A Mathematical Puzzle: Determining the Remaining Fraction of a Cake

In the whimsical world of mathematics, a delightful puzzle arises when we encounter a scenario involving the consumption of a cake. Specifically, if Tlabo ate 2}{5} of a cake and gave 1}{2} of it to Tanaka, then what fraction of the cake remains?

Breaking Down the Problem

To solve this, we need to subtract the portions that Tlabo ate and shared from the whole cake. Here's a step-by-step guide to determining the remaining fraction of the cake:

Step 1: Identify the fractions eaten and given away

Tlabo ate 2}{5} of the cake and gave away 1}{2} to Tanaka. These fractions need to be added together to find the total portion consumed.

Step 2: Find a common denominator

Both fractions, 2}{5} and 1}{2}, need a common denominator to be added. The least common multiple of 5 and 2 is 10, which serves as our common denominator.

Step 3: Convert the fractions

Convert both fractions to have a denominator of 10:

$$frac{2}{5} frac{2 times 2}{5 times 2} frac{4}{10}$$ $$frac{1}{2} frac{1 times 5}{2 times 5} frac{5}{10}$$

Step 4: Add the fractions

Add the two fractions to find the total portion of the cake consumed:

$$frac{4}{10} frac{5}{10} frac{9}{10}$$

This means that Tlabo ate and gave away a combined total of 9}{10} of the cake.

Step 5: Calculate the remaining fraction

To find the fraction of the remaining cake, subtract the consumed portion from the whole cake, which is represented by 1 or 10}{10}:

$$frac{10}{10} - frac{9}{10} frac{1}{10}$$

Therefore, the fraction of the cake that remains is 1}{10}.

Alternative Calculations

Another way to approach this problem is to use an algorithm that involves the subtraction of fractions:

Let y 1 - frac{2}{5} - frac{1}{2} Express both fractions with a common denominator: frac{2}{5} frac{4}{10} and frac{1}{2} frac{5}{10} Add the fractions: frac{4}{10} frac{5}{10} frac{9}{10} Subtract the sum from the whole cake: 1 - frac{9}{10} frac{1}{10}

Thus, the fraction of the cake that remains is 1}{10}, which is equal to the decimal 0.1.

Conclusion

This problem demonstrates the practical application of fraction arithmetic in real-life scenarios. Such puzzles not only sharpen our mathematical skills but also provide a fun and engaging way to understand fractions and their operations.

Finding the remaining fraction of a cake, while seemingly simple, involves critical thinking and a thorough understanding of fractions. Mastering these skills can be beneficial in numerous other areas of mathematics and real-world problem-solving.

Key takeaways:

Fraction addition and subtraction Common denominators Subtraction of fractions from a whole Conversion of fractions to decimals