The Broken Wine Glasses Problem: A Logical Dilemma Solved

Imagine the following scenario: you have 100 wine glasses, and you offer your servant a reward of 2 paise for each glass delivered safely and a penalty of 9 paise for each glass broken. At the end of the day, the servant receives Rs. 2.40. The question is, how many glasses did the servant break?

Solving the Problem

Let's start by defining our variables:

u Number of unbroken glasses b Number of broken glasses

We can establish two equations based on this scenario:

3u - 9b 240 (since 3 paise are earned for each unbroken glass and 9 paise are deducted for each broken glass) u b 100 (since there are 100 glasses in total)

Multiplying the second equation by 3 gives us:

3u 3b 300

Subtracting the first equation from the modified second equation:

3u 3b - (3u - 9b) 300 - 240

12b 60

b 5

Substituting b 5 back into the second equation:

u 5 100

u 95

Verification:

3(95) - 9(5) 285 - 45 240 (as expected)

Alternative Method

In another scenario, if 2 paise is offered for safely delivered while 9 paise is forfeited when broken, the method can be simplified to:

Let the number of broken glasses be x

The number of unbroken glasses 100 - x

Amount received 3(100 - x) - 9x 240

300 - 3x - 9x 240

300 - 12x 240

60 12x

x 5 (glasses were broken)

Further Exploration

Let's consider the third method, which offers 3 paise for every unbroken glass and deducts 9 paise for every broken glass:

Assume he broke x glasses

The number of unbroken glasses 100 - x

3(100 - x) - 9x 240

300 - 3x - 9x 240

12x 60

x 5 (glasses were broken)

A more intuitive approach is:

If all 100 glasses were intact, the servant would receive 300 paise (300% of 1 paise per glass). For each glass that goes from intact to broken, it results in a loss of 12 paise (9 paise deduction 3 paise forfeiture).

The servant's actual payment is 240 paise, which is 60 paise less than 300 paise. Therefore, the number of broken glasses can be calculated by dividing the difference by 12 paise.

60 / 12 5 (glasses were broken)

Conclusion

Solving such logical problems is an excellent exercise for developing analytical and problem-solving skills. By methodically breaking down the problem and using basic algebra, we can determine that the servant broke 5 wine glasses, leaving 95 unbroken.