Try this beautiful problem from the PRMO, 2019 based on Lines and Angles.

Lines and Angles – PRMO 2019


On a clock, there are two instants between 12 noon and 1 pm when the hour hand and the minute hand are at right angles. the difference in minutes between these two instants written as a+\(\frac{b}{c}\) where a b c are positive integers with \(b\lt c\) and \(\frac{b}{c} \) in the reduced form. find a+b+c.

  • is 107
  • is 51
  • is 840
  • cannot be determined from the given information

Key Concepts


Lines

Algebra

Angles

Check the Answer


But try the problem first…

Answer: is 51.

Source
Suggested Reading

PRMO, 2019, Question 7

Higher Algebra by Hall and Knight

Try with Hints


First hint

Minute hand turns 6 (in degrees) in 1 min and hour hand turns half (in degree) in 1 min

Second Hint

x min after 12 is 6x -\(\frac{x}{2}\)=90 or 270 (in degrees) then x =16\(\frac{4}{11}\) or 49\(\frac{1}{11}\)

Final Step

Difference 32\(\frac{8}{11}\) then a+b+c=32+8+11=51.

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