CAT 2021 Quant was dominated by Arithmetic followed by Algebra. In Arithmetic, the questions were dominated by topics like **Speed-time-distance**, **Mixture and Alligations**. This year, there was a surprise. The questions from **Geometry** were relatively on the lower side as compared to the previous years. There were 8 TITA Qs this year. Overall this section was at a medium level of difficulty.

Ten objects o1, o2, ..., o10 were distributed among Amar, Barat, Charles, Disha, and Elise. Each item went to exactly one person. Each person got exactly two of the items, and this pair of objects is called her/his bundle.

The following table shows how each person values each object.

The value of any bundle by a person is the sum of that person's values of the objects in that bundle. A person X envies another person Y if X values Y's bundle more than X's own bundle.

For example, hypothetically suppose Amar's bundle consists of o1 and o2, and Barat's bundle consists of o3 and o4. Then Amar values his own bundle at 4 + 9 = 13 and Barat's bundle at 9 + 3 = 12. Hence Amar does not envy Barat. On the other hand, Barat values his own bundle at 7 + 5 = 12 and Amar's bundle at 5 + 9 = 14. Hence Barat envies Amar.

The following facts are known about the actual distribution of the objects among the five people.

1. If someone's value for an object is 10, then she/he received that object.

2. Objects o1, o2, and o3 were given to three different people.

3. Objects o1 and o8 were given to different people.

4. Three people value their own bundles at 16. No one values her/his own bundle at a number higher than 16.

5. Disha values her own bundle at an odd number. All others value their own bundles at an even number.

6. Some people who value their own bundles less than 16 envy some other people who value their own bundle at 16. No one else envies others.

Question 3 : What is Amar's value for his own bundle?

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Amar values Barat’s bundle at 17. So, Amar could not have scored 16, otherwise he would have envied Barat. So, Charles scores 16. Charles, Barat and Elise score 16 each. Charles 16 should come from two 8’s. O1 cannot comes with O2, O3 or O8. O2 and O3 cannot come together. So, O3 should be accompanied by O2 or O3.

If Charles had got O8 and O2, Barat would have valued Charles bundle at 17 and envied it. This cannot happen. So, Charles should have gotten O8 and O3.

Now, we have o1, o2, o4, o5, o6 remaining – two for Amar, 2 for Disha and 1 for Elise. Elise should have gotten o1 or o5. Amar should have got some two of o2, o4, o5, o6.
Disha should have got one of o1, o2 and one of o4, o5, o6.
Let us break this into two possibilities - E getting o1 or E getting o5.

This gets more interesting now.

1. In this combo, Amar has to get 3 + 7, or 3 + 3. Amar values Disha’s o2 at 9. So, Amar will end up valuing Disha’s bundle at higher than this own. Or, Amar will envy Disha. This is not possible.

Now Amar has to get one of 04 or 06 and Disha has to get the other. If Amar gets o4, his score would be 12. Disha’s score would be 11 and Disha would value Amar’s bundle at 13 (8 for o2 and 5 for o4). This would make Disha envious of Amar, which is not possible. So, Amar should get o6 and Disha 04.

The question is **" What is Amar's value for his own bundle? " **

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