Brain Teasers and the Mystery of the Missing Rupee: A Mathematical Exploration
The story of the two ladies and their oranges is a classic brain teaser that has long puzzled many. It brings into light how subtle differences in sales strategies can lead to unexpected results, and the importance of meticulous calculations in understanding earnings. Let's delve into the details of this fascinating puzzle and uncover the mystery of the missing rupee.
The Initial Sales Scenario
We begin with the first lady who had 30 oranges and was selling them at a rate of 2 for 1 rupee. To break down her sales:
She sold 2 oranges for 1 rupee, meaning each orange cost frac12; rupee. To make 15 rupees, she sold 30 oranges (15 times 2 30).The second lady had a similar situation, only she was selling oranges at 3 for 1 rupee. Her sales can be broken down as follows:
She sold 3 oranges for 1 rupee, meaning each orange cost frac13; rupee. To make 10 rupees, she also sold 30 oranges (10 times 3 30).Combining their sales, they had 60 oranges in total and made 25 rupees (15 rupees 10 rupees).
Combined Sales the Next Day
The next day, they combined their 60 oranges and decided to sell them at a rate of 5 for 2 rupees. The calculation is as follows:
They sold 60 oranges in total (60/5 12). Since 5 oranges were sold for 2 rupees, their total earnings were 12 times 2, which is 24 rupees.The confusion arises when they claim they should have made 25 rupees based on their previous sales. However, their combined earnings of 24 rupees reflect the new rate at which they sold the oranges. The discrepancy is not due to any missing rupee; rather, it is a result of misunderstanding the total sales amount based on varying selling prices.
The Incorrect Calculation: Where Did the 1 Rupee Go?
The controversy stems from the incorrect assumption that the average cost of an orange from the first day can be directly applied to the second day's sales. Let's break it down:
Originally, they sold 30 oranges for frac13; rupee each, resulting in an average cost of 0.41666 rupees per orange. The next day, they sold 60 oranges at a rate of 5 for 2 rupees, resulting in an average cost of 0.4 rupees per orange. This difference is accounted for by the additional rupee they earned due to the new pricing strategy.Thus, the first day's average cost of 0.41666 rupees per orange was only applicable to the previous rates. The new rate of 0.4 rupees per orange was more advantageous, and this explained the 1 rupee difference in their total earnings.
Understanding the Discrepancy
The discrepancy is not a missing rupee but a misunderstanding of the total sales amount. The first day’s total earnings of 25 rupees cannot be directly compared to the second day's earnings of 24 rupees because the selling prices were different. This puzzle highlights the importance of careful calculation and the impact of varying prices on total earnings.
Conclusion
There is no missing rupee; the discrepancy arises from a misinterpretation of their earnings based on varying selling prices. This brain teaser serves as a reminder of the complexities involved in sales calculations and the importance of keeping track of changing rates and quantities.