What angle does minute hand and hour hand clock inclined to each other at $2:25$

What angle does minute hand and hour hand clock inclined to each other at $2:25$

My Attempt:

The hour hand of a clock rotate throufh an angle of $30°$ in $1$ hour,

So,

At $2$o'clock, $2$ hrs$\equiv 60°$

Again,

The minute hand of a clock rotate through an angle of $6°$ in $1$ minute.

So, At $2:25$, i.e, $25$min$\equiv 150°$

What should I do further?

Time 2 hrs 25 minutes. Converting into hours $2\frac{25}{60} = 2 \frac5{12} = \frac {29}{12}$

Angle made by hour hand in $\frac {29}{12}$ hrs $= \frac{360}{12} \times \frac{29}{12} = \frac {145}{2} = 72.5°$

Angle made by minute hand in 25 min $= \frac{360}{60} \times 25 = 150°$

Difference in angle $= 150° - 72.5° = 77.5°$

You're correct in thinking that the minute hand will be at 150°, but the hour hand moves from 2:00 to 2:25, no?

You can think about 2:25 as being 2.416666 hours. Thus, 2.41666 x 30° = 72.5°.

So, minute hand at 150°, hour hand at 72.5°, what is the angle between the two?

• how did you get $2.416666...$?? – pi-π Apr 6 '17 at 15:54
• 0.41666 = 25/60. This is because the hour hand lies at 60° at 2:00, but as time goes on the hour hand continues to move. So, at 2:25, it is 25/60 = .41666 of an hour further. Each hour it moves 30°, so from 2:00 to 2:25 it moves .41666 x 30° = 12.5°. – mizichael Apr 6 '17 at 15:56
• I would have done this just slightly differently: In the 25 minutes after 2 the hour hand will have moved 25/60= 5/12 of the 30 degree so an additional (5/12)(30)= 25/2 degrees. – user247327 Apr 6 '17 at 15:57
• @user247327 I think this is just a matter of preference. – mizichael Apr 6 '17 at 16:01

I'd envision it like so:

The blue arc represents a rotation of $2/12$ of a full rotation, or $60^{\circ}$ since a full rotation is $360^{\circ}$.

The green segment represents a rotation in the opposite direction of $(1/12)-(25/60)\times(1/12)$ of a full rotation or $17.5^{\circ}$; the $1/12$ represents the rotation from the three o'clock hour back to the two o'clock hour, while the $(25/60)\times(1/12)$ represents the proportion that the hour hand moves as the time changes from 2:00 to 2:25.

Taken together, the angle between the two is $60^{\circ}+17.5^{\circ}$ or $77.5^{\circ}$.

• [+1] for the clock... and the explanation. – Jean Marie Apr 6 '17 at 18:42

You're so close!

As you say, the hour hand moves 30 degrees in 1 hour, hence 0.5° per minute, and therefore 12.5° from where it was at 2pm, which was at 60° from 'zero'. So at 2:25 it is at 72.5° from 'zero'.

And, like you say, the minute hand is at 150°. So just subtract 72.5° from 150°, which is 77.5° difference!

• I believe you're 1 degrees off – mrnovice Apr 6 '17 at 16:19
• Hello bro, 150 - 72.5 = 77.5 $\ne$ 78.5 – Kanwaljit Singh Apr 6 '17 at 16:19
• @mrnovice KanwaljitSingh Wow .... Blunder! Thanks for correcting me. Back to grade school ... – Bram28 Apr 6 '17 at 16:21
• @mrnovice what for what? – Kanwaljit Singh Apr 6 '17 at 16:26