Lines and Angles | PRMO 2019 | Question 7

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

Answer: is 51.

PRMO, 2019, Question 7

Higher Algebra by Hall and Knight

Try with Hints

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

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

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

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.