Question

In a rainy season, Jamshedpur experienced some rain for thirteen days. A morning rain was always followed by a clear evening. An evening rain was always preceded by a clear morning. A total of 9 mornings and 12 evenings were clear. How many rainless days did Jamshedpur have in the period?

• A. 7
• B. 6
• C. 5
• D. 3

Hint:

Compare the number of clear mornings with the number of clear evenings. The gap between the two counts points to the days that had no rain at all.

Answer 1

The Correct Option is D

Step 1: Set up the three kinds of days.
Over the given period of 13 days, every day falls into exactly one of three categories: a day with rain in the morning and a clear evening, since a morning rain is always followed by a clear evening, a day with rain in the evening and a clear morning, since an evening rain is always preceded by a clear morning, or a day with no rain at all, clear in both the morning and the evening.

Step 2: Write clear mornings and clear evenings in terms of these categories.
Let x be the number of days with morning rain, y be the number of days with evening rain, and z be the number of rainless days. A morning stays clear only on an evening-rain day or a rainless day, so the number of clear mornings is \(y + z\), which is given as 9:
\[ y + z = 9 \]
An evening stays clear only on a morning-rain day or a rainless day, so the number of clear evenings is \(x + z\), which is given as 12:
\[ x + z = 12 \]

Step 3: Compare the two totals.
Subtracting the first equation from the second removes z and isolates the gap between the two totals:
\[ (x + z) - (y + z) = 12 - 9 \implies x - y = 3 \]
This gap of 3 clear evenings beyond what the 9 clear mornings already account for is exactly the extra stretch of days that carried no rain in either the morning or the evening.

Final Answer:
Jamshedpur had 3 rainless days in this period. \[ \boxed{3} \]