Four institutes, A, B, C, and D, had contracts with four vendors W, X, Y, and Z during the ten calendar years from 2010 to 2019. The contracts were either multi-year contracts running for several consecutive years or single-year contracts. No institute had more than one contract with the same vendor. However, in a calendar year, an institute may have had contracts with multiple vendors, and a vendor may have had contracts with multiple institutes. It is known that over the decade, the institutes each got into two contracts with two of these vendors, and each vendor got into two contracts with two of these institutes.
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
EXPLANATION
answer 3
Institutes A, B, C, D Vendors W, X, Y, Z
Contracts Awarded are multiyear Contracts (consecutive years) or single Year Contract.
No institute had more than one contract with the single vendor.
Each Institute →→ Two contracts →→ two vendors
Each Vendor →→ two contracts →→ two institutes
V. Exactly 4 multi year contract (A-3years, B-7 years, C-3 years, D-4 years). Exactly 4 single year contracts. In total 8 contracts.
I. Vendor Z had at least one contract in ever year. This is only possible if he had both the contracts which are multi years contract. As 7 + 1 = 8 not possible. Only possibility is (7years + 3 years) or (3 years + 7 years) = 10 years.
II. Vendor X →→ Six years 2010, 2011, 2012, 2013, 2014, 2015. To have contract for theses six years, only possibility is left with multi years contracts (4 years + 3 years or 3 years + 4 years) with 1 year overlapping contract. PI. Note (3years + 3 years) is not possible for vendor X as one 3 years is already awarded to vendor Z{(7years + 3 years ) or (3 years + 7 years) = 10 years}.
So all the 4 multiyear contracts are awarded to Z (3 years, 7 years) & X (3 years, 4 years). So we are left with 4 single year contracts.
III. Vendor Y had contracts in 2010 & 2019. Both of these contracts are single year contracts. Vendor W had contracts (two contracts) only in year 2012. So both of these are single year contracts.
IV. There are 5 contracts in 2012.
VI. Institute C had one or more contracts in 2012, but no contract in 2011.
VII. Institute B & D each had exactly one contract in 2012. Institute D did not have any contract in 2010.
Out of 5 contracts in 2012 {B, D, W, W and C at least one contract). Hence C has exactly one contract in 2012.
For Vendor X, 4 year contract is with D. It Can vary from (2011-2014) or (2012-2015). No contract can be awarded to him after 2015 (point II). Only possibility for 3 year contract is with A only, as C do not have any contract in 2011. Therefore Vendor X has a 3 year contract with institute A from 2010-2012 & a 4 year contract with institute D from 2012 to 2015 (so as to have one or more contract from 2012 to 2015).
Vendor Z can be allotted contracts with only Institute B & C Only. As vendor Z has at least one contract in every year ( point I) , thus the only possibility left is first 7 year contract with institute B and then 3 year contract with institute C ( as institute C do not have any contract in 2011).
In 2012 there are 5 contracts. Three contracts are already assigned. Remaining two are single year contracts of W in 2012. Also Institute C has at least one contract in 2012. {No institute had more than one contract with the same vendor (initial condition)}. Hence both vendor W 1 years contract cannot be with Institute C. Hence Exactly one contract of vendor W is with institute C. Further, Point VII, Institutes B and D each had exactly one contract in 2012 which is already assigned. Hence second 1 year contract of vendor W is with institute A in 2012 (only possibility).
Institute A has allotted two contracts to 2 vendors {X & W}. Institute C had allotted contracts to two vendors {W & Z}. Hence Vendor Y can have a 1 year contract with only Institute B in 2010, (only possibility left). Hence vendor Y had 1 year contract in 2019 with Institute D (only possibility left).
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
EXPLANATION
OPTION: 1
Institutes A, B, C, D Vendors W, X, Y, Z
Contracts Awarded are multiyear Contracts (consecutive years) or single Year Contract.
No institute had more than one contract with the single vendor.
Each Institute →→ Two contracts →→ two vendors
Each Vendor →→ two contracts →→ two institutes
V. Exactly 4 multi year contract (A-3years, B-7 years, C-3 years, D-4 years). Exactly 4 single year contracts. In total 8 contracts.
I. Vendor Z had at least one contract in ever year. This is only possible if he had both the contracts which are multi years contract. As 7 + 1 = 8 not possible. Only possibility is (7years + 3 years) or (3 years + 7 years) = 10 years.
