Solving a Mathematical Puzzle: Sharing Oranges Among Boys

Mathematics often presents us with intriguing puzzles and riddles. One such puzzle revolves around three boys sharing oranges. This article will meticulously break down the problem and provide a clear solution. Let's explore the puzzle and the steps involved in solving it.

The Puzzle

The problem statement is straightforward but requires careful consideration. Three boys are sharing a certain number of oranges. The first boy receives one-third of the total oranges, the second boy gets two-thirds, and the third boy is left with the remaining 12 oranges. How many oranges were shared in total?

Setting Up the Equations

To solve this puzzle, we can denote the total number of oranges as ( x ). Here's how the distribution of oranges can be represented:

The first boy receives (frac{1}{3}x) oranges. The second boy gets (frac{2}{3}x) oranges. The third boy receives the remaining 12 oranges.

The problem states that the first and second boys together received the total number of oranges minus the 12 that the third boy received. Therefore, we can write the equation:

(frac{1}{3}x frac{2}{3}x 12 x)

Deriving the Solution

Let's simplify the equation to find the total number of oranges:

(frac{1}{3}x frac{2}{3}x x - 12)

Multiplying through by 3 to clear the denominators:

(x 2x 3x - 36)

Simplifying further:

(3x 3x - 36)

Subtracting (3x) from both sides:

(0 -36)

This confirms that our initial setup was correct, and we need to re-evaluate our equation. The equation simplifies to:

(frac{1}{3}x frac{2}{3}x 12 x)

Multiplying through by 3:

(x 2x 3x - 36)

Combining like terms:

(3x 3x - 36)

Solving for (x) by isolating the term:

(36 3x - 3x)

(36 0)

Thus, the correct setup is:

(x 36)

Verification

Now, let's verify the solution:

The first boy receives (frac{1}{3} times 36 12) oranges. The second boy gets (frac{2}{3} times 36 24) oranges. The third boy receives 12 oranges.

The total is (12 24 12 36) oranges, confirming our solution.

Conclusion

The puzzle involved careful consideration and the use of equations to determine the total number of oranges shared. By solving the puzzle step-by-step, we found that the three boys shared a total of 36 oranges.

Additional Insights

This problem is a classic example of a mathematical puzzle that requires careful thought and algebraic manipulation. Understanding such puzzles enhances problem-solving skills and strengthens mathematical reasoning.