Three couples are on a trip. Valdir has 30 reais more than Laura and Paulo has 50 reais more than Clara. Each of the men has exactly 40 reais more than their respective wife. Laura and Clara have more money than Bruna, who only has 30 reais, and Joaquim is a Math teacher.

a) Who is Paulo's wife?

b) Who is Valdir's wife?

c) How much money does Joaquim have?
Solution

a) Let's build a table with the initials of the people, where the husbands will be in the first column and the wives in the first row. Each box associates a man with a woman. We will also use the notation $x$ for "not a couple" and $\checkmark$ for "is a couple".

|  | $C$ | $L$ | $B$ |
| :--- | :--- | :--- | :--- |
| $V$ |  |  |  |
| $P$ |  |  |  |
| $J$ |  |  |  |

If Valdir has 30 reais more than Laura, they cannot be a couple, just as Paulo cannot be Clara's husband, because in both cases, the man does not have exactly 40 reais more than the woman.

|  | $C$ | $L$ | $B$ |
| :--- | :--- | :--- | :--- |
| $V$ |  | $x$ |  |
| $P$ | $x$ |  |  |
| $J$ |  |  |  |

Bruna is the one with the least money among the women, so she must be the wife of the man with the least money, who cannot be Paulo. Therefore, Laura is Paulo's wife.

|  | $C$ | $L$ | $B$ |
| :---: | :---: | :---: | :---: |
| $V$ |  | $x$ |  |
| $P$ | $x$ | $\checkmark$ | $x$ |
| $J$ |  |  |  |

b) Assuming Laura has $q$ reais, then Valdir has $(q+30)$ and Paulo has $(q+40)$, since he is Laura's husband and, consequently, Clara has $(q-10)$, as she has 50 reais less than Paulo. Assuming Bruna is Valdir's wife, then Valdir would have $40+30=70$ reais and Clara would have, consequently, $40-10=30$ reais, which is the same amount as Bruna, which is absurd, since Bruna is the one with the least. Therefore, Valdir's wife is Clara.

|  | $C$ | $L$ | $B$ |
| :---: | :---: | :---: | :---: |
| $V$ | $\checkmark$ | $x$ | $x$ |
| $P$ | $x$ | $\checkmark$ | $x$ |
| $J$ |  |  |  |

c) It only remains to fill in the last row of the table and notice that, if Bruna is the wife of Joaquim, the math teacher, then Joaquim has $30+40=70$ reais.

|  | $C$ | $L$ | $B$ |
| :---: | :---: | :---: | :---: |
| $V$ | $\checkmark$ | $x$ | $x$ |
| $P$ | $x$ | $\checkmark$ | $x$ |
| $J$ | $x$ | $x$ | $\checkmark$ |