A reservation clerk worked 12.6 hours one day. She spent twice as much time entering new reservations as she did verifying old ones and a half as much time calling to confirm reservations as verifying old ones. How much time did she spend entering new reservations?

x + 2x + 0.5x = 12.6

3.5x = 12.6

x = 3.6

2 * 3.6 = 7.2

To find out how much time the reservation clerk spent entering new reservations, we need to determine the time she spent in total on all three activities: entering new reservations, verifying old ones, and calling to confirm reservations.

Let's assign variables to represent the time spent on each activity. Let's say:
x = time spent entering new reservations
y = time spent verifying old reservations
z = time spent calling to confirm reservations

According to the given information, the reservation clerk spent twice as much time entering new reservations as she did verifying old ones and a half as much time calling to confirm reservations as verifying old ones. This can be represented as two equations:
x = 2y (equation 1)
z = 0.5y (equation 2)

We also know that the reservation clerk worked 12.6 hours in total. This can be expressed as an equation:
x + y + z = 12.6 (equation 3)

Now, we have a system of three equations (equations 1, 2, and 3). We can solve this system by substituting the values from equation 1 and equation 2 into equation 3:

2y + y + 0.5y = 12.6
3.5y = 12.6
y = 12.6 / 3.5
y ≈ 3.6 hours

According to equation 1, x = 2y, so:
x = 2 * 3.6
x ≈ 7.2 hours

Therefore, the reservation clerk spent approximately 7.2 hours entering new reservations.