Clock
1.An accurate clock shows 8 o'clock in the morning. Through how may degrees will the hour hand rotate when the clock shows 2 o'clock in the afternoon?

Explanation:

Solution:
$\begin{array}{rl}& \text{Angle}\phantom{\rule{thinmathspace}{0ex}}\text{traced}\phantom{\rule{thinmathspace}{0ex}}\text{be}\phantom{\rule{thinmathspace}{0ex}}\text{the}\phantom{\rule{thinmathspace}{0ex}}\text{hour}\phantom{\rule{thinmathspace}{0ex}}\text{hand}\phantom{\rule{thinmathspace}{0ex}}\text{in}\phantom{\rule{thinmathspace}{0ex}}\text{6}\phantom{\rule{thinmathspace}{0ex}}\text{hours}\\ & ={\left(\frac{360}{12}×6\right)}^{\circ }={180}^{\circ }\end{array}$

2.The reflex angle between the hands of a clock at 10.25 is:

Explanation:

Solution:
$\begin{array}{rl}& \text{Angle}\phantom{\rule{thinmathspace}{0ex}}\text{traced}\phantom{\rule{thinmathspace}{0ex}}\text{by}\phantom{\rule{thinmathspace}{0ex}}\text{hour}\phantom{\rule{thinmathspace}{0ex}}\text{hand}\phantom{\rule{thinmathspace}{0ex}}\text{in}\phantom{\rule{thinmathspace}{0ex}}\frac{125}{12}hrs\\ & ={\left(\frac{360}{12}×\frac{125}{12}\right)}^{\circ }=312\frac{{1}^{\circ }}{2}\\ & \text{Angle}\phantom{\rule{thinmathspace}{0ex}}\text{traced}\phantom{\rule{thinmathspace}{0ex}}\text{by}\phantom{\rule{thinmathspace}{0ex}}\text{minute}\phantom{\rule{thinmathspace}{0ex}}\text{hand}\phantom{\rule{thinmathspace}{0ex}}\text{in}\phantom{\rule{thinmathspace}{0ex}}\text{25}\phantom{\rule{thinmathspace}{0ex}}min\\ & ={\left(\frac{360}{60}×25\right)}^{\circ }={150}^{\circ }\\ & \therefore \text{Reflex}\phantom{\rule{thinmathspace}{0ex}}\text{angle}\\ & ={360}^{\circ }-{\left(312\frac{1}{2}-150\right)}^{\circ }\\ & ={360}^{\circ }-162\frac{{1}^{\circ }}{2}\\ & =197\frac{1}{2}\end{array}$

3.A clock is started at noon. By 10 minutes past 5, the hour hand has turned through:

Explanation:

Solution:
$\begin{array}{rl}& \text{Angle}\phantom{\rule{thinmathspace}{0ex}}\text{traced}\phantom{\rule{thinmathspace}{0ex}}\text{by}\phantom{\rule{thinmathspace}{0ex}}\text{hour}\phantom{\rule{thinmathspace}{0ex}}\text{hand}\phantom{\rule{thinmathspace}{0ex}}\text{in}\phantom{\rule{thinmathspace}{0ex}}12\phantom{\rule{thinmathspace}{0ex}}hrs={360}^{\circ }\\ & \text{Angle}\phantom{\rule{thinmathspace}{0ex}}\text{traced}\phantom{\rule{thinmathspace}{0ex}}\text{by}\phantom{\rule{thinmathspace}{0ex}}\text{hour}\phantom{\rule{thinmathspace}{0ex}}\text{hand}\phantom{\rule{thinmathspace}{0ex}}\text{in}\phantom{\rule{thinmathspace}{0ex}}5\phantom{\rule{thinmathspace}{0ex}}hrs\phantom{\rule{thinmathspace}{0ex}}10\phantom{\rule{thinmathspace}{0ex}}min.\phantom{\rule{thinmathspace}{0ex}}i.e.,\\ & \frac{31}{6}hrs={\left(\frac{360}{12}×\frac{31}{6}\right)}^{\circ }={155}^{\circ }\end{array}$

4.How much does a watch lose per day, if its hands coincide every 64 minutes?

Explanation:

