Tapas, Avi and Rishi shared a cake. Tapas had 1/2 of it, Rishi had 1/3 of it and Avi had the rest. What was Avi’s share of the cake?
1/6
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
| Person | Share (Fraction) |
|---|---|
| Tapas | \( \frac{1}{2} \) (or \( \frac{3}{6} \)) |
| Rishi | \( \frac{1}{3} \) (or \( \frac{2}{6} \)) |
| Avi | \( \frac{1}{6} \) |
| Total | \( \frac{3}{6} + \frac{2}{6} + \frac{1}{6} = \frac{6}{6} = 1 \) |
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
| Operation | Description | Example |
|---|---|---|
| Adding Fractions (Different Denominators) | Find a common denominator (LCM), convert fractions, then add numerators. | \( \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \) |
| Subtracting Fractions (from Whole Number) | Write the whole number as a fraction with the same denominator as the fraction being subtracted, then subtract numerators. | \( 1 - \frac{5}{6} = \frac{6}{6} - \frac{5}{6} = \frac{1}{6} \) |
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.
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