Question

What is the number of degrees in the acute angle formed by the hands of a clock at 6:44?

Answer 1

[asy]

unitsize(0.8inch);

for (int i=0 ; i<=11 ;++i)

{

draw((rotate(i*30)*(0.8,0)) -- (rotate(i*30)*(1,0)));

label(format("%d",i+1),(rotate(60 - i*30)*(0.68,0)));

}

draw(Circle((0,0),1),linewidth(1.1));

draw(rotate(186)*(0.7,0)--(0,0)--(rotate(-22)*(0,-0.5)),linewidth(1.2));

[/asy]

There are 12 hours on a clock, so each hour mark is $360^\circ/12 = 30^\circ$ from its neighbors. At 6:44, the minute hand points at minute 44, which is $\frac45$ of the way from hour 8 to hour 9. Therefore, the minute hand is $\frac45\cdot 30^\circ = 24^\circ$ past hour 8. The hour hand is $\frac{44}{60} = \frac{11}{15}$ of the way from hour 6 to hour 7, so it is $\frac{11}{15}\cdot 30^\circ = 22^\circ$ past hour 6. This means that the hour hand is $30^\circ -22^\circ = 8^\circ$ from hour 7. Since hours 7 and 8 are $30^\circ$ apart, the total angle between the two hands is $8^\circ + 30^\circ + 24^\circ = \boxed{62^\circ}$.

Answer 2

The minute hand has moved 44 minutes. Hence, it is at: 44 60 × 360 ∘ = 264 ∘ mark. Hence, the angle between the hands = 264 − 202 = 62