A fruit seller has oranges, apples, and bananas in the ratio 3:6:7. If the number of oranges is a multiple of both 5 and 6, then the minimum number of fruits the seller has is ______________.
A fruit seller has oranges, apples, and bananas in the ratio 3:6:7. If the number of oranges is a multiple of both 5 and 6, then the minimum number of fruits the seller has is ______________.
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Solution and Explanation
Step 1: Express the number of fruits in terms of a common variable. Let the number of oranges, apples, and bananas be \( 3x \), \( 6x \), and \( 7x \) respectively, where \( x \) is a common multiplier.
Step 2: Find the least value of \( x \). The number of oranges is a multiple of both 5 and 6, so we find the least common multiple of 5 and 6, which is 30. Therefore, \( 3x \) must be a multiple of 30. \[ 3x = 30k \quad \Rightarrow \quad x = 10k \]
Step 3: Find the minimum number of fruits. The minimum number of fruits occurs when \( k = 1 \), so: \[ x = 10 \] Thus, the number of oranges is \( 3x = 30 \), the number of apples is \( 6x = 60 \), and the number of bananas is \( 7x = 70 \). The total number of fruits is: \[ 30 + 60 + 70 = 160 \] Thus, the minimum number of fruits the seller has is: \[ \boxed{160} \]
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