The Madhura Fruits Company is packing four types of fruits into boxes. There are 126 oranges, 162 apples, 198 guavas and 306 pears. The fruits must be packed in such a way that a given box must have only one type of fruit and must contain the same number of fruit units as any other box.
What is the minimum number of boxes that must be used?
C
Step-by-Step Solution
Key idea: This is an HCF and distribution problem. The trigger is packing items into boxes of equal size with no leftovers, while minimizing the total number of boxes.
Step 1: Understand the conditions.
- Each box contains only one type of fruit.
- Every box has the same number of fruits.
- We need to minimize the number of boxes.
Step 2: Relate to HCF.
To minimize the number of boxes, we must maximize the number of fruits per box. The number of fruits per box must perfectly divide the total count of each fruit type. Therefore, the capacity of each box is the Highest Common Factor (HCF) of the quantities.
Step 3: Calculate the HCF.
Quantities: 126, 162, 198, 306.
Prime factorizations:
The common factors are and .
.
So, each box contains 18 fruits.
Step 4: Calculate the number of boxes for each fruit.
Oranges: boxes.
Apples: boxes.
Guavas: boxes.
Pears: boxes.
Step 5: Find the total number of boxes.
Total boxes = .
Answer: 44 (Option C).