\circ 8'=\left( 2640+8 \right)'=2648'$. And we know that $1{}^\circ =60$ seconds. Therefore, we can write $2648'=2648\times 60''=158880$ seconds.
Now, let us consider that in the first part there are x sexagesimal seconds and in the second part there are x centesimal seconds. So, we know that terms of like units can only be added. So, for that, we will keep x sexagesimal seconds as it is and convert x centesimal seconds into sexagesimal. Now, we know that $1{}^\circ =11111.11$ centesimal seconds and therefore, we can write,
1 centesimal second $={{\left( \dfrac{1}{11111.11} \right)}^{{}^\circ }}$
So, we can write, x centesimal second $={{\left( \dfrac{x}{11111.11} \right)}^{{}^\circ }}$
Now, we also know that $1{}^\circ =3600\text{ seconds}$. So, we can write,
${{\left( \dfrac{x}{11111.11} \right)}^{{}^\circ }}=\dfrac{x\times 3600}{11111.11}$ seconds, which implies that, x centesimal seconds $=\dfrac{3600x}{11111.11}$ sexagesimal seconds. According to the question, we have been given that $44{}^\circ 8'$ is divided. So, we can say that the sum of seconds of both parts will be equal to $44{}^\circ 8'$ in seconds. So, we can write as,
$x+\dfrac{x\times 3600}{11111.11}=158880$
Simplifying further, we get,
$\begin{aligned} & \dfrac{11111.11x+3600x}{11111.11}=158880 \\\ & \Rightarrow 14711.11x=158880\times 11111.11 \\\ & \Rightarrow x=\dfrac{158880\times 11111.11}{14711.11} \\\ & \Rightarrow x=119999.99 \\\ & \Rightarrow x\simeq 120000 \\\ \end{aligned}$
And therefore, we can say that there are 120000 sexagesimal seconds in the first part and 120000 centesimal seconds in the second part. And to write sexagesimal seconds and centesimal seconds, we will use the conversions of $1{}^\circ =3600\text{ seconds}$ and $1{}^\circ =11111.11\text{ seconds}$. So, we get,
Sexagesimal seconds as, $\dfrac{120000{}^\circ }{3600}=33{}^\circ 40'$ and centesimal seconds as, $\dfrac{120000{}^\circ }{11111.11}=10{}^\circ 28'$.
Hence, $44{}^\circ 8'$ can be divided as $33{}^\circ 40'$ and $10{}^\circ 28'$ such that the sexagesimal second and the centesimal second will be equal.

Note: While solving this question, there are high possibilities of error by incorrect calculation, by considering the wrong values of $1{}^\circ$, that is $1{}^\circ =3600$ sexagesimal seconds and $1{}^\circ =11111.11$ centesimal seconds. Also, we should remember that terms with like units can only be added.