# ACT Math : Clock Math

## Example Questions

1 3 Next →

### Example Question #24 : Circles

What is the angle of the major arc between the minute and hour hands of a clock reading ? Assume a  display (not a military clock).

Explanation:

To find angular distance between the minute and hour hand, first find the position of each. Using  as a reference (both hands straight up), we can calculate the difference in degree more easily.

First, remember that a circle contains , and therefore for the minute hand, each minute past the hour takes up  of angular distance.

For the hour hand, each hour in a 12-hour cycle takes up  of angular distance, which means each minute takes up  of distance for the hour hand.

Thus, our distance from  (our reference angle) in degrees for the minute hand can be expressed as , where  is the number of minutes that have passed.

Likewise, our distance from  in degrees for the hour hand can be expressed as , where  is again the number of minutes that have passed.

Then, all we need to do is find the positive difference between these two measurements, , and we have our angle.

This looks more complicated than it is! Fortunately, once our equation is set up, the problem is as easy as plug and play. First, let's find the measure of the minor arc between the minute hand and the reference. We can ignore hours (as each hour returns the minute hand to our reference angle) and just look at minutes.

Now, find the same distance, but for the hour hand. Here, we must consider both hours and minutes by using a fraction:

Lastly, we find the difference between these two references (remembering that our answer should be positive):

The problem asked for the major arc, so our answer is actually

Thus, the hands are  apart at .

### Example Question #25 : Circles

What is the angle of the major arc between the minute and hour hands of a clock reading ? Assume a  display (not a military clock).

Explanation:

To find angular distance between the minute and hour hand, first find the position of each. Using  as a reference (both hands straight up), we can calculate the difference in degree more easily.

First, remember that a circle contains , and therefore for the minute hand, each minute past the hour takes up  of angular distance.

For the hour hand, each hour in a 12-hour cycle takes up  of angular distance, which means each minute takes up  of distance for the hour hand.

Thus, our distance from  (our reference angle) in degrees for the minute hand can be expressed as , where  is the number of minutes that have passed.

Likewise, our distance from  in degrees for the hour hand can be expressed as , where  is again the number of minutes that have passed.

Then, all we need to do is find the positive difference between these two measurements, , and we have our angle.

This looks more complicated than it is! Fortunately, once our equation is set up, the problem is as easy as plug and play. First, let's find the measure of the minor arc between the minute hand and the reference. We can ignore hours (as each hour returns the minute hand to our reference angle) and just look at minutes.

Now, find the same distance, but for the hour hand. Here, we must consider both hours and minutes by using a fraction:

Lastly, we find the difference between these two references (remembering that our answer should be positive):

Thus, the hands are  apart at .

### Example Question #26 : Circles

What is the angle of the major arc between the minute and hour hands of a clock reading ? Assume ab display (not a military clock).

Explanation:

To find angular distance between the minute and hour hand, first find the position of each. Using  as a reference (both hands straight up), we can calculate the difference in degree more easily.

First, remember that a circle contains , and therefore for the minute hand, each minute past the hour takes up  of angular distance.

For the hour hand, each hour in a 12-hour cycle takes up  of angular distance, which means each minute takes up  of distance for the hour hand.

Thus, our distance from  (our reference angle) in degrees for the minute hand can be expressed as , where  is the number of minutes that have passed.

Likewise, our distance from  in degrees for the hour hand can be expressed as , where  is again the number of minutes that have passed.

Then, all we need to do is find the positive difference between these two measurements, , and we have our angle.

This looks more complicated than it is! Fortunately, once our equation is set up, the problem is as easy as plug and play. First, let's find the measure of the minor arc between the minute hand and the reference. We can ignore hours (as each hour returns the minute hand to our reference angle) and just look at minutes.

Now, find the same distance, but for the hour hand. Here, we must consider both hours and minutes by using a fraction:

Lastly, we find the difference between these two references (remembering that our answer should be positive):

The problem asked for the major arc, so our answer is actually

Thus, the hands are  apart at .

### Example Question #491 : Plane Geometry

A watch's hands move from  to . What total angular distance does the minute hand move?

Explanation:

First, remember that a circle contains , and therefore for the minute hand, each minute past the hour takes up  of angular distance.

Thus, our total distance from our reference angle can be found as , where  is the number of minutes that have elapsed.

In this case, we simply find the difference of the two times, remembering to first convert hours to minutes:

Now, plug our answer in minutes into our equation:

So, our minute hand has moved  in total.

### Example Question #492 : Plane Geometry

A watch's hands move from  to . What total angular distance does the minute hand move?

Explanation:

First, remember that a circle contains , and therefore for the minute hand, each minute past the hour takes up  of angular distance.

Thus, our total distance from our reference angle can be found as , where  is the number of minutes that have elapsed.

In this case, we simply find the difference of the two times, remembering to first convert hours to minutes:

Now, plug our answer in minutes into our equation:

So, our minute hand has moved  in total.

### Example Question #11 : How To Find The Angle Of Clock Hands

A watch's hands move from  to . What total angular distance does the minute hand move?

Explanation:

First, remember that an hour hand moves  for each hour that passes, and therefore for the hour hand, each minute past the hour takes up  of angular distance.

Thus, our total distance from our reference angle can be found as , where  is the number of minutes that have elapsed.

In this case, we simply find the difference of the two times, remembering to first convert hours to minutes:

Now, plug our answer in minutes into our equation:

So, our hour hand has moved  in total.

### Example Question #21 : Circles

A clock reads  pm.  What is the angle formed between the minute and hour hand on the clock?

Explanation:

When the clock reads , the hour hand is on the , and the minute hand is on the   If we think about this as a fraction, there are twelve spots the hour hand can be on, which we means we are on the  position.

Since there are  in a circle, the angle can simply be found by multiplying this fraction by the number of degrees in a circle:

Alternatively, if the clock reads , the angle the clock reads is visually  of the entire clock, which has .

.

### Example Question #31 : Circles

On a standard analog clock, what is the angle between the hands when the clock reads ? Give the smaller of the two angles.