Calculating the Remaining Portion of a Cake: A Fractional Challenge

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Introduction to the Problem

In a culinary scenario, two friends, Jenny and John, decide to divide a cake equally among themselves. Jenny eats 2/5 of the cake, and John eats 2/7 of the cake. To find out how much of the cake remains, we need to subtract the fractions eaten by Jenny and John from the whole cake.

Understanding the Problem

The entire cake can be represented as 1 (or 4/4). We need to determine the portions eaten by Jenny and John first.

Jenny's Portion

Jenny ate 2/5 of the cake.

John's Portion

John ate 2/7 of the cake.

Solving the Problem

To find the total portion eaten by Jenny and John, we need to add the fractions 2/5 and 2/7. However, since the denominators are different, we must find a common denominator before adding the fractions.

Finding a Common Denominator

The least common multiple (LCM) of 5 and 7 is 35. We will convert both fractions to have a denominator of 35.

For 2/5:

2/5 2 * 7 / (5 * 7) 14/35

For 2/7:

2/7 2 * 5 / (7 * 5) 10/35

Now, we can add the two fractions:

14/35 10/35 24/35

Calculating the Remaining Portion

Finally, to find the portion of the cake that remains, we subtract the total portion eaten from the whole cake (1 or 35/35).

1 - 24/35 35/35 - 24/35 11/35

So, the portion of the cake that remains is 11/35.

Alternative Methods

Another way to solve this problem involves converting fractions to a common denominator and then performing the subtraction. For instance, if we use 15 as the common denominator:

Jenny ate 2/5, which is equivalent to 6/15 (since 5 * 3 15 and 2 * 3 6).

John ate 1/3, which is equivalent to 5/15 (since 3 * 5 15 and 1 * 5 5).

Adding the fractions:

6/15 5/15 11/15

This shows that they ate 11/15 of the cake, leaving:

15/15 - 11/15 4/15

Therefore, 4/15 of the cake remains.

Conclusion

By using the method of finding a common denominator or converting fractions, we can accurately determine the remaining portion of the cake after Jenny and John have eaten their shares. Understanding and mastering fraction operations are essential skills in solving real-world mathematical problems.