CAT 2021 Slot 3 · DILR · Set 1: Testing Bottles for Impurity

The full set and all 4 questions (Q1–Q4) from CAT 2021 Slot 3, with answers and solutions.

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 · Caselet DI

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?

A
1 ml
B
Less than 1 ml
C
10 ml
D
10 ml or more

Pick an option, then check.

Answer & solution

Answer: Option D

Solving the set

Key idea: the device detects impurity only when it is at least 10% of the tested liquid, so by choosing how much of each bottle to mix, one test can answer a yes/no question about several bottles at once.

Question	Situation	Best test plan	Tests
Volume of I in B	5 ml A (pure) + 5 ml B shows I	Impurity ≥ 10% of 10 ml ⇒ ≥ 1 ml I in 5 ml of B ⇒ ≥ 20% of B ⇒ ≥ 10 ml of I in B	—
All pure?	Each bottle is only P or only I	Mix equal amounts of all four: any I bottle makes ≥ 25% impurity	1
Find the 20% bottle	Three pure, one 20% I	Mix two bottles 1 : 1 (10% if the impure one is there); then test one bottle of the indicated pair	2
One or two pure?	Rest 15% I	Mix equal amounts of all four: 3 impure ⇒ 11.25% (detected), 2 impure ⇒ 7.5% (not detected)	1

This question

From the table: 5 ml of pure A + 5 ml of B shows impurity, so the 10 ml mixture has at least 1 ml of I, all from B's 5 ml. B is at least 20% impure, i.e., at least 10 ml of I in 50 ml.

Hence, the answer is option 4.

Question 2 · Caselet DI

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 & solution

Answer: 1

Solving the set

Question	Situation	Best test plan	Tests
Volume of I in B	5 ml A (pure) + 5 ml B shows I	Impurity ≥ 10% of 10 ml ⇒ ≥ 1 ml I in 5 ml of B ⇒ ≥ 20% of B ⇒ ≥ 10 ml of I in B	—
All pure?	Each bottle is only P or only I	Mix equal amounts of all four: any I bottle makes ≥ 25% impurity	1
Find the 20% bottle	Three pure, one 20% I	Mix two bottles 1 : 1 (10% if the impure one is there); then test one bottle of the indicated pair	2
One or two pure?	Rest 15% I	Mix equal amounts of all four: 3 impure ⇒ 11.25% (detected), 2 impure ⇒ 7.5% (not detected)	1

This question

One test of an equal mix of all four bottles: any bottle of pure I makes the impurity at least 25%, which is detected; if all are P, nothing is detected.

Hence, the answer is 1.

Question 3 · Caselet DI

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 & solution

Answer: 2

Solving the set

Question	Situation	Best test plan	Tests
Volume of I in B	5 ml A (pure) + 5 ml B shows I	Impurity ≥ 10% of 10 ml ⇒ ≥ 1 ml I in 5 ml of B ⇒ ≥ 20% of B ⇒ ≥ 10 ml of I in B	—
All pure?	Each bottle is only P or only I	Mix equal amounts of all four: any I bottle makes ≥ 25% impurity	1
Find the 20% bottle	Three pure, one 20% I	Mix two bottles 1 : 1 (10% if the impure one is there); then test one bottle of the indicated pair	2
One or two pure?	Rest 15% I	Mix equal amounts of all four: 3 impure ⇒ 11.25% (detected), 2 impure ⇒ 7.5% (not detected)	1

This question

A single test has only two outcomes, so it cannot pick out one of four bottles. Two tests suffice: test bottles 1 + 2 mixed 1 : 1 (impurity 10% if one of them is the bad bottle), then test one bottle of the indicated pair.

Hence, the answer is 2.

Question 4 · Caselet DI

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?

Pick an option, then check.

Answer & solution

Answer: Option D

Solving the set

Question	Situation	Best test plan	Tests
Volume of I in B	5 ml A (pure) + 5 ml B shows I	Impurity ≥ 10% of 10 ml ⇒ ≥ 1 ml I in 5 ml of B ⇒ ≥ 20% of B ⇒ ≥ 10 ml of I in B	—
All pure?	Each bottle is only P or only I	Mix equal amounts of all four: any I bottle makes ≥ 25% impurity	1
Find the 20% bottle	Three pure, one 20% I	Mix two bottles 1 : 1 (10% if the impure one is there); then test one bottle of the indicated pair	2
One or two pure?	Rest 15% I	Mix equal amounts of all four: 3 impure ⇒ 11.25% (detected), 2 impure ⇒ 7.5% (not detected)	1

This question

Mixing equal amounts of all four bottles: with 3 impure bottles the impurity is 11.25% (detected); with 2 impure bottles it is 7.5% (not detected). So one test tells whether there are one or two pure bottles.

Hence, the answer is option 4.

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