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