The following facts are also known about these contracts.
I. Vendor Z had at least one contract in every year.
II. Vendor X had one or more contracts in every year up to 2015, but no contract in any year after that.
III. Vendor Y had contracts in 2010 and 2019. Vendor W had contracts only in 2012.
IV. There were five contracts in 2012.
V. There were exactly four multi-year contracts. Institute B had a 7-year contract, D had a 4- year contract, and A and C had one 3-year contract each. The other four contracts were single-year contracts.
VI. Institute C had one or more contracts in 2012 but did not have any contract in 2011.
VII. Institutes B and D each had exactly one contract in 2012. Institute D did not have any contract in 2010.
In which of the following years were there two or more contracts?
- 2018
- 2017
- 2015
- 2016
The Correct Option is C
Solution and Explanation
From IV: A, B, C, and D have one long-term contract each: 3, 7, 3, and 4 years respectively. All other contracts are one-year contracts.
From I, Z has at least one contract every year. The only possible combinations are 7+3 or 7+4 years, with the 7-year contract being from B.
From III, Vendor W had contracts only in 2012. From VII, Institutes B and D each had exactly one contract in 2012, meaning W received contracts from A and C.
From II, Vendor X had contracts every year up to 2015 but none after that. From VI and VII, C and D had no contracts in 2011 and 2010 respectively. Thus, A must have had a 3-year contract with X from 2010-2012. For 2013-2015, X could not have a contract with B, so it must be with either C or D.
- Case 1: If X had a 3-year contract with C from 2013-2015, D couldn't have had a contract in 2012, making this case invalid.
- Case 2: If X had a 4-year contract with D from 2012-2015, C must have had a 3-year contract with Z from 2017-2019, ensuring Z has at least one contract every year.
- It is known that over the decade, each institute had two contracts with two different vendors, and each vendor had two contracts with two different institutes. Therefore:
- A didn't have any contracts from 2013-2019 since it had contracts with X and W from 2010-2012.
- C didn't have any contracts in 2010, 2013, 2014, 2015, and 2016.
- From III, Vendor Y had contracts in 2010 and 2019. In 2010, neither D nor C had any contracts, and A had already fulfilled its two different vendor contracts. Thus, Y had a contract with B in 2010, meaning B had no contracts in 2017, 2018, and 2019.
- For Y, the only possible contract left is with D, meaning D had no contracts in 2011, 2016, 2017, and 2018.
Now, the table looks like:
| 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | |
| A | X | X | X,W | N | N | N | N | N | N | N |
| B | Z,Y | Z | Z | Z | Z | Z | Z | N | N | N |
| C | N | N | W | N | N | N | N | Z | Z | Z |
| D | N | N | X | X | X | X | N | N | N | Y |
N' represents no contract. Among the given options, only the year 2015 has two contracts, while the rest have only one contract in that year.
So, the correct option is (C): 2015
Which of the following is true?
- D had a contract with Y in 2019
- D had a contract with X in 2011
- B had a contract with Y in 2019
- B had a contract with Z in 2017
The Correct Option is A
Solution and Explanation
From IV: A, B, C, and D have one long-term contract each: 3, 7, 3, and 4 years respectively. All other contracts are one-year contracts.
From I, Z has at least one contract every year. The only possible combinations are 7+3 or 7+4 years, with the 7-year contract being from B.
From III, Vendor W had contracts only in 2012. From VII, Institutes B and D each had exactly one contract in 2012, meaning W received contracts from A and C.
From II, Vendor X had contracts every year up to 2015 but none after that. From VI and VII, C and D had no contracts in 2011 and 2010 respectively. Thus, A must have had a 3-year contract with X from 2010-2012. For 2013-2015, X could not have a contract with B, so it must be with either C or D.
- Case 1: If X had a 3-year contract with C from 2013-2015, D couldn't have had a contract in 2012, making this case invalid.
- Case 2: If X had a 4-year contract with D from 2012-2015, C must have had a 3-year contract with Z from 2017-2019, ensuring Z has at least one contract every year.
- It is known that over the decade, each institute had two contracts with two different vendors, and each vendor had two contracts with two different institutes. Therefore:
- A didn't have any contracts from 2013-2019 since it had contracts with X and W from 2010-2012.
- C didn't have any contracts in 2010, 2013, 2014, 2015, and 2016.
- From III, Vendor Y had contracts in 2010 and 2019. In 2010, neither D nor C had any contracts, and A had already fulfilled its two different vendor contracts. Thus, Y had a contract with B in 2010, meaning B had no contracts in 2017, 2018, and 2019.
- For Y, the only possible contract left is with D, meaning D had no contracts in 2011, 2016, 2017, and 2018.
