A credit rating agency scores six firms – Alpha, Beta, Gamma, Delta, Epsilon and Zeta – on four parameters: Liquidity (L), Solvency (S), Governance (G) and Growth (W). Each score is an integer from 1 to 5. A firm's total is the sum of its four scores, and the firm is rated:
- AAA if the total is 17 or more,
- AA if the total is 14 to 16,
- A if the total is 11 to 13,
- B if the total is 10 or less.
The table below shows the scores; some entries have been lost and are marked "?".
| Firm | L | S | G | W |
|---|---|---|---|---|
| Alpha | 5 | ? | 5 | 3 |
| Beta | ? | 5 | ? | 4 |
| Gamma | 4 | 3 | ? | 5 |
| Delta | 2 | ? | 3 | ? |
| Epsilon | ? | 5 | 5 | ? |
| Zeta | 1 | 3 | 2 | 2 |
| Column total | 19 | 22 | 21 | 22 |
The following is also known:
1. No two firms have the same total.
2. Alpha and Epsilon are the only two firms rated AAA.
3. Beta's Governance score is strictly lower than each of its other three scores.
4. Gamma's total is exactly 2 more than Beta's total.
5. Epsilon's Liquidity and Growth scores are equal.
- AAA if the total is 17 or more,
- AA if the total is 14 to 16,
- A if the total is 11 to 13,
- B if the total is 10 or less.
The table below shows the scores; some entries have been lost and are marked "?".
| Firm | L | S | G | W |
|---|---|---|---|---|
| Alpha | 5 | ? | 5 | 3 |
| Beta | ? | 5 | ? | 4 |
| Gamma | 4 | 3 | ? | 5 |
| Delta | 2 | ? | 3 | ? |
| Epsilon | ? | 5 | 5 | ? |
| Zeta | 1 | 3 | 2 | 2 |
| Column total | 19 | 22 | 21 | 22 |
The following is also known:
1. No two firms have the same total.
2. Alpha and Epsilon are the only two firms rated AAA.
3. Beta's Governance score is strictly lower than each of its other three scores.
4. Gamma's total is exactly 2 more than Beta's total.
5. Epsilon's Liquidity and Growth scores are equal.
Question 1
Numerical answer
What is Alpha's Solvency score?
Show answer and solution
Answer: 4
From the column totals: Beta L + Epsilon L = 7, Alpha S + Delta S = 6, Beta G + Gamma G = 6, Delta W + Epsilon W = 8.
Gamma = 12 + Gamma G = 18 − Beta G; Beta = Beta L + 9 + Beta G. Clue 4 gives Beta L = 7 − 2·Beta G. Clue 3 needs Beta G < Beta L and Beta G < 4: Beta G = 1 (Beta L = 5) or Beta G = 2 (Beta L = 3). If Beta G = 1, Gamma = 17 would be AAA, contradicting clue 2. So Beta G = 2, Beta L = 3, Gamma G = 4, Beta = 14, Gamma = 16.
Then Epsilon L = 4 = Epsilon W (clue 5), Delta W = 4, Epsilon = 18. Alpha must be AAA: 13 + Alpha S ≥ 17 → Alpha S = 4 or 5; 5 would make Alpha 18 = Epsilon (clue 1 fails). So Alpha S = 4, Delta S = 2, Delta = 11.
Alpha's Solvency score is 4.
Gamma = 12 + Gamma G = 18 − Beta G; Beta = Beta L + 9 + Beta G. Clue 4 gives Beta L = 7 − 2·Beta G. Clue 3 needs Beta G < Beta L and Beta G < 4: Beta G = 1 (Beta L = 5) or Beta G = 2 (Beta L = 3). If Beta G = 1, Gamma = 17 would be AAA, contradicting clue 2. So Beta G = 2, Beta L = 3, Gamma G = 4, Beta = 14, Gamma = 16.
Then Epsilon L = 4 = Epsilon W (clue 5), Delta W = 4, Epsilon = 18. Alpha must be AAA: 13 + Alpha S ≥ 17 → Alpha S = 4 or 5; 5 would make Alpha 18 = Epsilon (clue 1 fails). So Alpha S = 4, Delta S = 2, Delta = 11.
Alpha's Solvency score is 4.
Question 2
Single correct option
How many of the six firms are rated AA?
- A. 1
- B. 2
- C. 3
- D. 4
Show answer and solution
Answer: (B) 2
Deduced grid:
| Firm | L | S | G | W | Total | Rating |
|---|---|---|---|---|---|---|
| Alpha | 5 | 4 | 5 | 3 | 17 | AAA |
| Beta | 3 | 5 | 2 | 4 | 14 | AA |
| Gamma | 4 | 3 | 4 | 5 | 16 | AA |
| Delta | 2 | 2 | 3 | 4 | 11 | A |
| Epsilon | 4 | 5 | 5 | 4 | 18 | AAA |
| Zeta | 1 | 3 | 2 | 2 | 8 | B |
Only Beta (14) and Gamma (16) fall in the 14–16 band, so 2 firms are rated AA. Key step: Beta's Governance cannot be 1, because that would push Gamma to 17 and create a third AAA firm.
| Firm | L | S | G | W | Total | Rating |
|---|---|---|---|---|---|---|
| Alpha | 5 | 4 | 5 | 3 | 17 | AAA |
| Beta | 3 | 5 | 2 | 4 | 14 | AA |
| Gamma | 4 | 3 | 4 | 5 | 16 | AA |
| Delta | 2 | 2 | 3 | 4 | 11 | A |
| Epsilon | 4 | 5 | 5 | 4 | 18 | AAA |
| Zeta | 1 | 3 | 2 | 2 | 8 | B |
Only Beta (14) and Gamma (16) fall in the 14–16 band, so 2 firms are rated AA. Key step: Beta's Governance cannot be 1, because that would push Gamma to 17 and create a third AAA firm.
