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CAT 2022 Slot 1 — QA Question 3

RatioEasy

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

Answer & solution

Answer: 111

Solution

Easy

Model the queue with one variable from the 3:53:5 ratio, add 11 for Pinky herself, apply the "fewer than 300300" cap to bound the variable, then read off the largest valid count ahead of her.

1

Set up the total. Let 3x3x stand ahead of Pinky and 5x5x behind her:

total=3x+5x+1⏟Pinky=8x+1\text{total}=3x+5x+\underbrace{1}_{\text{Pinky}}=8x+1
2

Apply the cap (total <300<300) and solve for xx:

8x+1<300 ⇒ x<2998=37.375 ⇒ xmax⁡=378x+1<300\ \Rightarrow\ x<\frac{299}{8}=37.375\ \Rightarrow\ x_{\max}=37
3

People ahead of Pinky =3x=3x, maximised at x=37x=37:

3×37=1113\times37=111
111\mathbf{111}

The count ahead is always a multiple of 33. Largest multiple of 8x+18x+1 under 300300 comes from the largest integer xx with 8x<2998x<299, i.e. x=37x=37, giving 3x=1113x=111.

CAT 2022 Slot 1 QA Q3: Pinky is standing in a queue at a ticket counter. Suppose the ratio of the number of persons standing ahead of — Solution | TheCATExam