CAT 2022 Slot 2 · DILR · Set 4: Products in a Speciality Supermarket

The full set and all 5 questions (Q16–Q20) from CAT 2022 Slot 2, with answers and solutions.

A speciality supermarket sells 320 products. Each of these products was either a cosmetic product or a nutrition product. Each of these products was also either a foreign product or a domestic product. Each of these products had at least one of the two approvals - FDA or EU.

The following facts are also known:

1. There were equal numbers of domestic and foreign products.

2. Half of the domestic products were FDA approved cosmetic products.

3. None of the foreign products had both the approvals, while 60 domestic products had both the approvals.

4. There were 140 nutrition products, half of them were foreign products.

5. There were 200 FDA approved products. 70 of them were foreign products and 120 of them were cosmetic products.

Question 16 · Venn Diagram DI

How many foreign products were FDA approved cosmetic products?
Answer & solution

Answer: 40

Solving the set

Step 1 – Totals. 320 products: 160 domestic, 160 foreign (fact 1). Nutrition 140 (70 foreign, 70 domestic), so cosmetics 180 (90 foreign, 90 domestic).

Step 2 – FDA approved: 200 in all, 70 foreign and 130 domestic; 120 cosmetic and 80 nutrition (fact 5).

Step 3 – Domestic FDA cosmetics = 160/2 = 80 (fact 2). Foreign FDA cosmetics = 120 − 80 = 40. Foreign FDA nutrition = 70 − 40 = 30. Domestic FDA nutrition = 80 − 30 = 50.

Step 4 – Every product has at least one approval, so products without FDA are EU-only: foreign cosmetics 90 − 40 = 50, foreign nutrition 70 − 30 = 40, domestic cosmetics 90 − 80 = 10, domestic nutrition 70 − 50 = 20.

Step 5 – No foreign product has both approvals; 60 domestic products have both, split between domestic FDA cosmetics (80) and domestic FDA nutrition (50).

Cosmetic

Nutrition

Total

Foreign – FDA only

40

30

70

Foreign – EU only

50

40

90

Domestic – FDA (incl. both)

80

50

130

Domestic – EU only

10

20

30

Total

180

140

320

This question

From Step 3, foreign FDA approved cosmetics = 120 − 80 = 40.

Hence, the answer is 40.

Question 17 · Venn Diagram DI

How many cosmetic products did not have FDA approval?
  1. A
    10
  2. B
    Cannot be determined
  3. C
    50
  4. D
    60
Pick an option, then check.
Answer & solution

Answer: Option D

Solving the set

Step 1 – Totals. 320 products: 160 domestic, 160 foreign (fact 1). Nutrition 140 (70 foreign, 70 domestic), so cosmetics 180 (90 foreign, 90 domestic).

Step 2 – FDA approved: 200 in all, 70 foreign and 130 domestic; 120 cosmetic and 80 nutrition (fact 5).

Step 3 – Domestic FDA cosmetics = 160/2 = 80 (fact 2). Foreign FDA cosmetics = 120 − 80 = 40. Foreign FDA nutrition = 70 − 40 = 30. Domestic FDA nutrition = 80 − 30 = 50.

Step 4 – Every product has at least one approval, so products without FDA are EU-only: foreign cosmetics 90 − 40 = 50, foreign nutrition 70 − 30 = 40, domestic cosmetics 90 − 80 = 10, domestic nutrition 70 − 50 = 20.

Step 5 – No foreign product has both approvals; 60 domestic products have both, split between domestic FDA cosmetics (80) and domestic FDA nutrition (50).

Cosmetic

Nutrition

Total

Foreign – FDA only

40

30

70

Foreign – EU only

50

40

90

Domestic – FDA (incl. both)

80

50

130

Domestic – EU only

10

20

30

Total

180

140

320

This question

Cosmetics without FDA approval = 180 − 120 = 60 (50 foreign + 10 domestic, all EU-only).

Hence, the answer is option 4.

Question 18 · Venn Diagram DI

Which among the following options best represents the number of domestic cosmetic products that had both the approvals?
  1. A
    At least 10 and at most 60
  2. B
    At least 10 and at most 80
  3. C
    At least 20 and at most 70
  4. D
    At least 20 and at most 50
Pick an option, then check.
Answer & solution

Answer: Option A

Solving the set

Step 1 – Totals. 320 products: 160 domestic, 160 foreign (fact 1). Nutrition 140 (70 foreign, 70 domestic), so cosmetics 180 (90 foreign, 90 domestic).

Step 2 – FDA approved: 200 in all, 70 foreign and 130 domestic; 120 cosmetic and 80 nutrition (fact 5).

Step 3 – Domestic FDA cosmetics = 160/2 = 80 (fact 2). Foreign FDA cosmetics = 120 − 80 = 40. Foreign FDA nutrition = 70 − 40 = 30. Domestic FDA nutrition = 80 − 30 = 50.

