What is the first time after 12 00 that the hour hand and the minute hand of a clock are together?

By the time the minute hand has gone all the way round the clock and is back at 12, one hour later (i.e., at 1 o'clock), the hour hand has moved to 1. Five minutes later, the minute hand reaches 1, but they don't quite meet there because the hour hand would have moved 1/12 of the distance between 1 and 2, to indicate 5 minutes past 1. So the next time after 12 that the minute hand is directly over the hour hand is a bit after 1:05 (we'll calculate the exact time a bit later).

Similarly, the next time it happens is a bit after 2:10. Then a bit after 3:15, and so on.

Note that each time they meet, the number of minutes past the hour keeps increasing, so the hour hand would have moved closer to the next number. After 11 o'clock, the minute hand has to travel all the way and by the time they meet it is has to be 12 o'clock again, since we know what the clock looks like at that time. So the two hands overlap 11 times in a 12 hour period. So, in a 24 hour period, they would overlap 22 times.

To answer the second part of the question, let's try to figure out the little bit of extra time the minute hand needs to catch up to the hour hand after every 1:05 hours. Well, after 12 o'clock there are eleven occasions when the two hands match up, and since the clock hands move at constant speeds, those 11 events are spread equally apart around the clock face, so they are 1/11th of an hour apart. That's 5.454545 minutes apart. In other words they meet every after every 1 hour and 5.454545 minutes.

The precise times they overlap (in hours) would be 1 + 1/11, 2 + 2/11, 3+ 3/11, all the way up to 11 + 11/11, which is 12 o'clock again.

Q. An analog clock is showing 2 o'clock right now. At what time will the hour and minute hands overlap next?

Q. What is the angle between the hour and minute hands at 4:10?

What is the angle formed by the minute hand and the hour hand at 4:45?

Possible Answers:

Correct answer:

Explanation:

The angle measure between any two consecutive numbers on a clock is

Call the "12" point on the clock the zero-degree point.

At 4:45, the minute hand is at the "9" - that is, at the

mark. The hour hand is three-fourths of the way from the "4" to the "5; that is,

Therefore, the angle between the hands is

, the desired measure.

What is the angle between the hour hand and the minute hand at 4:40?

Possible Answers:

Correct answer:

Explanation:

At 4:40, the minute hand is on the 8, and the hour hand is two-thirds of the way from the 4 to the 5. That is, the hands are three and one-third number positions apart. Each number position is thirty degrees around the clock, so the hands form an angle of

It is 4 o’clock. What is the measure of the angle formed between the hour hand and the minute hand?

Possible Answers:

Correct answer:

Explanation:

At four o’clock the minute hand is on the 12 and the hour hand is on the 4. The angle formed is 4/12 of the total number of degrees in a circle, 360.

4/12 * 360 = 120 degrees

The hour hand on a clockface points to the

, and the minute hand points to the

. How many degrees is the angle between the minute and hour hands?

Possible Answers:

Correct answer:

Explanation:

There are

degrees in one complete revolution of a circle. There are

minutes in one hour.

Create a fraction out of these two quantities to use later as a conversion rate:

Between the

and

there are

minutes, so multiply this by the conversion rate to solve for the number of degrees:

What is the measure of the smaller angle formed by the hands of an analog watch if the hour hand is on the 10 and the minute hand is on the 2?

Possible Answers:

Explanation:

A analog clock is divided up into 12 sectors, based on the numbers 1–12. One sector represents 30 degrees (360/12 = 30). If the hour hand is directly on the 10, and the minute hand is on the 2, that means there are 4 sectors of 30 degrees between then, thus they are 120 degrees apart (30 * 4 = 120).

Thomas is trying to determine the angle between the hands of his clock. Right now it reads

pm, what angle do the clock hands make?

Possible Answers:

Correct answer:

Explanation:

You can think of a clock in two ways:

1. Out of 12 hours, or

2. In terms of a circle with

If you try to solve it in terms of #1:

Goal: Find the angle measurement between the hour and the minute hands. We only want to find the degrees between the hours of 9 and 12

So we are looking at 3 hours out of the 12 total hours on a clock.

As a fraction:

So that means that the clock hands are making an angle that is 1/4 of the clock (which is a circle). So knowing that a circle has

in it,

1/4 of a circle is

____________________________________________________________

2. If you think of the clock as a circle first you can determine the angle that the clock hands create very quickly.

Since there are

in a circle, every hour that passes is a movement of

. So knowing that, the clock will be moving 3 hours:

What is the measure of the larger angle formed by the hands of a clock at

Possible Answers:

Correct answer:

Explanation:

Like any circle, a clock contains a total of

. Because the clock face is divided into

equal parts, we can find the number of degrees between each number by doing

. At 5:00 the hour hand will be at 5 and the minute hand will be at 12. Using what we just figured out, we can see that there is an angle of

between the two hands. We are looking for the larger angle, however, so we must now do

What is the measure, in degrees, of the acute angle formed by the hands of a 12-hour clock that reads exactly 3:10?

Possible Answers:

Explanation:

The entire clock measures 360°. As the clock is divided into 12 sections, the distance between each number is equivalent to 30° (360/12). The distance between the 2 and the 3 on the clock is 30°. One has to account, however, for the 10 minutes that have passed. 10 minutes is 1/6 of an hour so the hour hand has also moved 1/6 of the distance between the 3 and the 4, which adds 5° (1/6 of 30°). The total measure of the angle, therefore, is 35°.

Find the angle in degrees between clock hands at 3:30.

Possible Answers:

Correct answer:

Explanation:

On first glance, this problem may seem simple in that the angle between the 3 and 6 on a clock is one quarter of a circle or 90 degrees. However, you must take into account that in the half hour that has passed from 3:00 to 3:30, the hour hand has moved half the distance between the 3 and 4. To find the degrees, simply divide the total number of degrees in a circle by 12 to find the degrees between each consecutive number, and then multiply that number by 2.5 because you have half the distance to the 4, and the the full distance to the 5 and the full distance to the 6. Thus,

A clock shows that the time is 9:00am. What is the angle between the minute and the hour hands?

Possible Answers:

Correct answer:

Explanation:

By dividing the clock into pieces, we can determine that the angle between the two hands is

Within a clock, just like any circle, there are 360 total degrees.

Within an clock, there are 60 total minutes.

Each minute that passes, the minute hand advances 6 degrees. The hour hand advances .5 degrees. But since it is 9am on the dot, we will just be using the minute hand to count the degrees.

Since the minute hand is covering a total of 15 minutes in between it and the hour hand, we can do the math:

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