Solving Riddles: A Mathematical Approach to Counting Fruits in a Bag

Did you know that solving brain teasers can not only be fun but also enhance your problem-solving skills? Today, we are exploring a classic riddle involving counting fruits. By breaking down the problem into smaller, manageable parts, we can solve it using a systematic approach. Let's dive in!

Problem Statement

In a bag, there are all apples except four, all oranges except five, and all mangoes except seven. The goal is to determine the total number of fruits in the bag. Let's denote the number of apples, oranges, and mangoes as A, O, and M, respectively.

Breaking Down the Problem

The given statements can be translated into mathematical equations:

All apples except four: O M 4 All oranges except five: A M 5 All mangoes except seven: A O 7

These equations form a system of linear equations which we can solve step-by-step.

Solving the System of Equations

Step 1: Express M in terms of O from Equation 1:

M 4 - O

Step 2: Substitute M into Equation 2:

A 4 - O 5

This simplifies to:

A - O 1 —- Equation 4

Step 3: Substitute A from Equation 4 into Equation 3:

From Equation 4, we can express A in terms of O as:

A O 1

Now substituting this into Equation 3:

2O - 1 7

This simplifies to:

2O 8

O 4

Step 4: Find A and M:

Now that we have O 4, we can find A and M:

Using O 4 in Equation 4:

A 4 1 5

Using O 4 in Equation 1:

M 4 - 4 0

Conclusion: Total Fruits in the Bag

Thus, the total number of fruits in the bag is:

A O M 5 4 0 9

However, let's validate our solution using a simpler approach:

Alternative Approach

If we denote the total number of fruits as x, then:

There are x - 4 apples There are x - 5 oranges There are x - 7 mangoes

Adding these together:

x - 4 x - 5 x - 7 3x - 16

Setting this equal to the total number of fruits:

3x - 16 x

This simplifies to:

2x 16

x 8

So, there were 8 fruits in the bag: 4 apples, 3 oranges, and 1 mango.

Conclusion

By using a step-by-step approach and validating through an alternative method, we have determined that the total number of fruits in the bag is 8. This problem showcases the power of logical thinking and systematic problem-solving.