Question

How do you convert $${45}^{\circ }{50}^{\prime }{50}^{″}$$ to decimal degree notation?

Answer 1

Hint: We use the concept of conversion of minutes and seconds in terms of degree. Find the conversion and add all the values obtained in terms of degree.
* Minute is denoted by (‘) and seconds is denoted by (‘’)
* 1 degree contains 60 minutes and 1 minute contains 60 seconds.
* Unitary method helps us to calculate the value of a single unit by dividing the value of multiple units by the number of units given.
* Unitary method helps us to calculate the value of multiple units by multiplying the value of a single unit to the number of units given.

Complete step by step solution:
We have to convert $${45}^{\circ }{50}^{\prime }{50}^{″}$$to decimal degree notation … (1)
We know that the term given above means 45 degrees, 50 minutes and 50 seconds.
First we convert seconds into minutes:
We know 1 minute has 60 seconds,
So, 60 seconds $$=1$$minute
Use unitary method to calculate value of 1 second
$$\Rightarrow 1$$second $$={\displaystyle \frac{1}{60}}$$minutes
Now again use unitary method to find the value of 50 seconds
 $$\Rightarrow 50$$seconds $$={\displaystyle \frac{1}{60}}\times 50$$minutes
$$\Rightarrow 50$$seconds $$={\displaystyle \frac{5}{6}}$$minutes … (2)
So, the value of $${45}^{\circ }{50}^{\prime }{50}^{″}$$can be written as:
$$\Rightarrow {45}^{\circ }{50}^{\prime }{50}^{″}={45}^{\circ }+{50}^{\prime }+{{\displaystyle \frac{5}{6}}}^{\prime }$$
Take LCM and solve right hand side of the equation
$$\Rightarrow {45}^{\circ }{50}^{\prime }{50}^{″}={45}^{\circ }+{\left({\displaystyle \frac{300+5}{6}}\right)}^{\prime }$$
$$\Rightarrow {45}^{\circ }{50}^{\prime }{50}^{″}={45}^{\circ }+{{\displaystyle \frac{305}{6}}}^{\prime }$$ … (3)
Now we convert the value of minutes in degrees.
We know 60 minutes $$=1$$degree
Use unitary method to calculate value of 1 minute
$$\Rightarrow 1$$minute$$={\displaystyle \frac{1}{60}}$$degree
Now again use unitary method to find the value of $${\displaystyle \frac{305}{6}}$$ minutes
 $$\Rightarrow {\displaystyle \frac{305}{6}}$$minutes $$={\displaystyle \frac{1}{60}}\times {\displaystyle \frac{305}{6}}$$degrees
$$\Rightarrow {\displaystyle \frac{305}{6}}$$minutes $$={\displaystyle \frac{305}{360}}$$degrees … (4)
Substitute the value of 50’ from equation (4) in equation (3)
$$\Rightarrow {45}^{\circ }{50}^{\prime }{50}^{″}={45}^{\circ }+{\left({\displaystyle \frac{305}{360}}\right)}^{\circ }$$ … (5)
Calculate the decimal value of the fraction inside the bracket on right hand side of the equation
$$\Rightarrow {45}^{\circ }{50}^{\prime }{50}^{″}={45}^{\circ }+{\left(0.847\right)}^{\circ }$$
Add the terms in right hand side of the equation
$$\Rightarrow {45}^{\circ }{50}^{\prime }{50}^{″}={45.847}^{\circ }$$
$$\therefore$$ The value of $${45}^{\circ }{50}^{\prime }{50}^{″}$$in decimal degree notation is $${45.847}^{\circ }$$

Note: Many students make the mistake of multiplying the values while converting the minutes and seconds to degrees, as they see no sign in between the terms when written altogether. Keep in mind we add the values of seconds and minutes to the value of degree in order to get the complete value in degrees.