Clocks Questions and Answers

Answer: B

The angle is 7.5°.

At 3:15 the minute hand points exactly at 3. The hour hand, though, has moved a quarter of the way from 3 to 4, since a quarter of the hour has passed. One hour on the dial is \(30^\circ\):

\[ \frac{15}{60} \times 30^\circ = 7.5^\circ \]

General formula: angle \(= |30H - 5.5M| = |90 - 82.5| = 7.5^\circ\). 0° is the trap of assuming the hour hand stays on 3; 15° and 22.5° come from using the wrong rate of 1° or 1.5° per minute.

Answer: B

The right time is 11:00 p.m.

The clock loses 16 minutes a day, so it shows 23 h 44 min (1424 minutes) for every 24 true hours (1440 minutes). From 5 a.m. on day 1 to 10 p.m. on day 4 the clock shows 3 days 17 hours, which is 89 hours.

\[ \begin{aligned} \text{true time} &= 89 \times \frac{1440}{1424} \\ &= 90\ \text{hours} \end{aligned} \]

90 hours after 5 a.m. on day 1 is 11 p.m. on day 4. The clock is behind by an hour, so the true time is later than 10 p.m.; 11:15 and the midnight options overshoot.

Answer: D

The hour hand will point North-east.

At 4.30 the minute hand is on 6 and the hour hand is halfway between 4 and 5, which is 45° clockwise from the 3 (and 135° from the 12).

On a normally held clock the 6 points south, but here the minute hand points east, so the whole dial has been turned 90° anticlockwise. The hour hand normally points south-east (135°), and turning it 90° anticlockwise brings it to 45°, which is north-east.

South-east is where the hour hand would point if the dial were not turned.

Answer: B

The hands are at right angles 44 times in a day.

The minute hand gains 330° on the hour hand every hour, so it gains a full circle 11 times in 12 hours. In each of those 11 laps it passes through a right angle twice (once 90° ahead, once 90° behind), so the hands are at right angles 22 times in 12 hours.

\[ \begin{aligned} 12\text{ hours} &= 11 \times 2 = 22 \\ 24\text{ hours} &= 22 \times 2 = 44 \end{aligned} \]

22 is the count for 12 hours only. 24 assumes two right angles in every hour of a 12-hour period, which overcounts: around 3 o'clock and 9 o'clock one right angle is shared between neighbouring hours, so two hours lose one each.

Answer: A

In a clock, 12 hours form 360° degree. So one hour forms 30°. Therefore 7 hours form 210° (30° X 7) degree. In a minute, hour hand moves 0.5° degree, so in 10 minutes it moves 0.5° X 10 = 5°
Thus, by 10 Minutes past 7, hour hand has turn through 210° + 5° = 215°

Answer: B

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 must gain only 5 min. space
55 min. spaces are gained in 60 min.
5 min. spaces are gained in (60/55 × 5) min = 5 (5/11) min. 
Required time = 5 (5/11) min. past 7.

Answer: B

Correct answer: 315°

First find the ordinary angle between the hands.

\[\text{Hour hand} = 4 \times 30 + 30 \times 0.5 = 120 + 15 = 135°\]

\[\text{Minute hand} = 30 \times 6 = 180°\]

The hands are \(180 - 135 = 45°\) apart, so the reflex angle is

\[360 - 45 = 315°\]

Answer: A

Correct answer: 38 (2/11) min past 1

At 1 o’clock the hour hand is \(30°\) ahead. To be opposite, the minute hand must pass it and then lead by \(180°\), a total relative gain of \(30 + 180 = 210°\).

\[t = \frac{210}{5.5} = \frac{420}{11} = 38\tfrac{2}{11}\ \text{minutes}\]

So the hands are opposite at \(38\tfrac{2}{11}\) minutes past 1.

Answer: D

At 3 o'clock ,the moment hand is 15 min. spaces a section
Structure the hour hand. To be coincident,
it must pick up 15 min. spaces.
55 min. are picked up in 60 min.
15 min. are picked up in (60/55*15)min.=180/11 min.
∴ The hands are correspondent at 180/11 min. past 3.

Answer: B

In each twelve-hour period the hands are at right angles 22 times - twice in most hours, but the crossings near 3 and 9 o'clock overlap so that two are lost.

\[ 22 \times 2 = 44 \text{ times in 24 hours} \]

The count is not 48, because between 2–3 and 3–4 (and again around 9) one right angle is shared rather than occurring twice.