CAT 2023 Slot 3 · DILR · Set 4: Project and Test Scores of Six Students

The full set and all 5 questions (Q16–Q20) from CAT 2023 Slot 3, with answers and solutions.

There are only three female students - Amala, Koli and Rini - and only three male students - Biman, Mathew and Shyamal - in a course. The course has two evaluation components, a project and a test. The aggregate score in the course is a weighted average of the two components, with the weights being positive and adding to 1.

The projects are done in groups of two, with each group consisting of a female and a male student. Both the group members obtain the same score in the project.

The following additional facts are known about the scores in the project and the test.

1. The minimum, maximum and the average of both project and test scores were identical - 40, 80 and 60, respectively.

2. The test scores of the students were all multiples of 10; four of them were distinct and the remaining two were equal to the average test scores.

3. Amala’s score in the project was double that of Koli in the same, but Koli scored 20 more than Amala in the test. Yet Amala had the highest aggregate score.

4. Shyamal scored the second highest in the test. He scored two more than Koli, but two less than Amala in the aggregate.

5. Biman scored the second lowest in the test and the lowest in the aggregate.

6. Mathew scored more than Rini in the project, but less than her in the test.

Question 16 · Caselet DI

What was Rini’s score in the project?
Answer & solution

Answer: 60

Solving the set

Step 1 – Project scores. Each female–male pair shares a score, so there are three pair scores, each counted twice. Min 40, max 80 and average 60 ⇒ the pair scores are 40, 60, 80. Amala = 2 × Koli ⇒ Amala 80, Koli 40, Rini 60. Mathew scored more than Rini in the project ⇒ Mathew 80 (Amala's partner).

Step 2 – Test scores. Six multiples of 10 with min 40, max 80, average 60 (sum 360), two of them equal to 60 and four distinct: 40 + 80 + a + b = 240 ⇒ a + b = 120 with a, b distinct and not 60 ⇒ 50 and 70. Test scores: 40, 50, 60, 60, 70, 80. Shyamal has the second highest (70); Biman the second lowest (50).

Step 3 – Test weight. Shyamal's aggregate is 2 more than Koli's and 2 less than Amala's, so Amala − Koli = 4. Project gap 80 − 40 = 40, test gap −20: 40(1 − w) − 20w = 4 ⇒ w = 0.6 (project weight 0.4).

Step 4 – Amala, Koli, Rini and Mathew share the test scores 40, 60, 60, 80, with Koli = Amala + 20 and Mathew < Rini.

Case

Amala test

Koli test

Mathew test

Rini test

Shyamal's project needed

Valid?

1

40

60

60

80

30

No (must be 40 or 60)

2

60

80

40

60

60

Yes

In Case 2, Koli's aggregate = 0.4 × 40 + 0.6 × 80 = 64, so Shyamal needs 66 = 0.4 × P + 0.6 × 70 ⇒ P = 60: Shyamal is Rini's partner and Biman is Koli's partner.

Student

Project

Test

Aggregate = 0.4 × Project + 0.6 × Test

Amala

80

60

32 + 36 = 68

Mathew

80

40

32 + 24 = 56

Rini

60

60

24 + 36 = 60

Shyamal

60

70

24 + 42 = 66

Koli

40

80

16 + 48 = 64

Biman

40

50

16 + 30 = 46

Check: Amala (68) is the highest and Biman (46) the lowest, as required.

This question

From Step 1, Rini's project score is 60.

Hence, the answer is 60.

Question 17 · Caselet DI

What was the weight of the test component?
  1. A
    0.60
  2. B
    0.50
  3. C
    0.75
  4. D
    0.40
Pick an option, then check.
Answer & solution

Answer: Option A

Solving the set

Step 1 – Project scores. Each female–male pair shares a score, so there are three pair scores, each counted twice. Min 40, max 80 and average 60 ⇒ the pair scores are 40, 60, 80. Amala = 2 × Koli ⇒ Amala 80, Koli 40, Rini 60. Mathew scored more than Rini in the project ⇒ Mathew 80 (Amala's partner).

Step 2 – Test scores. Six multiples of 10 with min 40, max 80, average 60 (sum 360), two of them equal to 60 and four distinct: 40 + 80 + a + b = 240 ⇒ a + b = 120 with a, b distinct and not 60 ⇒ 50 and 70. Test scores: 40, 50, 60, 60, 70, 80. Shyamal has the second highest (70); Biman the second lowest (50).

