Introduction / Context:
This problem is a rate and work puzzle disguised as a logic question. It states that nineteen persons can complete nineteen programs in nineteen hours, and asks how long the same group will take to complete fifty two programs when they also take one fifteen minute break. The puzzle tests your ability to interpret work rate correctly and avoid the common mistake of overcomplicating the relationship between persons, programs, and time.
Given Data / Assumptions:
- 19 persons complete 19 programs in 19 hours.
- All 19 persons work at the same constant rate and all programs are of equal difficulty.
- The group works together and their combined rate remains fixed.
- They will now complete 52 similar programs under the same conditions.
- They take exactly one break of 15 minutes during the total working time.
Concept / Approach:
The main concept is combined work rate. Instead of focusing individually on each person, determine how many programs the entire team finishes per hour. From 19 persons finishing 19 programs in 19 hours, you can deduce how many programs are completed per hour by the full group. After that, you can scale up to 52 programs, and finally add the single 15 minute break to the total working time. The puzzle is simpler than it appears once you identify that the team completes exactly one program per hour.
Step-by-Step Solution:
Step 1: Use the initial information to find the group rate. They complete 19 programs in 19 hours.
Step 2: Divide total programs by total hours to get the combined rate: 19 programs / 19 hours equals 1 program per hour.
Step 3: Apply this rate to the new target. To complete 52 programs at 1 program per hour, the group needs 52 working hours.
Step 4: Now account for the break. The group takes one 15 minute break that is added to the working time.
Step 5: Convert the break to hours. Fifteen minutes is one quarter of an hour, so 0.25 hours.
Step 6: Add this break to the 52 working hours to get total elapsed time: 52 plus 0.25 equals 52.25 hours.
Step 7: Express 0.25 hours as 15 minutes. Therefore, the total time needed is 52 hours 15 minutes.
Verification / Alternative check:
As a quick check, imagine how many programs the group finishes over a smaller time span. In 10 hours, they finish 10 programs at 1 program per hour. In 19 hours, they finish 19 programs, matching the initial condition. This confirms the group rate is consistent. For 52 programs, sticking to the same rate gives 52 hours. Adding only one break of 15 minutes, not multiple breaks, ensures you do not accidentally add too much idle time. This straightforward reasoning matches the option 52 hrs 15 min exactly.
Why Other Options Are Wrong:
23 hours: This is less than half of the initial time needed for 19 programs, so it cannot be enough time for 52 programs at the same rate.
47 hrs 15 min: This underestimates the time because even without a break they need 52 hours of effective work.
22 hrs 15 min: This is even shorter than 23 hours and clearly impossible, since 19 programs alone already take 19 hours.
Common Pitfalls:
Many learners misread the initial statement and think that each person finishes one program in 19 hours, then perform unnecessary multiplications. Others assume that the break occurs after each program, which is not stated in the question, leading to a much larger time. The safe method is to interpret the sentence literally: as a team, they finish 19 programs in 19 hours. From there, the rate is one program per hour, and only a single 15 minute break is added, giving the simple result of 52 hours 15 minutes.
Final Answer:
They will need a total of
52 hrs 15 min to complete 52 programs, including the single 15 minute break.
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