How many minutes are left until 6:00 if, fifty minutes ago, the time was four times as many minutes past 3:00 as the number of minutes now remaining until 6:00?

Introduction / Context:
This is a classic clock and algebra puzzle that links a past time with the number of minutes remaining until a future time. The question connects the unknown number of minutes remaining until 6:00 with how many minutes past 3:00 it was fifty minutes ago. Such puzzles are common in aptitude tests because they test both your understanding of time intervals and your ability to set up and solve a simple linear equation using those intervals.

Given Data / Assumptions:

- Let x be the number of minutes from now until 6:00. - Fifty minutes ago, the time was (current time) minus 50 minutes. - Six o clock corresponds to 6:00, and three o clock corresponds to 3:00. - At that earlier time, the minutes past 3:00 are given to be four times x. - We assume a normal 12 hour clock with no irregularities.

Concept / Approach:
The key idea is to express all times in minutes counted from a fixed reference, here from 3:00. By letting x be the unknown minutes until 6:00, we can compute the actual clock time now, then compute the clock time fifty minutes before now, and finally express how many minutes that earlier time was past 3:00. The condition in the question translates into one linear equation in x, which we solve using basic algebra. Once x is found, we compare it to the answer choices.

Step-by-Step Solution:
Step 1: Let x be the number of minutes from now until 6:00. Step 2: The current time is 6:00 minus x minutes. Step 3: Fifty minutes ago, the time was 6:00 minus x minus 50 minutes. Step 4: Count minutes from 3:00. In minutes, 6:00 is 3 hours after 3:00, that is 180 minutes after 3:00. Step 5: The time fifty minutes ago is therefore 180 minus x minus 50 = 130 - x minutes after 3:00. Step 6: The condition says that at that moment, the minutes past 3:00 equal four times the minutes now remaining until 6:00, so 130 - x = 4x. Step 7: Solve the equation 130 - x = 4x which gives 130 = 5x and hence x = 26 minutes. Step 8: Therefore, there are 26 minutes left until 6:00.

Verification / Alternative check:
Compute the actual times. If x = 26, the current time is 5:34. Fifty minutes earlier, the time was 4:44. From 3:00 to 4:44 is 1 hour 44 minutes which is 60 + 44 = 104 minutes. Four times 26 is 104, which matches the statement that fifty minutes ago it was four times as many minutes past 3:00 as it is now to 6:00. This confirms that the algebraic solution is consistent with the actual clock times and therefore correct.

Why Other Options Are Wrong:
If we test 18, 20, 22, or 24 minutes in place of x, the computed time fifty minutes earlier does not give minutes past 3:00 equal to four times the assumed x. For example, with x = 24, the current time is 5:36 and fifty minutes earlier it is 4:46, which is 106 minutes after 3:00, not equal to 4 * 24 = 96. Similar mismatches occur for 18, 20 and 22, so they do not satisfy the condition.

Common Pitfalls:
A frequent error is to mix up the direction of time shifts, for example adding 50 instead of subtracting it for the earlier time, or forgetting to convert all times to a consistent reference like minutes after 3:00. Another pitfall is misinterpreting the statement and accidentally making x four times the past minutes instead of the other way around. Writing the equation carefully in symbolic form before solving helps avoid these traps.

Final Answer:
The number of minutes remaining until 6:00 is 26 minutes.