Pinky is standing in a queue at a ticket counter. Suppose the ratio of the number of persons standing ahead of Pinky to the number of persons standing behind her in the queue is 3:5. If the total number of persons in the queue is less than 300, then the maximum possible number of persons standing ahead of Pinky is
Correct Answer: 111
Approach Solution - 1
Step 1: Let Variables Represent the Scenario
Let the number of persons ahead of Pinky be \( 3x \) Let the number of persons behind Pinky be \( 5x \) (since the ratio is \( 3:5 \))
Step 2: Total Number of Persons in the Queue
Total number of persons in the queue is: \[ 3x + 1 + 5x = 8x + 1 \] We're told: \[ 8x + 1 < 300 \]
Step 3: Solve the Inequality
Subtract 1 from both sides: \[ 8x < 299 \] Divide by 8: \[ x < \frac{299}{8} = 37.375 \Rightarrow \text{Maximum integer value of } x = 37 \]
Step 4: Calculate Persons Ahead of Pinky
Using \( x = 37 \): \[ \text{Persons ahead of Pinky} = 3x = 3 \times 37 = \boxed{111} \]
Final Answer:
\[ \boxed{\text{Maximum persons ahead of Pinky} = 111} \]
Approach Solution -2
Step 1: Assume Representation
Let Pinky be surrounded by:
- \(3a\) people ahead of her
- \(5a\) people behind her
Total number of people in the queue (including Pinky) is: \[ 3a + 5a + 1 = 8a + 1 \]
Step 2: Apply the Total Constraint
Given: total people in the queue is less than 300: \[ 8a + 1 < 300 \Rightarrow 8a < 299 \Rightarrow a < 37.375 \] Since \(a\) must be a whole number, the maximum possible value of \(a\) is: \[ a = 37 \]
Step 3: Calculate People Ahead of Pinky
Using \( a = 37 \): \[ \text{People ahead of Pinky} = 3a = 3 \times 37 = \boxed{111} \]
Final Answer:
\[ \boxed{111} \] is the maximum number of people that might be standing in front of Pinky.
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