Solving the Mathematical Puzzle of Tobias's Erasers and Pencils

Mathematics can seem like a challenging subject, but it can also be a fun way to engage with problems that have clear and logical solutions. One such problem involves Tobias and his collection of erasers and pencils. Let's break down the puzzle and see how we can solve it step by step.

The Initial Setup

Tobias has 64 erasers and 3 times as many pencils as erasers.

Step 1: Determine the Initial Number of Pencils

Since Tobias has 3 times as many pencils as he does erasers, we can calculate:

No. of pencils  3 * 64  192

So, initially, Tobias has 192 pencils.

The Puzzle: Giving Away Equal Numbers of Pencils and Erasers

Tobias gives an equal number of erasers and pencils to his sister. Let's denote the number of each item he gives away as x.

Step 2: Determine the Remaining Amounts

After giving away x erasers and x pencils, the remaining amounts can be expressed as:

Remaining number of erasers 64 - x Remaining number of pencils 192 - x

Step 3: Apply the Given Condition

The problem states that after giving away the items, Tobias has 9 times as many pencils as erasers. This gives us the equation:

192 - x  9(64 - x)

Let's solve this equation step by step.

Step 4: Solve the Equation

192 - x  9(64 - x)
192 - x  576 - 9x
8x  384
x  48

Thus, Tobias gave away 48 pencils and 48 erasers to his sister.

Step 5: Calculate the Remaining Amounts

After giving away 48 erasers:

Remaining erasers  64 - 48  16

After giving away 48 pencils:

Remaining pencils  192 - 48  144

Hence, Tobias is left with 144 pencils in the end.

Conclusion

Through careful algebraic manipulation, we arrived at the solution that Tobias has 144 pencils in the end. This problem is a great way to practice algebraic thinking and problem-solving skills.

Using mathematical puzzles like this one is not only fun but also an effective way to understand and master mathematical concepts. Whether you're a student or someone just looking to challenge your brain, these puzzles can provide valuable practice and enjoyment.