# ACT Math : Clock Math

## Example Questions

### Example Question #1 : How To Find The Angle Of Clock Hands

How many degrees are in each hour-long section of an analog clock with 8 equally spaced numbers on the face?

Explanation:

If creating a picture helps, draw a circle and place 8 at the top where 12 normally is. Then put 2, 4, and 6 at the positions of 3, 6, and 9 on a normal 12-hour analog clock. 1, 3, 5, and 7 go halfway in between each even number.

Now, because each section is equally spaced, and because there are 8 sections we simply divide the total number of degrees in a circle () by the number of sections (8). Thus:

### Example Question #2 : How To Find The Angle Of Clock Hands

What is the angle between the hands of a standard 12-hour digital clock when it is 8:15? (note, give the smaller of the two angles, the one between the hands going clockwise

Explanation:

When the clock reads 8:15 the minute hand is on the 3 and the hour hand is just past the 8.

Each section of the clock is

degrees.

From 3 to 8 then there are 150 degrees. However, the hour hand has moved a quarter of the way between the 8 and the 9, or a quarter of 30 degrees.  and so

### Example Question #3 : How To Find The Angle Of Clock Hands

What is the measure of the angle between the hands of a clock at ? (compute the angle going clockwise from the hour hand to the minute hand)

Explanation:

Each section of the clock is , and by  the hour hand has gone three quarters of the way between the  and the . Thus there are  between the hour hand and the  numeral. The minute hand is on the , and there are  between the  and the . So in total there are  between the hands

### Example Question #4 : How To Find The Angle Of Clock Hands

On a  analog clock (there is a  where the  normally is, a  in the normal position of the , with  where the  is on a standard  analog, the  where the  is on a standard analog, and  and  are at the spots normally occupied by  and  respectively), what is the angle between the hands when the clock reads

Explanation:

The number of degrees between each numeral on the clock face is equal to the number of degrees in a circle divided by the number of sections:

At  the  hand has gone  way through the  between the and the . Thus there are only  left between it and the 3. There are 120 degrees between the 3 and the 5, where the minute hand is, so the total amount of degrees between the hands is:

### Example Question #5 : How To Find The Angle Of Clock Hands

What is the angle between the clock hands when the clock reads 6:30?

Explanation:

Remember there are  in each hour long section of the clockface

.

When the clock reads 6:30 the minute hand is on the 6, and the hour hand is halfway between the 6 and 7.

Thus the number of degrees between the hands is

### Example Question #6 : How To Find The Angle Of Clock Hands

On a  analog clock (with a  where the  normally is and a  where the  normally is) what is the angle between the hands when the clock reads ? (Give the smaller of the two angles)

Explanation:

When the clock reads  on this clock, the hour hand will be  of the way between the  and the . Since there are , evenly sized, sections of this clock each section has:

. And . At  the minute hand will be one-quarter of the way around the entire dial.
Thus the hands are  from each other

### Example Question #1 : How To Find The Angle Of Clock Hands

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

72°

55°

65°

60°

35°

35°

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°.

### Example Question #21 : Circles

What is the angle of the minor 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 #22 : Circles

What is the angle of the minor 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 #21 : Circles

What is the angle of the minor 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 .