What is the measure of the smaller angle between the hour and the

It helped me to visualize the clock. At first I kept leaving the hour hand unmoved at 6:00. Now, calculated the movement for the minute hand as 360/60 = 6 angle change per minute. 6 times 40 = 240 minus 360/12 = 30 angle change per hour for the hour hand. Therefore, 240 - 200 (30 times 6 + 30(2/3) = 40 degrees. The 30(2/3) is derived from the 2/3 of one hour change from 6:00 to 6:40.

I drew a diagram of a clock and wrote in the angles around the clock.

From the 9 tick to the 6 tick (clockwise) measures 270º.

Each hourly tick measures 30 (each fourth of the clock measures 90, there are two ticks in each fourth, which makes it 90/3). We know that the minute hand is on the 40, which is the 8 tick. This is 30º from the 9.

For the hourly hand, we know that the hour is 2/3 the way through (40/60). If we split each tick distance by 3, it gives us 10º for every 20 minutes. There are 40 minutes elapsed, which makes 20º.

270º (angle from 9 to 6)
30º (angle from 8 to 9)
20º (angle from 6 to 2/3rds to 7).

320º total for the outside angle. Subtract that from 360º gives us a result of 40º. Answer C.

6.40 means that we have 1/3*30=10 degree from short hand to 7 + 30 degree from 7 to long hand which is in 8 = 40 degree

C

These questions can be easily done once you understand the concept of speed of hour hand and speed of minute hand.

An hour hand covers 360 degrees (the full circle) in 12 hours ( = 720 mins). So it covers 1/2 degree every minute.
A minute hand covers 360 degrees every hour ( = 60 mins). So it covers 360/60 = 6 degrees every minute.
So the minute hand covers 5(1/2) = 11/2 degrees more than hour hand every minute. So the relative speed of the minute hand with reference to the hour hand is 11/2 degrees every minute. Now just use the relative speed concepts and you can find the angle between the two hands at any given time, even if the time given is not very easy to calculate.

Say what is the angle between them at 6:22?
At 6 o clock, the angle between them is 180 degrees. Say this is the distance between them initially - 180 degrees
In 22 minutes, the minute hand covers 11/2 * 22 = 121 degrees of distance between them.
So distance between them now = 180 - 121 = 59 degrees.

Similarly, take the case of 6:40
At 6 o clock, the angle between them is 180 degrees.
In 40 minutes, the minute hand covers 11/2 * 40 = 220 degrees of distance between them.
So distance between them now = 220 - 180 = 40 degrees.

Check out this post for more examples of this concept: https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2012/09 ... clockwork/

I have seen a lot of people here showing the full reasoning behind the answer. I know a shortcut that I think will help you save time.
Using the formula [(11/2)*min - 30*hour], we can solve any clock-related question.

Since the time is 6:40, we get [(11/2)*40 - 30*6] = 220 degrees, but the question asks for the smaller angle so we subtract 180, i.e 220-180=40 degrees.