Solution:
$\begin{array}{rl}& 55\phantom{\rule{thinmathspace}{0ex}}min.\phantom{\rule{thinmathspace}{0ex}}\text{spaces}\phantom{\rule{thinmathspace}{0ex}}\text{are}\phantom{\rule{thinmathspace}{0ex}}\text{covered}\phantom{\rule{thinmathspace}{0ex}}\text{in}\phantom{\rule{thinmathspace}{0ex}}60\phantom{\rule{thinmathspace}{0ex}}min\\ & 60\phantom{\rule{thinmathspace}{0ex}}min.\phantom{\rule{thinmathspace}{0ex}}\text{spaces}\phantom{\rule{thinmathspace}{0ex}}\text{are}\phantom{\rule{thinmathspace}{0ex}}\text{covered}\phantom{\rule{thinmathspace}{0ex}}\text{in}\\ & =\left(\frac{60}{55}×60\right)\phantom{\rule{thinmathspace}{0ex}}min.\\ & =65\frac{5}{11}\phantom{\rule{thinmathspace}{0ex}}min.\\ & \text{Loss}\phantom{\rule{thinmathspace}{0ex}}\text{in}\phantom{\rule{thinmathspace}{0ex}}64\phantom{\rule{thinmathspace}{0ex}}min.\\ & =\left(65\frac{5}{11}-64\right)=\frac{16}{11}\phantom{\rule{thinmathspace}{0ex}}min.\\ & \text{Loss}\phantom{\rule{thinmathspace}{0ex}}\text{in}\phantom{\rule{thinmathspace}{0ex}}24\phantom{\rule{thinmathspace}{0ex}}hrs\\ & =\left(\frac{16}{11}×\frac{1}{64}×24×60\right)\phantom{\rule{thinmathspace}{0ex}}min\\ & =32\frac{8}{11}\phantom{\rule{thinmathspace}{0ex}}min\end{array}$

5.At what time between 7 and 8 o'clock will the hands of a clock be in the same straight line but, not together?

Explanation:

Solution:
When the hands of the clock are in the same straight line but not together, they are 30 minute spaces apart.
At 7 o'clock, they are 25 min. spaces apart.
∴ Minute hand will have to gain only 5 min. spaces.
55 min. spaces are gained in 60 min.
5 min. spaces are gained in
$\begin{array}{rl}& \left(\frac{60}{55}×5\right)\phantom{\rule{thinmathspace}{0ex}}min=5\frac{5}{11}\phantom{\rule{thinmathspace}{0ex}}min\\ & \therefore \text{Required}\phantom{\rule{thinmathspace}{0ex}}\text{time}=5\frac{5}{11}\phantom{\rule{thinmathspace}{0ex}}min.\phantom{\rule{thinmathspace}{0ex}}past\phantom{\rule{thinmathspace}{0ex}}7\end{array}$

6.At what time between 5.30 and 6 will the hands of a clock be at right angles?

Explanation:

Solution:
At 5 o'clock, the hands are 25 min. spaces apart.
To be at right angles and that too between 5.30 and 6, the minute hand has to gain (25 + 15) = 40 min. spaces.
55 min. spaces are gained in 60 min.
40 min. spaces are gained in
$\begin{array}{rl}& \left(\frac{60}{55}×40\right)\phantom{\rule{thinmathspace}{0ex}}min=43\frac{7}{11}\phantom{\rule{thinmathspace}{0ex}}min\\ & \therefore \text{Required}\phantom{\rule{thinmathspace}{0ex}}\text{time}=43\frac{7}{11}\phantom{\rule{thinmathspace}{0ex}}min.\phantom{\rule{thinmathspace}{0ex}}past\phantom{\rule{thinmathspace}{0ex}}5\end{array}$

7.The angle between the minute hand and the hour hand of a clock when the time is 4.20, is:

Explanation:

Solution:
$\begin{array}{rl}& \text{Angle}\phantom{\rule{thinmathspace}{0ex}}\text{traced}\phantom{\rule{thinmathspace}{0ex}}\text{by}\phantom{\rule{thinmathspace}{0ex}}\text{hour}\phantom{\rule{thinmathspace}{0ex}}\text{hand}\phantom{\rule{thinmathspace}{0ex}}\text{in}\phantom{\rule{thinmathspace}{0ex}}\frac{13}{3}\phantom{\rule{thinmathspace}{0ex}}\text{hrs}\\ & ={\left(\frac{360}{12}×\frac{13}{3}\right)}^{\circ }={130}^{\circ }\\ & \text{Angle}\phantom{\rule{thinmathspace}{0ex}}\text{traced}\phantom{\rule{thinmathspace}{0ex}}\text{by}\phantom{\rule{thinmathspace}{0ex}}\text{min}\text{.}\phantom{\rule{thinmathspace}{0ex}}\text{hand}\phantom{\rule{thinmathspace}{0ex}}\text{in}\phantom{\rule{thinmathspace}{0ex}}\text{20}\phantom{\rule{thinmathspace}{0ex}}\text{min}\\ & ={\left(\frac{360}{60}×20\right)}^{\circ }={120}^{\circ }\\ & \therefore \text{Required}\phantom{\rule{thinmathspace}{0ex}}\text{angle}\\ & ={\left(130-120\right)}^{\circ }\\ & ={10}^{\circ }\end{array}$

8.At 3:40, the hour hand and the minute hand of a clock form an angle of:

Explanation:

Solution:

9.How many times are the hands of a clock at right angle in a day?

Explanation:

Solution:
In 12 hours, they are at right angles 22 times.
∴ In 24 hours, they are at right angles 44 times.

10.The angle between the minute hand and the hour hand of a clock when the time is 8.30, is: