Testing Pure and Impure Solutions

CAT 2021 Slot 3 · DILR · Medium · Logical Reasoning

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Passage / data set

Each of the bottles mentioned in this question contains 50 ml of liquid. The liquid in any bottle can be 100% pure content (P) or can have certain amount of impurity (I). Visually it is not possible to distinguish between P and I. There is a testing device which detects impurity, as long as the percentage of impurity in the content tested is 10% or more.

For example, suppose bottle 1 contains only P, and bottle 2 contains 80% P and 20% I. If content from bottle 1 is tested, it will be found out that it contains only P. If content of bottle 2 is tested, the test will reveal that it contains some amount of I. If 10 ml of content from bottle 1 is mixed with 20 ml content from bottle 2, the test will show that the mixture has impurity, and hence we can conclude that at least one of the two bottles has I. However, if 10 ml of content from bottle 1 is mixed with 5 ml of content from bottle 2, the test will not detect any impurity in the resultant mixture.

Question 1 of 4

5 ml of content from bottle A is mixed with 5 ml of content from bottle B. The resultant mixture, when tested, detects the presence of I. If it is known that bottle A contains only P, what BEST can be concluded about the volume of I in bottle B?

  1. A.

    10 ml or more

  2. B.

    Less than 1 ml

  3. C.

    10 ml

  4. D.

    1 ml

Answer

A

Explanation

In a 10 ml mixture (5 ml A + 5 ml B), the test detects impurity, meaning the impurity in the mixture is at least 10%×10 ml=1 ml10\% \times 10\text{ ml} = 1\text{ ml}. Since all impurity came from the 5 ml of bottle B, the concentration of I in bottle B is at least 1/5=20%1/5 = 20\%. Since bottle B contains 50 ml, the total volume of I in bottle B is at least 20%×50 ml=10 ml20\% \times 50\text{ ml} = 10\text{ ml} or more.

Question 2 of 4

There are four bottles. Each bottle is known to contain only P or only I. They will be considered to be "collectively ready for despatch" if all of them contain only P. In minimum how many tests, is it possible to ascertain whether these four bottles are "collectively ready for despatch"?

Answer

1

Explanation

Take equal amounts (e.g., 5 ml) from each of the 4 bottles and mix them into a 20 ml sample.

  • If all bottles contain only P, impurity = 0%0\%.
  • If at least one bottle contains pure I, impurity concentration in the sample is at least 520=25%≥10%\frac{5}{20} = 25\% \ge 10\%, which will be detected. Thus, 1 test is sufficient.

Question 3 of 4

There are four bottles. It is known that three of these bottles contain only P, while the remaining one contains 80% P and 20% I. What is the minimum number of tests required to definitely identify the bottle containing some amount of I?

Answer

2

Explanation

Number the bottles 1, 2, 3, 4.

  • Test 1: Mix 10 ml from bottle 1 and 20 ml from bottle 2 (Total = 30 ml).
    • If bottle 2 is impure: impurity in mixture = 20%×20/30=13.33%≥10%20\% \times 20 / 30 = 13.33\% \ge 10\% (Detected).
    • If bottle 1 is impure: impurity in mixture = 20%×10/30=6.67%<10%20\% \times 10 / 30 = 6.67\% < 10\% (Not detected).
  • With 2 properly designed pooled tests, one can uniquely identify which of the 4 bottles is impure.

Question 4 of 4

There are four bottles. It is known that either one or two of these bottles contain(s) only P, while the remaining ones contain 85% P and 15% I. What is the minimum number of tests required to ascertain the exact number of bottles containing only P?

  1. A.

    1

  2. B.

    4

  3. C.

    3

  4. D.

    2

Answer

A

Explanation

Mix equal volumes (e.g., 10 ml) from all 4 bottles into a 40 ml mixture.

  • Case 1: 1 bottle pure P, 3 bottles impure (15% I). Total impurity concentration = 0+15%+15%+15%4=11.25%≥10%\frac{0 + 15\% + 15\% + 15\%}{4} = 11.25\% \ge 10\%. (Test detects impurity)
  • Case 2: 2 bottles pure P, 2 bottles impure (15% I). Total impurity concentration = 0+0+15%+15%4=7.5%<10%\frac{0 + 0 + 15\% + 15\%}{4} = 7.5\% < 10\%. (Test does not detect impurity) Hence, 1 test is sufficient to determine whether there is 1 or 2 pure bottles.

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