Solving the Restaurant Bill Problem: A Mathematical Puzzle

Eight friends decided to dine together at a restaurant, planning to share the bill equally. However, the unfortunate situation arose when one friend forgot her money. Consequently, each of the remaining seven friends had to cover an additional $2.50 to cover the missing share. The question is: What was the total bill?

Understanding the Problem

Let's denote the total bill as B. If all eight friends had contributed, each would have paid:

[text{Share per person} frac{B}{8}]

Due to the missing money, the 7 friends ended up paying an extra amount. The actual share each of these 7 friends paid is:

[text{Share per person 7 friends} frac{B}{7}]

The problem states that each of the seven friends paid an extra $2.50. Therefore, we can set up the following equation:

[frac{B}{7} - frac{B}{8} 2.50]

Solving the Equation

To solve this equation, we need to find a common denominator for the fractions. The least common multiple of 7 and 8 is 56. So, we rewrite the equation with the common denominator:

[frac{8B}{56} - frac{7B}{56} 2.50]

This simplifies to:

[frac{B}{56} 2.50]

By multiplying both sides by 56, we get:

[B 2.50 times 56 140]

Conclusion

The total bill was $140.

Additional Insights

If the eighth friend's share was covered by the extra payment, each of the seven friends effectively paid an additional $2.50 to cover her part. This means that:

[7 times 2.50 17.50]

The total bill is then:

[8 times 17.50 140]

This solution not only resolves the immediate financial issue but also demonstrates the importance of mathematical reasoning in everyday situations. While the friends were likely credited for their meal, they might have enjoyed their dinner knowing they solved a problem that could have easily ruined their evening.

Note: The problem is a classic example of a puzzle that involves simple algebra but can be quite amusing and thought-provoking. It serves as a reminder of the practical applications of mathematical concepts.