Understanding the Cake Sharing Problem

The question describes how a cake was shared among three people: Tapan, Ravi, and Trisha. We are given the fractions of the cake that Tapan and Trisha received, and we need to find the fraction that Ravi received.

The total cake can be represented as a whole, which is 1.

Shares of Tapan and Trisha

According to the problem:

• Tapan had $\frac{1}{3}$ of the cake.
• Trisha had $\frac{1}{2}$ of the cake.
• Ravi had the rest of the cake.

Calculating the Combined Share of Tapan and Trisha

To find the rest of the cake that Ravi received, we first need to find out what fraction of the cake Tapan and Trisha had together. We do this by adding their shares:

Combined share = Tapan's share + Trisha's share

Combined share = $\frac{1}{3} + \frac{1}{2}$

To add fractions, they must have a common denominator. The least common multiple (LCM) of 3 and 2 is 6. We convert each fraction to have a denominator of 6:

• Tapan's share: $\frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6}$
• Trisha's share: $\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}$

Now, add the fractions with the common denominator:

Combined share = $\frac{2}{6} + \frac{3}{6} = \frac{2+3}{6} = \frac{5}{6}$

So, Tapan and Trisha together had $\frac{5}{6}$ of the cake.

Finding Ravi's Share of the Cake

Ravi had the rest of the cake. To find Ravi's share, we subtract the combined share of Tapan and Trisha from the total cake (which is 1 whole):

Ravi's share = Total cake - Combined share of Tapan and Trisha

Ravi's share = $1 - \frac{5}{6}$

To subtract the fraction from the whole number 1, we can write 1 as a fraction with the same denominator as the other fraction, which is 6. So, $1 = \frac{6}{6}$.

Ravi's share = $\frac{6}{6} - \frac{5}{6} = \frac{6-5}{6} = \frac{1}{6}$

Therefore, Ravi's share of the cake was $\frac{1}{6}$.

Verification

Let's check if the sum of all shares equals the whole cake (1):

Tapan's share + Trisha's share + Ravi's share = $\frac{1}{3} + \frac{1}{2} + \frac{1}{6}$

Using the fractions with the common denominator 6:

= $\frac{2}{6} + \frac{3}{6} + \frac{1}{6} = \frac{2+3+1}{6} = \frac{6}{6} = 1$

The sum of the shares is 1, which confirms our calculation for Ravi's share is correct.

Conclusion

Ravi's share of the cake was $\frac{1}{6}$.

Revision Table: Cake Share Calculations

Additional Information: Adding and Subtracting Fractions

When adding or subtracting fractions, it is essential to have a common denominator. The common denominator is typically the least common multiple (LCM) of the denominators of the fractions involved.

Steps for adding/subtracting fractions:

1. Find the LCM of the denominators. This will be the common denominator.
2. Convert each fraction into an equivalent fraction with the common denominator. To do this, multiply both the numerator and the denominator by the same number that makes the original denominator equal to the common denominator.
3. Add or subtract the numerators of the equivalent fractions.
4. Keep the common denominator the same.
5. Simplify the resulting fraction if possible.

For example, $\frac{1}{4} + \frac{1}{6}$. The LCM of 4 and 6 is 12.

$\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}$

$\frac{1}{6} = \frac{1 \times 2}{6 \times 2} = \frac{2}{12}$

$\frac{3}{12} + \frac{2}{12} = \frac{3+2}{12} = \frac{5}{12}$