Now, the table looks like:
| 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | |
| A | X | X | X,W | N | N | N | N | N | N | N |
| B | Z,Y | Z | Z | Z | Z | Z | Z | N | N | N |
| C | N | N | W | N | N | N | N | Z | Z | Z |
| D | N | N | X | X | X | X | N | N | N | Y |
N' represents no contract.
So, the correct option is (A): D had a contract with Y in 2019
In how many years during this period was there only one contract?
- 4
- 2
- 3
- 5
The Correct Option is C
Solution and Explanation
From IV: A, B, C, and D have one long-term contract each: 3, 7, 3, and 4 years respectively. All other contracts are one-year contracts.
From I, Z has at least one contract every year. The only possible combinations are 7+3 or 7+4 years, with the 7-year contract being from B.
From III, Vendor W had contracts only in 2012. From VII, Institutes B and D each had exactly one contract in 2012, meaning W received contracts from A and C.
From II, Vendor X had contracts every year up to 2015 but none after that. From VI and VII, C and D had no contracts in 2011 and 2010 respectively. Thus, A must have had a 3-year contract with X from 2010-2012. For 2013-2015, X could not have a contract with B, so it must be with either C or D.
- Case 1: If X had a 3-year contract with C from 2013-2015, D couldn't have had a contract in 2012, making this case invalid.
- Case 2: If X had a 4-year contract with D from 2012-2015, C must have had a 3-year contract with Z from 2017-2019, ensuring Z has at least one contract every year.
- It is known that over the decade, each institute had two contracts with two different vendors, and each vendor had two contracts with two different institutes. Therefore:
- A didn't have any contracts from 2013-2019 since it had contracts with X and W from 2010-2012.
- C didn't have any contracts in 2010, 2013, 2014, 2015, and 2016.
- From III, Vendor Y had contracts in 2010 and 2019. In 2010, neither D nor C had any contracts, and A had already fulfilled its two different vendor contracts. Thus, Y had a contract with B in 2010, meaning B had no contracts in 2017, 2018, and 2019.
- For Y, the only possible contract left is with D, meaning D had no contracts in 2011, 2016, 2017, and 2018.
Now, the table looks like:
| 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | |
| A | X | X | X,W | N | N | N | N | N | N | N |
| B | Z,Y | Z | Z | Z | Z | Z | Z | N | N | N |
| C | N | N | W | N | N | N | N | Z | Z | Z |
| D | N | N | X | X | X | X | N | N | N | Y |
'N' represents no contract. Only during 2016, 2017 and 2018 there was only 1 contract.
So, the correct option is (C): 3.
What BEST can be concluded about the number of contracts in 2010?
- exactly 4
- at least 4
- exactly 3
- at least 3
The Correct Option is C
Solution and Explanation
From IV: A, B, C, and D have one long-term contract each: 3, 7, 3, and 4 years respectively. All other contracts are one-year contracts.
From I, Z has at least one contract every year. The only possible combinations are 7+3 or 7+4 years, with the 7-year contract being from B.
From III, Vendor W had contracts only in 2012. From VII, Institutes B and D each had exactly one contract in 2012, meaning W received contracts from A and C.
From II, Vendor X had contracts every year up to 2015 but none after that. From VI and VII, C and D had no contracts in 2011 and 2010 respectively. Thus, A must have had a 3-year contract with X from 2010-2012. For 2013-2015, X could not have a contract with B, so it must be with either C or D.
- Case 1: If X had a 3-year contract with C from 2013-2015, D couldn't have had a contract in 2012, making this case invalid.
- Case 2: If X had a 4-year contract with D from 2012-2015, C must have had a 3-year contract with Z from 2017-2019, ensuring Z has at least one contract every year.
- It is known that over the decade, each institute had two contracts with two different vendors, and each vendor had two contracts with two different institutes. Therefore:
- A didn't have any contracts from 2013-2019 since it had contracts with X and W from 2010-2012.
- C didn't have any contracts in 2010, 2013, 2014, 2015, and 2016.
- From III, Vendor Y had contracts in 2010 and 2019. In 2010, neither D nor C had any contracts, and A had already fulfilled its two different vendor contracts. Thus, Y had a contract with B in 2010, meaning B had no contracts in 2017, 2018, and 2019.
- For Y, the only possible contract left is with D, meaning D had no contracts in 2011, 2016, 2017, and 2018.
Now, the table looks like:
| 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | |
| A | X | X | X,W | N | N | N | N | N | N | N |
| B | Z,Y | Z | Z | Z | Z | Z | Z | N | N | N |
| C | N | N | W | N | N | N | N | Z | Z | Z |
| D | N | N | X | X | X | X | N | N | N | Y |
'N' represents no contract. and the number of contracts in 2010 is 3.