II. Vendor X →→ Six years 2010, 2011, 2012, 2013, 2014, 2015. To have contract for theses six years, only possibility is left with multi years contracts (4 years + 3 years or 3 years + 4 years) with 1 year overlapping contract. PI. Note (3years + 3 years) is not possible for vendor X as one 3 years is already awarded to vendor Z{(7years + 3 years ) or (3 years + 7 years) = 10 years}.
So all the 4 multiyear contracts are awarded to Z (3 years, 7 years) & X (3 years, 4 years). So we are left with 4 single year contracts.
III. Vendor Y had contracts in 2010 & 2019. Both of these contracts are single year contracts. Vendor W had contracts (two contracts) only in year 2012. So both of these are single year contracts.
IV. There are 5 contracts in 2012.
VI. Institute C had one or more contracts in 2012, but no contract in 2011.
VII. Institute B & D each had exactly one contract in 2012. Institute D did not have any contract in 2010.
Out of 5 contracts in 2012 {B, D, W, W and C at least one contract). Hence C has exactly one contract in 2012.
For Vendor X, 4 year contract is with D. It Can vary from (2011-2014) or (2012-2015). No contract can be awarded to him after 2015 (point II). Only possibility for 3 year contract is with A only, as C do not have any contract in 2011. Therefore Vendor X has a 3 year contract with institute A from 2010-2012 & a 4 year contract with institute D from 2012 to 2015 (so as to have one or more contract from 2012 to 2015).
Vendor Z can be allotted contracts with only Institute B & C Only. As vendor Z has at least one contract in every year ( point I) , thus the only possibility left is first 7 year contract with institute B and then 3 year contract with institute C ( as institute C do not have any contract in 2011).
In 2012 there are 5 contracts. Three contracts are already assigned. Remaining two are single year contracts of W in 2012. Also Institute C has at least one contract in 2012. {No institute had more than one contract with the same vendor (initial condition)}. Hence both vendor W 1 years contract cannot be with Institute C. Hence Exactly one contract of vendor W is with institute C. Further, Point VII, Institutes B and D each had exactly one contract in 2012 which is already assigned. Hence second 1 year contract of vendor W is with institute A in 2012 (only possibility).
Institute A has allotted two contracts to 2 vendors {X & W}. Institute C had allotted contracts to two vendors {W & Z}. Hence Vendor Y can have a 1 year contract with only Institute B in 2010, (only possibility left). Hence vendor Y had 1 year contract in 2019 with Institute D (only possibility left).
In how many years during this period was there only one contract?
- 4
- 2
- 3
- 5
EXPLANATION
OPTION: 3
Institutes A, B, C, D Vendors W, X, Y, Z
Contracts Awarded are multiyear Contracts (consecutive years) or single Year Contract.
No institute had more than one contract with the single vendor.
Each Institute →→ Two contracts →→ two vendors
Each Vendor →→ two contracts →→ two institutes
V. Exactly 4 multi year contract (A-3years, B-7 years, C-3 years, D-4 years). Exactly 4 single year contracts. In total 8 contracts.
I. Vendor Z had at least one contract in ever year. This is only possible if he had both the contracts which are multi years contract. As 7 + 1 = 8 not possible. Only possibility is (7years + 3 years) or (3 years + 7 years) = 10 years.
II. Vendor X →→ Six years 2010, 2011, 2012, 2013, 2014, 2015. To have contract for theses six years, only possibility is left with multi years contracts (4 years + 3 years or 3 years + 4 years) with 1 year overlapping contract. PI. Note (3years + 3 years) is not possible for vendor X as one 3 years is already awarded to vendor Z{(7years + 3 years ) or (3 years + 7 years) = 10 years}.
So all the 4 multiyear contracts are awarded to Z (3 years, 7 years) & X (3 years, 4 years). So we are left with 4 single year contracts.
III. Vendor Y had contracts in 2010 & 2019. Both of these contracts are single year contracts. Vendor W had contracts (two contracts) only in year 2012. So both of these are single year contracts.
IV. There are 5 contracts in 2012.
VI. Institute C had one or more contracts in 2012, but no contract in 2011.
VII. Institute B & D each had exactly one contract in 2012. Institute D did not have any contract in 2010.
Out of 5 contracts in 2012 {B, D, W, W and C at least one contract). Hence C has exactly one contract in 2012.
For Vendor X, 4 year contract is with D. It Can vary from (2011-2014) or (2012-2015). No contract can be awarded to him after 2015 (point II). Only possibility for 3 year contract is with A only, as C do not have any contract in 2011. Therefore Vendor X has a 3 year contract with institute A from 2010-2012 & a 4 year contract with institute D from 2012 to 2015 (so as to have one or more contract from 2012 to 2015).
