### All ACT Math Resources

## 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?

**Possible Answers:**

**Correct answer:**

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

**Possible Answers:**

**Correct answer:**

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)

**Possible Answers:**

**Correct answer:**

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 ?

**Possible Answers:**

**Correct answer:**

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?

**Possible Answers:**

**Correct answer:**

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)

**Possible Answers:**

**Correct answer:**

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?

**Possible Answers:**

72°

55°

65°

60°

35°

**Correct answer:**

35°

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).

**Possible Answers:**

**Correct answer:**

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).

**Possible Answers:**

**Correct answer:**

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).

**Possible Answers:**

**Correct answer:**

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 .