Solving the Pie Problem: How Much Pie is Left?

Solving the Pie Problem: How Much Pie is Left?

Imagine a situation where a pie is being shared among three individuals. The first two individuals take two-thirds of the pie, and one of them returns a sixth of the pie. Later, a third person takes a quarter of the remaining pie. How much of the original pie is left?

Step-by-Step Breakdown

Let's break down the problem step-by-step to find the answer.

Initial Steps

The first two individuals take two-thirds of the pie. This can be represented mathematically as:

[ frac{2}{3} ]

One of them then returns a sixth of the pie. The fraction that is returned is:

[ frac{1}{6} ]

To calculate the amount of pie left after this return, we perform the following subtraction:

[ frac{2}{3} - frac{1}{6} ]

To perform this subtraction, we need a common denominator. The least common multiple (LCM) of 3 and 6 is 6. Converting ( frac{2}{3} ) to sixths:

[ frac{2}{3} frac{4}{6} ]

Subtracting ( frac{1}{6} ) from ( frac{4}{6} ) gives:

[ frac{4}{6} - frac{1}{6} frac{3}{6} frac{1}{2} ]

So, after the return, they have ( frac{1}{2} ) of the pie left.

Subsequent Actions

A third person then takes a quarter of the remaining pie. This can be represented as:

[ frac{1}{4} ]

To find out how much is left after the third person takes their share, we subtract ( frac{1}{4} ) from ( frac{1}{2} ):

[ frac{1}{2} - frac{1}{4} ]

The least common multiple of 2 and 4 is 4. Converting ( frac{1}{2} ) to fourths:

[ frac{1}{2} frac{2}{4} ]

Subtracting ( frac{1}{4} ) from ( frac{2}{4} ) gives:

[ frac{2}{4} - frac{1}{4} frac{1}{4} ]

Thus, the amount of pie left is:

[ frac{1}{4} ]

Alternative Method Using Common Slices

Another method to solve this problem is by visualizing the pie cut into 12 slices.

Two-thirds of the pie is ( frac{2}{3} ) of 12 slices, which equals 8 slices. A sixth of the pie is ( frac{1}{6} ) of 12 slices, which equals 2 slices. A quarter of the pie is ( frac{1}{4} ) of 12 slices, which equals 3 slices.

Starting with a whole pie (12 slices), we perform the following subtractions:

[ 12 - 8 4 ]

After 8 slices are taken, 4 slices are left.

[ 4 - 2 2 ]

After returning 2 slices, 2 slices are left.

[ 2 - 3 -1 ]

However, since we can't take a negative number of slices, we know that 1 slice is left.

This confirms our previous solution where the amount left is ( frac{1}{4} ) of the pie.

Final Answer

Therefore, the amount of pie left is:

[ frac{1}{4} ]

Conclusion

In summary, by following the steps of finding the common denominator and performing the necessary subtractions, we can conclude that ( frac{1}{4} ) of the pie is left. This problem demonstrates the practical application of fractions and common denominators in real-world scenarios.