How many minutes are left until 6:00 if, 50 minutes ago, the time was 4 times as many minutes past 3:00 as the number of minutes now remaining until 6:00?

Introduction / Context:
This question is a variant of a standard time puzzle involving relationships between minutes past a fixed hour and minutes remaining to another fixed hour. It uses a condition relating the time fifty minutes ago to the current time left until 6:00. Solving such problems systematically builds skill in combining basic algebra with time calculations and is very useful for competitive examinations where time and work or time and distance concepts are frequently tested.

Given Data / Assumptions:

- Let x be the number of minutes from now until 6:00. - Fifty minutes ago, the time is 50 minutes earlier than the current time. - Six o clock is 6:00 and three o clock is 3:00. - Fifty minutes ago, the minutes past 3:00 equal four times x. - We assume a standard 12 hour clock with uniform time flow.

Concept / Approach:
The approach is to express all relevant times in minutes counted from a common reference, which in this case is 3:00. By defining x as the minutes remaining to 6:00, we can express the current time and the time fifty minutes ago in terms of minutes after 3:00, then use the verbal condition to write an equation. This equation is linear in x and can be solved with normal algebra techniques. The final result is then checked against the multiple choice options.

Step-by-Step Solution:
Step 1: Let x denote the number of minutes now remaining 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. Six o clock is 180 minutes after 3:00. Step 5: The time fifty minutes ago is therefore 180 - x - 50 = 130 - x minutes after 3:00. Step 6: The condition states that 130 - x equals 4 times x, because at that earlier time the minutes past 3:00 are four times the minutes now remaining until 6:00. Step 7: Solve 130 - x = 4x to get 130 = 5x and x = 26 minutes. Step 8: Therefore, there are 26 minutes left until 6:00.

Verification / Alternative check:
With 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 equals 104 minutes. Four times 26 is 104, so the condition given in the problem is perfectly satisfied. This consistency between the algebraic solution and the concrete clock times confirms that 26 minutes is the correct answer.

Why Other Options Are Wrong:
Testing 24 minutes would make the current time 5:36 and the time fifty minutes earlier 4:46, which is 106 minutes past 3:00, not 4 * 24 = 96. Similar mismatches occur for 22, 20, and 18 minutes. None of these choices satisfy the equality that the minutes past 3:00 fifty minutes ago are four times the minutes now remaining until 6:00. Only 26 minutes maintains the correct proportional relationship in both algebraic and numerical checks.

Common Pitfalls:
Learners often reverse the relationship and mistakenly set x equal to four times the earlier minutes, or they count from 6:00 instead of from 3:00, which produces incorrect equations. Another pitfall is misinterpreting the phrase fifty minutes ago and wrongly adding 50 instead of subtracting it. Writing down a clear time line and explicitly converting everything into minutes from a single base time helps to avoid these errors and makes such puzzles much more manageable.

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