Solving the Cake Sharing Problem with Fractions

This problem involves understanding fractions and how to subtract them from a whole to find the remaining part. We are given the shares of Tapas and Rishi, and we need to find Avi's share of the cake.

Understanding the Shares

The total cake can be considered as one whole unit, which we represent as 1.

• Tapas's share is given as \( \frac{1}{2} \) of the cake.
• Rishi's share is given as \( \frac{1}{3} \) of the cake.
• Avi's share is the rest of the cake after Tapas and Rishi have taken theirs.

Calculating the Combined Share of Tapas and Rishi

To find out how much cake Tapas and Rishi took together, we need to add their fractions:

\( \text{Combined share} = \text{Tapas's share} + \text{Rishi's share} \)

\( \text{Combined share} = \frac{1}{2} + \frac{1}{3} \)

To add these fractions, we need a common denominator. The least common multiple (LCM) of 2 and 3 is 6.

• Convert \( \frac{1}{2} \) to a fraction with a denominator of 6: \( \frac{1}{2} \times \frac{3}{3} = \frac{3}{6} \)
• Convert \( \frac{1}{3} \) to a fraction with a denominator of 6: \( \frac{1}{3} \times \frac{2}{2} = \frac{2}{6} \)

Now, add the fractions with the common denominator:

\( \text{Combined share} = \frac{3}{6} + \frac{2}{6} = \frac{3 + 2}{6} = \frac{5}{6} \)

So, Tapas and Rishi together had \( \frac{5}{6} \) of the cake.

Finding Avi's Share of the Cake

Avi had the rest of the cake. The rest is what remains after the combined share of Tapas and Rishi is taken from the whole cake (which is 1).

\( \text{Avi's share} = \text{Total cake} - \text{Combined share} \)

\( \text{Avi's share} = 1 - \frac{5}{6} \)

To subtract the fraction from the whole number 1, we can express 1 as a fraction with the same denominator as \( \frac{5}{6} \). The denominator is 6, so \( 1 = \frac{6}{6} \).

\( \text{Avi's share} = \frac{6}{6} - \frac{5}{6} = \frac{6 - 5}{6} = \frac{1}{6} \)

Therefore, Avi's share of the cake was \( \frac{1}{6} \).

Summary of Shares

This confirms that all the shares add up to the whole cake.

Conclusion

By calculating the combined share of Tapas and Rishi and subtracting it from the whole cake, we found that Avi's share is \( \frac{1}{6} \).

Revision Table: Fraction Operations Review

Additional Information: Understanding Fractions in Sharing

Fractions represent parts of a whole. When sharing an item like a cake, the whole is usually represented by the number 1. Each person's share is a fraction of this whole. To find the remaining part, you subtract the known parts from the whole.

• The denominator of a fraction tells you how many equal parts the whole is divided into.
• The numerator tells you how many of those parts you have.
• Adding or subtracting fractions requires them to have the same denominator (a common denominator) so you are combining or separating parts of the same size.
• Finding the least common multiple (LCM) helps in finding the smallest common denominator, simplifying calculations.

This problem is a classic example of applying fraction addition and subtraction to solve real-world sharing scenarios.