How to efficiently approach this puzzle/logical reasoning problem?

Question

Answer the following question based on the information given below.

Five colleagues from different divisions of a company met in the club discussing their Sunday's winnings at cards games.

Conditions :

1. Mathur and the person from Engineering Division together had won Rs1500.

2. Sastry and Saxena together won Rs1400.

3. Saxena and Senior Vice President together won Rs1200.

4. Verma and the Production manger had together won Rs1000.

5. The General Manager and the International Trading Division person together won Rs900.

6. The Foods Division man and the Soaps Division person together won Rs700.

7. The Vice President and Oil Seeds Division person together won Rs600.

8. Rao and the Soaps Division person together won Rs400.

9. The Deputy General Manager together with the only person in Churidar-kurta won Rs800.

10. Two people in three-piece suit together won Rs1100.

11. The person in Safari-suit has won more than the person in two-piece suit.

Question. Who has won the maximum amount?

I tried solving this problem by forming a table with five columns (i.e. Name, Division, Amount won, Designation and Dress), but I'm stuck with a lot of equations.

For example: Let's say a,b,c,d and e are the amount won by Mathur, Saxena, Sastry, Rao and Verma .

Senior Vice President can be selected from a,d,e --> 3C1 ways.

Production Manager can be selected from a,b,c,d --> 4C1 ways.

Similarly, for all other columns there are number of ways.

I'm stumped here. Please help.

Answer 1

Ten different sums of pairs are given, and there are only ten pairs, so you know all the sums, and this allows you to deduce the ranks of the summed amounts (in fact also the amounts, but you don't need those).

Let $x_i$ be the $i$-th greatest amount. We have $x_1+x_2=1500$ and $x_1+x_3=1400$, and thus $x_2-x_3=100$. We also have $x_5+x_4=400$ and $x_5+x_3=600$ and thus $x_3-x_4=200$.

So the amounts are $x_1$, $x_2$, $x_2-100$, $x_2-300$, $x_5$. So $x_1+x_4=x_1+x_2-300=1200$ and $x_5+x_2=x_5+x_4+300=700$.

That leaves the sums $1100$, $1000$, $900$, $800$ for $x_1+x_5$, $x_2+x_3$, $x_2+x_4$ and $x_3+x_4$. Since $x_2+x_3$ and $x_2+x_4$ must have a difference of $200$ and $x_3+x_4$ is even less, it follows that $x_2+x_3=1100$, $x_2+x_4=900$, $x_3+x_4=800$ and $x_1+x_5=1000$.

To summarise:

\begin{align} x_1+x_2&=1500\;,\\ x_1+x_3&=1400\;,\\ x_1+x_4&=1200\;,\\ x_2+x_3&=1100\;,\\ x_1+x_5&=1000\;,\\ x_2+x_4&=900\;,\\ x_3+x_4&=800\;,\\ x_2+x_5&=700\;,\\ x_3+x_5&=600\;,\\ x_4+x_5&=400\;.\\ \end{align}

From $2$. and $3$., Saxena won $x_1$. If that answers the question "Who has won the maximum amount?", we're done. Else you can readily deduce the remaining information on this basis.