Question 3
Numerical answer
What is Delta's total score?
Show answer and solution
Answer: 11
Delta W = 22 − (3 + 4 + 5 + 2) − Epsilon W = 8 − Epsilon W. Clue 5 with Beta L = 3 (see the deduction via clues 3 and 4) gives Epsilon L = 4 = Epsilon W, so Delta W = 4.
Alpha S must make Alpha AAA but not tie with Epsilon's 18 (clue 1), so Alpha S = 4 and Delta S = 6 − 4 = 2.
Delta = 2 + 2 + 3 + 4 = 11 (rating A).
Alpha S must make Alpha AAA but not tie with Epsilon's 18 (clue 1), so Alpha S = 4 and Delta S = 6 − 4 = 2.
Delta = 2 + 2 + 3 + 4 = 11 (rating A).
Question 4
Single correct option
Suppose the agency changes its method so that the Governance score is counted twice in the total (all other scores counted once). How many firms would then have a total of 20 or more?
- A. 2
- B. 3
- C. 4
- D. 5
Show answer and solution
Answer: (B) 3
Deduced grid:
| Firm | L | S | G | W | Total | Rating |
|---|---|---|---|---|---|---|
| Alpha | 5 | 4 | 5 | 3 | 17 | AAA |
| Beta | 3 | 5 | 2 | 4 | 14 | AA |
| Gamma | 4 | 3 | 4 | 5 | 16 | AA |
| Delta | 2 | 2 | 3 | 4 | 11 | A |
| Epsilon | 4 | 5 | 5 | 4 | 18 | AAA |
| Zeta | 1 | 3 | 2 | 2 | 8 | B |
New total = old total + G: Alpha 17 + 5 = 22, Beta 14 + 2 = 16, Gamma 16 + 4 = 20, Delta 11 + 3 = 14, Epsilon 18 + 5 = 23, Zeta 8 + 2 = 10.
Firms at 20 or more: Alpha, Gamma, Epsilon → 3. Note Gamma reaches exactly 20, so it must be included.
| Firm | L | S | G | W | Total | Rating |
|---|---|---|---|---|---|---|
| Alpha | 5 | 4 | 5 | 3 | 17 | AAA |
| Beta | 3 | 5 | 2 | 4 | 14 | AA |
| Gamma | 4 | 3 | 4 | 5 | 16 | AA |
| Delta | 2 | 2 | 3 | 4 | 11 | A |
| Epsilon | 4 | 5 | 5 | 4 | 18 | AAA |
| Zeta | 1 | 3 | 2 | 2 | 8 | B |
New total = old total + G: Alpha 17 + 5 = 22, Beta 14 + 2 = 16, Gamma 16 + 4 = 20, Delta 11 + 3 = 14, Epsilon 18 + 5 = 23, Zeta 8 + 2 = 10.
Firms at 20 or more: Alpha, Gamma, Epsilon → 3. Note Gamma reaches exactly 20, so it must be included.
Question 5
Single correct option
Which of the following statements is FALSE?
- A. Epsilon's Liquidity score is 4.
- B. Beta's Liquidity score equals Gamma's Solvency score.
- C. Delta's Growth score is higher than Beta's Growth score.
- D. Gamma's Governance score equals Epsilon's Growth score.
Show answer and solution
Answer: (C) Delta's Growth score is higher than Beta's Growth score.
Deduced grid:
| Firm | L | S | G | W | Total | Rating |
|---|---|---|---|---|---|---|
| Alpha | 5 | 4 | 5 | 3 | 17 | AAA |
| Beta | 3 | 5 | 2 | 4 | 14 | AA |
| Gamma | 4 | 3 | 4 | 5 | 16 | AA |
| Delta | 2 | 2 | 3 | 4 | 11 | A |
| Epsilon | 4 | 5 | 5 | 4 | 18 | AAA |
| Zeta | 1 | 3 | 2 | 2 | 8 | B |
Check each option:
- Epsilon L = 4 — true.
- Beta L = 3 and Gamma S = 3 — true.
- Delta W = 4 and Beta W = 4 — equal, so 'higher' is false.
- Gamma G = 4 and Epsilon W = 4 — true.
| Firm | L | S | G | W | Total | Rating |
|---|---|---|---|---|---|---|
| Alpha | 5 | 4 | 5 | 3 | 17 | AAA |
| Beta | 3 | 5 | 2 | 4 | 14 | AA |
| Gamma | 4 | 3 | 4 | 5 | 16 | AA |
| Delta | 2 | 2 | 3 | 4 | 11 | A |
| Epsilon | 4 | 5 | 5 | 4 | 18 | AAA |
| Zeta | 1 | 3 | 2 | 2 | 8 | B |
Check each option:
- Epsilon L = 4 — true.
- Beta L = 3 and Gamma S = 3 — true.
- Delta W = 4 and Beta W = 4 — equal, so 'higher' is false.
- Gamma G = 4 and Epsilon W = 4 — true.
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