Directions: Read the following passage and answer the given questions. There are five pouches (A, B, C, D, and E) which contain candies of four flavors (Orange, Pineapple, Caramel, and Coffee). Below table shows the number of candies of four different flavors in five pouches: | | Orange candy | Pineapple candy | Caramel candy | Coffee candy | |---|---|---|---|---| | Pouch A | 12 | --- | --- | --- | | Pouch B | --- | 24 | --- | 10 | | Pouch C | --- | --- | 16 | --- | | Pouch D | --- | --- | 10 | --- | | Pouch E | --- | 10 | --- | --- | Some information is given below: - When one candy is picked at random from pouch A, then the probability of getting one coffee candy is $1/3$ and when one candy is picked at random from pouch C, then the probability of getting neither caramel nor coffee is $1/2$. - In pouch E, number of orange candies is same as number of caramel candies. Total number of caramel candies in all five pouches together is $69$. - The number of pineapple candies in pouches C and D are equal and when two candies are picked at random from pouch A, then the probability of getting both being pineapple candies is $1/22$. - When one candy is picked at random from pouch B, then the probability of getting either orange candy or caramel candy is $25/42$. Total number of caramel candies and coffee candies are same in all five pouches together. Number of caramel candies in pouch D is $2$ more than that in pouch A. - When one candy is picked at random from pouch C, then the probability that the candy picked is not caramel flavor is $4/5$. When two candies are picked at random from pouch D, then the probability that none of the candies is orange flavor is $17/35$. - When one candy is picked at random from pouch E, then the probability of getting one orange candy is $1/6$. When one candy is picked at random from pouch D, then probability of getting coffee candy is $1/5$. When one candy is picked at random from pouch E, then the probability of getting one pineapple candy is $1/3$. If 'K' pineapple candies are taken out from pouch D and put into pouch E, and then '2K' caramel candies and 'K' coffee candies are taken out from pouch E and put into pouch D. One candy is picked at random from pouch D, then the probability of getting orange flavor candy is $1/4$. What is the value of 'K'?
Aptitude
Probability
Difficulty: Hard
Choose an option
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A30
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B7
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C10
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D2
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E5
Answer
Correct Answer: 5
Explanation
### Concept & Probability Logic
To solve this Data Interpretation problem, we first determine the missing values in the table by forming equations from the given probabilities.
$$P(\text{Event}) = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}}$$
### Step-by-Step Solution
**Step 1: Complete the Candy Distribution Table**
* **Pouch E:** $P(\text{pineapple}) = 1/3 \implies 10/T_E = 1/3 \implies T_E = 30$. Since $P(\text{orange}) = 1/6$, $O_E = 5$. Given $O_E = Ca_E$, $Ca_E = 5$. $Co_E = 30-20 = 10$.
* **Pouch D:** $Ca_D = 10$. $P(\text{coffee}) = 1/5 \implies Co_D/T_D = 1/5$. Testing multiples of 5 with $P(\text{none is orange}) = 17/35$ yields $T_D = 50$, $O_D = 15$. Thus, $P_D = 15$, $Co_D = 10$.
* **Pouch C:** $Ca_C = 16$. $P(\text{not caramel}) = 4/5 \implies T_C = 80$. $P(\text{neither Ca nor Co}) = 1/2 \implies O_C + P_C = 40$. Since $P_C = P_D = 15$, $O_C = 25$. $Co_C = 24$.
* **Pouch A:** $Ca_D = Ca_A + 2 \implies Ca_A = 8$. $P(\text{coffee}) = 1/3 \implies Co_A/T_A = 1/3$. Using $P(\text{two pineapple}) = 1/22$ gives $T_A = 45$, $P_A = 10$, $Co_A = 15$.
* **Pouch B:** Total Caramel = 69 $\implies Ca_B = 30$. $P(\text{orange or caramel}) = 25/42 \implies T_B = 84$, leaving $O_B = 20$.
**Step 2: Apply the Transfer Operations**
* Pouch D initially has 50 candies ($O=15, P=15, Ca=10, Co=10$).
* Move $K$ pineapple to E: Pouch D loses $K$ candies.
* Move $2K$ caramel and $K$ coffee from E to D: Pouch D gains $3K$ candies.
* Net change in Pouch D = $+2K$.
* New Total in Pouch D = $50 + 2K$.
* The number of Orange candies remains unchanged at 15.
**Step 3: Solve for K**
* New Probability of getting an Orange candy = $1/4$.
* $\frac{15}{50 + 2K} = \frac{1}{4}$
* $60 = 50 + 2K$
* $2K = 10 \implies K = 5$.
### Exam Strategy & Shortcut
You only need to track the total number of candies in Pouch D and the number of orange candies in Pouch D. Since no orange candies were moved, the numerator stays exactly 15, drastically simplifying the algebra to a single fast equation.
### Common Pitfall
A common mistake is meticulously tracking every flavor dynamically when only the total count and the specific target flavor (orange) are relevant to the final question.
### Final Answer
Therefore, the correct answer is **5**.