Step 3 – Test weight. Shyamal's aggregate is 2 more than Koli's and 2 less than Amala's, so Amala − Koli = 4. Project gap 80 − 40 = 40, test gap −20: 40(1 − w) − 20w = 4 ⇒ w = 0.6 (project weight 0.4).

Step 4 – Amala, Koli, Rini and Mathew share the test scores 40, 60, 60, 80, with Koli = Amala + 20 and Mathew < Rini.

Case

Amala test

Koli test

Mathew test

Rini test

Shyamal's project needed

Valid?

1

40

60

60

80

30

No (must be 40 or 60)

2

60

80

40

60

60

Yes

In Case 2, Koli's aggregate = 0.4 × 40 + 0.6 × 80 = 64, so Shyamal needs 66 = 0.4 × P + 0.6 × 70 ⇒ P = 60: Shyamal is Rini's partner and Biman is Koli's partner.

Student

Project

Test

Aggregate = 0.4 × Project + 0.6 × Test

Amala

80

60

32 + 36 = 68

Mathew

80

40

32 + 24 = 56

Rini

60

60

24 + 36 = 60

Shyamal

60

70

24 + 42 = 66

Koli

40

80

16 + 48 = 64

Biman

40

50

16 + 30 = 46

Check: Amala (68) is the highest and Biman (46) the lowest, as required.

This question

From Step 3, the test weight is 0.6.

Hence, the answer is option 1.

Question 18 · Caselet DI

What was the maximum aggregate score obtained by the students?
  1. A
    68
  2. B
    80
  3. C
    62
  4. D
    66
Pick an option, then check.
Answer & solution

Answer: Option A

Solving the set

Step 1 – Project scores. Each female–male pair shares a score, so there are three pair scores, each counted twice. Min 40, max 80 and average 60 ⇒ the pair scores are 40, 60, 80. Amala = 2 × Koli ⇒ Amala 80, Koli 40, Rini 60. Mathew scored more than Rini in the project ⇒ Mathew 80 (Amala's partner).

Step 2 – Test scores. Six multiples of 10 with min 40, max 80, average 60 (sum 360), two of them equal to 60 and four distinct: 40 + 80 + a + b = 240 ⇒ a + b = 120 with a, b distinct and not 60 ⇒ 50 and 70. Test scores: 40, 50, 60, 60, 70, 80. Shyamal has the second highest (70); Biman the second lowest (50).

Step 3 – Test weight. Shyamal's aggregate is 2 more than Koli's and 2 less than Amala's, so Amala − Koli = 4. Project gap 80 − 40 = 40, test gap −20: 40(1 − w) − 20w = 4 ⇒ w = 0.6 (project weight 0.4).

Step 4 – Amala, Koli, Rini and Mathew share the test scores 40, 60, 60, 80, with Koli = Amala + 20 and Mathew < Rini.

Case

Amala test

Koli test

Mathew test

Rini test

Shyamal's project needed

Valid?

1

40

60

60

80

30

No (must be 40 or 60)

2

60

80

40

60

60

Yes

In Case 2, Koli's aggregate = 0.4 × 40 + 0.6 × 80 = 64, so Shyamal needs 66 = 0.4 × P + 0.6 × 70 ⇒ P = 60: Shyamal is Rini's partner and Biman is Koli's partner.

Student

Project

Test

Aggregate = 0.4 × Project + 0.6 × Test

Amala

80

60

32 + 36 = 68

Mathew

80

40

32 + 24 = 56

Rini

60

60

24 + 36 = 60

Shyamal

60

70

24 + 42 = 66

Koli

40

80

16 + 48 = 64

Biman

40

50

16 + 30 = 46

Check: Amala (68) is the highest and Biman (46) the lowest, as required.

This question

From the final table, the highest aggregate is Amala's: 0.4 × 80 + 0.6 × 60 = 68.

Hence, the answer is option 1.

Question 19 · Caselet DI

What was Mathew’s score in the test?
Answer & solution

Answer: 40

Solving the set

Step 1 – Project scores. Each female–male pair shares a score, so there are three pair scores, each counted twice. Min 40, max 80 and average 60 ⇒ the pair scores are 40, 60, 80. Amala = 2 × Koli ⇒ Amala 80, Koli 40, Rini 60. Mathew scored more than Rini in the project ⇒ Mathew 80 (Amala's partner).

