Noah’s teacher asked him to express 58 degrees, 59 minutes, and 40 seconds in degrees only without using the calculator. Noah forgot the procedure, so he could not answer his teacher’s question. His teacher requested that his colleagues Liam, Anthony, and Jacob help him. Liam said that the answer was 157 degrees, Anthony said the answer was 58.9944 degrees, and Jacob said that the answer was 59.65 degrees. Whose answer is correct? Is it option (A) Anthony, option (B) Jacob? Or is it option (C) Liam?
In this question, three different students attempt to convert 58 degrees, 59 minutes, and 40 seconds into degrees. We need to determine which of the three options is correct. Is it 157 degrees, 58.9944 degrees, or 59.65 degrees? And we can do this by converting the angle into degrees ourselves. And we can do this by recalling a minute is one sixtieth of a degree and a second is one sixtieth of a minute. And this gives us a formula for converting an angle given in degrees, minutes, and seconds into one given in degrees.
𝑑 degrees, 𝑚 minutes, 𝑠 seconds will be equal to 𝑑 plus 𝑚 over 60 plus 𝑠 divided by 3600 degrees. And this follows directly from the definitions of minutes and seconds. A minute is one sixtieth of a degree and a second is one sixtieth of a minute, which means a second is one three thousand six hundredth of a degree.
Therefore, we can substitute 𝑑 is 58, 𝑚 is 59, and 𝑠 is 40 into this formula to find the angle in degrees. We get that 58 degrees, 59 minutes, and 40 seconds is equal to 58 plus 59 over 60 plus 40 divided by 3600 degrees. We could then evaluate this expression by using a calculator; however, Noah’s teacher wanted him to do this without using a calculator. So, we should also try and do this without using a calculator.
To do this, we first notice the number of minutes we’re given and the number of seconds we’re given are both less than 60. This means the combined number of degrees in minutes and seconds will be less than one full degree. So, our answer will be 58 point some value. This is actually enough to eliminate two of our options. The answer can’t be 157 degrees or 59 degrees because these start with a number which is not 58.
So, we could then mark that Anthony is correct. However, we should also check that this holds true. Since if we weren’t given any options, we might conclude all three were incorrect. We could, of course, do this by using a calculator. However, let’s attempt to verify this solution without using a calculator. First, we’re going to need to evaluate our expression. And to do this, we should simplify the final fraction, 40 divided by 3600. First, we can cancel the shared factor of 10 in the numerator and denominator. And similarly, we can cancel the shared factor of four in the numerator and denominator by noticing 360 over four is 90. This gives us 58 plus 59 over 60 plus one over 90 degrees.
We now want to add the second and third terms together. And to do this, we need to write them with a common denominator. We can do this by noticing we can write both fractions with a common denominator of 180. We multiply 59 over 60 by three divided by three to get 177 divided by 80 and one over 90 by two divided by two to get two divided by 180. We can then just add the numerators together to get 179 divided by 180.
Therefore, the angle in degrees, minutes, and seconds can be rewritten as 58 plus 179 divided by 180 degrees. And this is not an easy expression to evaluate without a calculator. The easiest possible way would be to notice it’s equal to 59 minus one divided by 180. To write this as a decimal, we would need to know that one divided by 180 is 0.005 recurring. This would then allow us to calculate the answer. It’s 58.994 recurring degrees, which to four decimal places is 58.9944 degrees.
Therefore, we were able to show the correct option was Anthony.