Bill has 5 apples and 5 bananas. He can put only 5 pieces of fruit in a bowl. He starts a table to show all the ways he can put fruit in the bowl. 3. Generalize What will be the same in each row of the table?
The Correct Answer and Explanation is:
Correct Answer:
In each row of the table, the total number of fruits will always be 5.
Explanation:
Bill has 5 apples and 5 bananas available. He wants to explore all the combinations of apples and bananas he can place in a bowl, with the condition that the bowl can only hold 5 pieces of fruit in total.
To do this, he creates a table where each row represents a different combination of apples and bananas that add up to 5. For example:
- 0 apples, 5 bananas
- 1 apple, 4 bananas
- 2 apples, 3 bananas
- 3 apples, 2 bananas
- 4 apples, 1 banana
- 5 apples, 0 bananas
What is common in every row is the sum of the fruits: apples + bananas = 5.
This consistency exists because the bowl has a fixed capacity — it cannot hold more or less than 5 pieces of fruit. Therefore, while the number of apples and bananas may vary, their total will always equal 5.
This idea reflects a fundamental concept in combinatorics: when selecting items under a fixed total constraint, you explore the different combinations of parts that make up the whole. In this case, the total is fixed at 5, so every valid entry in the table must satisfy that rule.
In general, when solving similar problems:
- Fix the total number (e.g., 5 pieces of fruit).
- Vary the individual components (apples and bananas).
- Ensure the sum remains constant in every case.
So, in summary, the number that will remain the same in every row of the table is the total number of fruits in the bowl — 5.