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Math Help - Measure of angle formed by minute and hour hands

  1. #1
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    Measure of angle formed by minute and hour hands

    Hi,

    What is the measure of the angle formed by the minute and hour hands of a clock at 1:50?

    (A) 90^o
    (B) 95^o
    (C) 105^o
    (D) 115^o
    (E) 120^o

    Answer: D
    My guess was 90^o after mentally flipping it. My approach was worthless and remains worthless for questions of this type.
    I seek a better approach to this question and an explanation of this approach. I ask kindly.
    Last edited by Hellbent; November 6th 2010 at 05:52 PM.
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  2. #2
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    Quote Originally Posted by Hellbent View Post
    Hi,

    What is the measure of the angle formed by the minute and hour hands of a clock at 1:50?

    (A) 90^o
    (B) 95^o
    (C) 105^o
    (D) 115^o
    (E) 120^o

    Answer: C
    My guess was 90^o after mentally flipping it. My approach was worthless and remains worthless for questions of this type.
    I seek a better approach to this question and an explanation of this approach. I ask kindly.
    A 12-hour clock has 12 subdivisions, each of which is 30^o

    At 1:50, the minute hand is pointing at 10, so it is (2)30^0 left of 12.

    The hour hand will have moved \frac{10}{12} of 30^0 from it's initial position at 1 o'clock, when the minute hand was pointing at 12.

    This leaves it 30^0+\left(\frac{5}{6}\right)30^0 to the right of 12.
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  3. #3
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    Quote Originally Posted by Archie Meade View Post
    A 12-hour clock has 12 subdivisions, each of which is 30^o

    At 1:50, the minute hand is pointing at 10, so it is (2)30^0 left of 12.

    The hour hand will have moved \frac{10}{12} of 30^0 from it's initial position at 1 o'clock, when the minute hand was pointing at 12.

    This leaves it 30^0+\left(\frac{5}{6}\right)30^0 to the right of 12.
    My understanding is much better. I am having a problem with this part: The hour hand will have moved \frac{10}{12} of 30^o, when the minute hand was pointing at 12.

    Wouldn't it have been the minute hand that moved \frac{10}{12} of 30^o? Seeing that it has moved from 12 to 10 - 300^o.
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  4. #4
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    Quote Originally Posted by Hellbent View Post
    My understanding is much better. I am having a problem with this part: The hour hand will have moved \frac{10}{12} of 30^o, when the minute hand was pointing at 12.

    Wouldn't it have been the minute hand that moved \frac{10}{12} of 30^o? Seeing that it has moved from 12 to 10 - 300^o.
    I didn't write the first post too well.

    Imagine the time is initially 1 o'clock. The minute hand is at 12 and the hour hand is at 1.
    Both hands move and the clock reads 1:50.

    Yes, the minute hand will have moved through 30^o ten times

    while the hour hand will have moved through \frac{30^o}{12} ten times.

    The hour hand moves through 30^0 for a 360^0 movement of the minute hand.

    I'm getting (D), not (C).
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  5. #5
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    Thanks.

    Sorry, typo, it is indeed (D.) 115^o.

    I just changed the values in this question to check my understanding:
    What is the measure of the angle formed by the minute and hour hands of a clock at 3:45?
    \frac{45}{60} = \frac{3}{4}

    90^o + (\frac{3}{4})30^o = 112.5^o Then adding another 90^o (intervening angle between 9 and 12) gives 202.5^o

    Would the approach be the same for non-multiples of 5. Say, 4:47?
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  6. #6
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    Quote Originally Posted by Hellbent View Post
    Thanks.

    Sorry, typo, it is indeed (D.) 115^o.

    I just changed the values in this question to check my understanding:
    What is the measure of the angle formed by the minute and hour hands of a clock at 3:45?
    \frac{45}{60} = \frac{3}{4}

    90^o + (\frac{3}{4})30^o = 112.5^o Then adding another 90^o (intervening angle between 9 and 12) gives 202.5^o

    Would the approach be the same for non-multiples of 5. Say, 4:47?
    Yes,

    Starting at 4 o'clock, the minute hand will swing through \frac{47}{60}360^o

    Then the hour hand will swing through \frac{47}{60}30^o

    You can simply think in terms of "fractions of an hour" and so "fractions of 360^o" and "fractions of 30^o"
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    Thanks.
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