So, the correct option is (C): 3.
Which institutes had multiple contracts during the same year?
- B and C only
- A only
- B only
- A and B only
The Correct Option is D
Solution and Explanation
From IV: A, B, C, and D have one long-term contract each: 3, 7, 3, and 4 years respectively. All other contracts are one-year contracts.
From I, Z has at least one contract every year. The only possible combinations are 7+3 or 7+4 years, with the 7-year contract being from B.
From III, Vendor W had contracts only in 2012. From VII, Institutes B and D each had exactly one contract in 2012, meaning W received contracts from A and C.
From II, Vendor X had contracts every year up to 2015 but none after that. From VI and VII, C and D had no contracts in 2011 and 2010 respectively. Thus, A must have had a 3-year contract with X from 2010-2012. For 2013-2015, X could not have a contract with B, so it must be with either C or D.
- Case 1: If X had a 3-year contract with C from 2013-2015, D couldn't have had a contract in 2012, making this case invalid.
- Case 2: If X had a 4-year contract with D from 2012-2015, C must have had a 3-year contract with Z from 2017-2019, ensuring Z has at least one contract every year.
- It is known that over the decade, each institute had two contracts with two different vendors, and each vendor had two contracts with two different institutes. Therefore:
- A didn't have any contracts from 2013-2019 since it had contracts with X and W from 2010-2012.
- C didn't have any contracts in 2010, 2013, 2014, 2015, and 2016.
- From III, Vendor Y had contracts in 2010 and 2019. In 2010, neither D nor C had any contracts, and A had already fulfilled its two different vendor contracts. Thus, Y had a contract with B in 2010, meaning B had no contracts in 2017, 2018, and 2019.
- For Y, the only possible contract left is with D, meaning D had no contracts in 2011, 2016, 2017, and 2018.
Now, the table looks like:
| 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | |
| A | X | X | X,W | N | N | N | N | N | N | N |
| B | Z,Y | Z | Z | Z | Z | Z | Z | N | N | N |
| C | N | N | W | N | N | N | N | Z | Z | Z |
| D | N | N | X | X | X | X | N | N | N | Y |
'N' represents no contract. A and B have multiple contract in a same year.
So, the correct option is (D): A and B only.
Which institutes and vendors had more than one contracts in any year?
- A, D, W, and Z
- B, W, X, and Z
- B, D, W, and X
- A, B, W, and X
The Correct Option is D
Solution and Explanation
From IV: A, B, C, and D have one long-term contract each: 3, 7, 3, and 4 years respectively. All other contracts are one-year contracts.
From I, Z has at least one contract every year. The only possible combinations are 7+3 or 7+4 years, with the 7-year contract being from B.
From III, Vendor W had contracts only in 2012. From VII, Institutes B and D each had exactly one contract in 2012, meaning W received contracts from A and C.
From II, Vendor X had contracts every year up to 2015 but none after that. From VI and VII, C and D had no contracts in 2011 and 2010 respectively. Thus, A must have had a 3-year contract with X from 2010-2012. For 2013-2015, X could not have a contract with B, so it must be with either C or D.
- Case 1: If X had a 3-year contract with C from 2013-2015, D couldn't have had a contract in 2012, making this case invalid.
- Case 2: If X had a 4-year contract with D from 2012-2015, C must have had a 3-year contract with Z from 2017-2019, ensuring Z has at least one contract every year.
- It is known that over the decade, each institute had two contracts with two different vendors, and each vendor had two contracts with two different institutes. Therefore:
- A didn't have any contracts from 2013-2019 since it had contracts with X and W from 2010-2012.
- C didn't have any contracts in 2010, 2013, 2014, 2015, and 2016.
- From III, Vendor Y had contracts in 2010 and 2019. In 2010, neither D nor C had any contracts, and A had already fulfilled its two different vendor contracts. Thus, Y had a contract with B in 2010, meaning B had no contracts in 2017, 2018, and 2019.
- For Y, the only possible contract left is with D, meaning D had no contracts in 2011, 2016, 2017, and 2018.
Now, the table looks like:
| 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | |
| A | X | X | X,W | N | N | N | N | N | N | N |
| B | Z,Y | Z | Z | Z | Z | Z | Z | N | N | N |
| C | N | N | W | N | N | N | N | Z | Z | Z |
| D | N | N | X | X | X | X | N | N | N | Y |
'N' represents no contract. A, B, W, X had more than one contracts in a single year.
So, the correct option is (D): A, B, W, and X.
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