Vendor Z can be allotted contracts with only Institute B & C Only. As vendor Z has at least one contract in every year ( point I) , thus the only possibility left is first 7 year contract with institute B and then 3 year contract with institute C ( as institute C do not have any contract in 2011).
In 2012 there are 5 contracts. Three contracts are already assigned. Remaining two are single year contracts of W in 2012. Also Institute C has at least one contract in 2012. {No institute had more than one contract with the same vendor (initial condition)}. Hence both vendor W 1 years contract cannot be with Institute C. Hence Exactly one contract of vendor W is with institute C. Further, Point VII, Institutes B and D each had exactly one contract in 2012 which is already assigned. Hence second 1 year contract of vendor W is with institute A in 2012 (only possibility).
Institute A has allotted two contracts to 2 vendors {X & W}. Institute C had allotted contracts to two vendors {W & Z}. Hence Vendor Y can have a 1 year contract with only Institute B in 2010, (only possibility left). Hence vendor Y had 1 year contract in 2019 with Institute D (only possibility left).
What BEST can be concluded about the number of contracts in 2010?
- exactly 4
- at least 4
- exactly 3
- at least 3
EXPLANATION
OPTION: 3
Institutes A, B, C, D Vendors W, X, Y, Z
Contracts Awarded are multiyear Contracts (consecutive years) or single Year Contract.
No institute had more than one contract with the single vendor.
Each Institute →→ Two contracts →→ two vendors
Each Vendor →→ two contracts →→ two institutes
V. Exactly 4 multi year contract (A-3years, B-7 years, C-3 years, D-4 years). Exactly 4 single year contracts. In total 8 contracts.
I. Vendor Z had at least one contract in ever year. This is only possible if he had both the contracts which are multi years contract. As 7 + 1 = 8 not possible. Only possibility is (7years + 3 years) or (3 years + 7 years) = 10 years.
II. Vendor X →→ Six years 2010, 2011, 2012, 2013, 2014, 2015. To have contract for theses six years, only possibility is left with multi years contracts (4 years + 3 years or 3 years + 4 years) with 1 year overlapping contract. PI. Note (3years + 3 years) is not possible for vendor X as one 3 years is already awarded to vendor Z{(7years + 3 years ) or (3 years + 7 years) = 10 years}.
So all the 4 multiyear contracts are awarded to Z (3 years, 7 years) & X (3 years, 4 years). So we are left with 4 single year contracts.
III. Vendor Y had contracts in 2010 & 2019. Both of these contracts are single year contracts. Vendor W had contracts (two contracts) only in year 2012. So both of these are single year contracts.
IV. There are 5 contracts in 2012.
VI. Institute C had one or more contracts in 2012, but no contract in 2011.
VII. Institute B & D each had exactly one contract in 2012. Institute D did not have any contract in 2010.
Out of 5 contracts in 2012 {B, D, W, W and C at least one contract). Hence C has exactly one contract in 2012.
For Vendor X, 4 year contract is with D. It Can vary from (2011-2014) or (2012-2015). No contract can be awarded to him after 2015 (point II). Only possibility for 3 year contract is with A only, as C do not have any contract in 2011. Therefore Vendor X has a 3 year contract with institute A from 2010-2012 & a 4 year contract with institute D from 2012 to 2015 (so as to have one or more contract from 2012 to 2015).
Vendor Z can be allotted contracts with only Institute B & C Only. As vendor Z has at least one contract in every year ( point I) , thus the only possibility left is first 7 year contract with institute B and then 3 year contract with institute C ( as institute C do not have any contract in 2011).
In 2012 there are 5 contracts. Three contracts are already assigned. Remaining two are single year contracts of W in 2012. Also Institute C has at least one contract in 2012. {No institute had more than one contract with the same vendor (initial condition)}. Hence both vendor W 1 years contract cannot be with Institute C. Hence Exactly one contract of vendor W is with institute C. Further, Point VII, Institutes B and D each had exactly one contract in 2012 which is already assigned. Hence second 1 year contract of vendor W is with institute A in 2012 (only possibility).
Institute A has allotted two contracts to 2 vendors {X & W}. Institute C had allotted contracts to two vendors {W & Z}. Hence Vendor Y can have a 1 year contract with only Institute B in 2010, (only possibility left). Hence vendor Y had 1 year contract in 2019 with Institute D (only possibility left).