Step 2 – Test scores. Six multiples of 10 with min 40, max 80, average 60 (sum 360), two of them equal to 60 and four distinct: 40 + 80 + a + b = 240 ⇒ a + b = 120 with a, b distinct and not 60 ⇒ 50 and 70. Test scores: 40, 50, 60, 60, 70, 80. Shyamal has the second highest (70); Biman the second lowest (50).

Step 3 – Test weight. Shyamal's aggregate is 2 more than Koli's and 2 less than Amala's, so Amala − Koli = 4. Project gap 80 − 40 = 40, test gap −20: 40(1 − w) − 20w = 4 ⇒ w = 0.6 (project weight 0.4).

Step 4 – Amala, Koli, Rini and Mathew share the test scores 40, 60, 60, 80, with Koli = Amala + 20 and Mathew < Rini.

Case

Amala test

Koli test

Mathew test

Rini test

Shyamal's project needed

Valid?

1

40

60

60

80

30

No (must be 40 or 60)

2

60

80

40

60

60

Yes

In Case 2, Koli's aggregate = 0.4 × 40 + 0.6 × 80 = 64, so Shyamal needs 66 = 0.4 × P + 0.6 × 70 ⇒ P = 60: Shyamal is Rini's partner and Biman is Koli's partner.

Student

Project

Test

Aggregate = 0.4 × Project + 0.6 × Test

Amala

80

60

32 + 36 = 68

Mathew

80

40

32 + 24 = 56

Rini

60

60

24 + 36 = 60

Shyamal

60

70

24 + 42 = 66

Koli

40

80

16 + 48 = 64

Biman

40

50

16 + 30 = 46

Check: Amala (68) is the highest and Biman (46) the lowest, as required.

This question

From Step 4 (Case 2), Mathew's test score is 40.

Hence, the answer is 40.

Question 20 · Caselet DI

Which of the following pairs of students were part of the same project team?

i) Amala and Biman

ii) Koli and Mathew

  1. A
    Only ii)
  2. B
    Only i)
  3. C
    Neither i) nor ii)
  4. D
    Both i) and ii)
Pick an option, then check.
Answer & solution

Answer: Option C

Solving the set

Step 1 – Project scores. Each female–male pair shares a score, so there are three pair scores, each counted twice. Min 40, max 80 and average 60 ⇒ the pair scores are 40, 60, 80. Amala = 2 × Koli ⇒ Amala 80, Koli 40, Rini 60. Mathew scored more than Rini in the project ⇒ Mathew 80 (Amala's partner).

Step 2 – Test scores. Six multiples of 10 with min 40, max 80, average 60 (sum 360), two of them equal to 60 and four distinct: 40 + 80 + a + b = 240 ⇒ a + b = 120 with a, b distinct and not 60 ⇒ 50 and 70. Test scores: 40, 50, 60, 60, 70, 80. Shyamal has the second highest (70); Biman the second lowest (50).

Step 3 – Test weight. Shyamal's aggregate is 2 more than Koli's and 2 less than Amala's, so Amala − Koli = 4. Project gap 80 − 40 = 40, test gap −20: 40(1 − w) − 20w = 4 ⇒ w = 0.6 (project weight 0.4).

Step 4 – Amala, Koli, Rini and Mathew share the test scores 40, 60, 60, 80, with Koli = Amala + 20 and Mathew < Rini.

Case

Amala test

Koli test

Mathew test

Rini test

Shyamal's project needed

Valid?

1

40

60

60

80

30

No (must be 40 or 60)

2

60

80

40

60

60

Yes

In Case 2, Koli's aggregate = 0.4 × 40 + 0.6 × 80 = 64, so Shyamal needs 66 = 0.4 × P + 0.6 × 70 ⇒ P = 60: Shyamal is Rini's partner and Biman is Koli's partner.

Student

Project

Test

Aggregate = 0.4 × Project + 0.6 × Test

Amala

80

60

32 + 36 = 68

Mathew

80

40

32 + 24 = 56

Rini

60

60

24 + 36 = 60

Shyamal

60

70

24 + 42 = 66

Koli

40

80

16 + 48 = 64

Biman

40

50

16 + 30 = 46

Check: Amala (68) is the highest and Biman (46) the lowest, as required.

This question

Teams: Amala–Mathew (80), Rini–Shyamal (60), Koli–Biman (40). Amala–Biman and Koli–Mathew are not teams.

Hence, the answer is option 3.

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