### All ACT Math Resources

## Example Questions

### 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).

**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):

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).

**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 #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).

**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):

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?

**Possible Answers:**

**Correct answer:**

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?

**Possible Answers:**

**Correct answer:**

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?

**Possible Answers:**

**Correct answer:**

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?

**Possible Answers:**

**Correct answer:**

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.

**Possible Answers:**

**Correct answer:**

To find the degrees of a clock hand, first find the angle between each hour-long sections. Since there are evenly spaced sections, we find that each section has an angle of: . at the hour hand has gone one-third of the way between the and . Thus there are two-thirds of between the hour hand and the . .

There are between and , where the minute hands is. Thus there's a total of between the hands.

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