A clock gains time at a steady rate. It is set to the correct time at 10: 00 AM. When the clock shows 2 : 00 PM, the correct time is
1: 36 PM. How many minutes fast will the clock be the next time the
correct time is 5 : 30 PM?

The clock will be 50 minutes fast

Step-by-step explanation:

Proportions

It's known a clock gains time at a steady state, or constant rate.

At 10:00 AM the clock is set at the correct time and at 1:36 PM the clock shows 2:00 PM. This means in that period of time, it has gained 24 minutes.

From 10:00 AM to 1:36 PM there are 3 hours and 36 minutes, or equivalently, 3*60+36 = 216 minutes

Our clock gains time at a rate of 24 minutes each 216 minutes. This gives the ratio 24:216 = 1:9.

When the new correct time is 5:30 PM, there will have passed 7 hours and 30 minutes since 10:00 AM, or equivalently, 7*60+30 = 450 minutes.

Since the clock gains time at a ratio 1:9, the time gained is 450/9=50 minutes.

The clock will show 50 minutes more than the correct time of 5:30 PM, that is, 6:20 PM.

The clock will be 50 minutes fast