Problem Overview: Sharing a Cake

The problem 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. Our goal is to find the fraction of the cake that Ravi received, which is the remaining portion.

Calculating Total Shares by Tapan and Trisha

First, let's find the total fraction of the cake shared by Tapan and Trisha. Tapan had $\frac{1}{4}$ of the cake, and Trisha had $\frac{2}{3}$ of the cake. To find their combined share, we add these two fractions:

Total shared = Tapan's share + Trisha's share

Total shared = $\frac{1}{4} + \frac{2}{3}$

To add fractions, we need a common denominator. The least common multiple (LCM) of 4 and 3 is 12. We convert each fraction to have a denominator of 12:

• $\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}$
• $\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}$

Now, add the converted fractions:

Total shared = $\frac{3}{12} + \frac{8}{12} = \frac{3 + 8}{12} = \frac{11}{12}$

So, Tapan and Trisha together shared $\frac{11}{12}$ of the cake.

Determining Ravi's Share of the Cake

The entire cake represents the whole, which is 1. Since Tapan and Trisha took $\frac{11}{12}$ of the cake, Ravi received the rest. To find Ravi's share, we subtract the total shared amount from the whole cake (1).

Ravi's share = Whole cake - Total shared

Ravi's share = $1 - \frac{11}{12}$

We can write 1 as a fraction with the same denominator, 12: $1 = \frac{12}{12}$.

Ravi's share = $\frac{12}{12} - \frac{11}{12} = \frac{12 - 11}{12} = \frac{1}{12}$

Therefore, Ravi's share of the cake was $\frac{1}{12}$.

This matches option 1.

Revision Table: Cake Sharing Fractions

Additional Information on Fraction Operations

This problem involved adding and subtracting fractions. Here's a quick reminder:

• Adding/Subtracting Fractions: To add or subtract fractions with different denominators, you must first find a common denominator. This is usually the least common multiple (LCM) of the denominators.
• Once fractions have the same denominator, you add or subtract the numerators and keep the denominator the same.
• Whole as a Fraction: A whole number (like 1) can be written as a fraction where the numerator and denominator are the same (e.g., $1 = \frac{4}{4} = \frac{12}{12}$). This is useful when subtracting a fraction from a whole number.