Step 4 – Every product has at least one approval, so products without FDA are EU-only: foreign cosmetics 90 − 40 = 50, foreign nutrition 70 − 30 = 40, domestic cosmetics 90 − 80 = 10, domestic nutrition 70 − 50 = 20.

Step 5 – No foreign product has both approvals; 60 domestic products have both, split between domestic FDA cosmetics (80) and domestic FDA nutrition (50).

Cosmetic

Nutrition

Total

Foreign – FDA only

40

30

70

Foreign – EU only

50

40

90

Domestic – FDA (incl. both)

80

50

130

Domestic – EU only

10

20

30

Total

180

140

320

This question

Let x = domestic cosmetics with both approvals; then domestic nutrition with both = 60 − x. Limits: x ≤ 80 and 60 − x ≤ 50 (so x ≥ 10), and x ≤ 60 (since 60 − x ≥ 0). So 10 ≤ x ≤ 60.

Hence, the answer is option 1.

Question 19 · Venn Diagram DI

If 70 cosmetic products did not have EU approval, then how many nutrition products had both the approvals?
  1. A
    50
  2. B
    30
  3. C
    10
  4. D
    20
Pick an option, then check.
Answer & solution

Answer: Option C

Solving the set

Step 1 – Totals. 320 products: 160 domestic, 160 foreign (fact 1). Nutrition 140 (70 foreign, 70 domestic), so cosmetics 180 (90 foreign, 90 domestic).

Step 2 – FDA approved: 200 in all, 70 foreign and 130 domestic; 120 cosmetic and 80 nutrition (fact 5).

Step 3 – Domestic FDA cosmetics = 160/2 = 80 (fact 2). Foreign FDA cosmetics = 120 − 80 = 40. Foreign FDA nutrition = 70 − 40 = 30. Domestic FDA nutrition = 80 − 30 = 50.

Step 4 – Every product has at least one approval, so products without FDA are EU-only: foreign cosmetics 90 − 40 = 50, foreign nutrition 70 − 30 = 40, domestic cosmetics 90 − 80 = 10, domestic nutrition 70 − 50 = 20.

Step 5 – No foreign product has both approvals; 60 domestic products have both, split between domestic FDA cosmetics (80) and domestic FDA nutrition (50).

Cosmetic

Nutrition

Total

Foreign – FDA only

40

30

70

Foreign – EU only

50

40

90

Domestic – FDA (incl. both)

80

50

130

Domestic – EU only

10

20

30

Total

180

140

320

This question

Cosmetics without EU approval are FDA-only cosmetics = 70. Foreign FDA cosmetics (40) are all FDA-only, so 30 domestic FDA cosmetics are FDA-only and 80 − 30 = 50 have both. Then domestic nutrition with both = 60 − 50 = 10, and foreign nutrition has none. Nutrition products with both approvals = 10.

Hence, the answer is option 3.

Question 20 · Venn Diagram DI

If 50 nutrition products did not have EU approval, then how many domestic cosmetic products did not have EU approval?
Answer & solution

Answer: 50

Solving the set

Step 1 – Totals. 320 products: 160 domestic, 160 foreign (fact 1). Nutrition 140 (70 foreign, 70 domestic), so cosmetics 180 (90 foreign, 90 domestic).

Step 2 – FDA approved: 200 in all, 70 foreign and 130 domestic; 120 cosmetic and 80 nutrition (fact 5).

Step 3 – Domestic FDA cosmetics = 160/2 = 80 (fact 2). Foreign FDA cosmetics = 120 − 80 = 40. Foreign FDA nutrition = 70 − 40 = 30. Domestic FDA nutrition = 80 − 30 = 50.

Step 4 – Every product has at least one approval, so products without FDA are EU-only: foreign cosmetics 90 − 40 = 50, foreign nutrition 70 − 30 = 40, domestic cosmetics 90 − 80 = 10, domestic nutrition 70 − 50 = 20.

Step 5 – No foreign product has both approvals; 60 domestic products have both, split between domestic FDA cosmetics (80) and domestic FDA nutrition (50).

Cosmetic

Nutrition

Total

Foreign – FDA only

40

30

70

Foreign – EU only

50

40

90

Domestic – FDA (incl. both)

80

50

130

Domestic – EU only

10

20

30

Total

180

140

320

This question

Nutrition products without EU = FDA-only nutrition = 50. Foreign FDA nutrition (30) are FDA-only, so 20 domestic FDA nutrition are FDA-only and 50 − 20 = 30 have both. Domestic cosmetics with both = 60 − 30 = 30, so domestic FDA cosmetics without EU = 80 − 30 = 50. (The 10 EU-only domestic cosmetics do have EU approval.)

Hence, the answer is 50.

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