On Angélica's fruit stand, there are 12 bananas, 1 pineapple, 4 oranges, 2 mangoes, and 3 papayas. The weight of 1 pineapple is the same as the weight of 1 orange, 1 mango, and 1 papaya, combined; the weight of 1 banana is half the weight of 1 papaya; 4 bananas weigh the same as 1 orange and 1 mango, combined; and 1 mango weighs $100 \mathrm{~g}$ more than 1 orange. If 1 pineapple weighs $600 \mathrm{~g}$, then:

a) How much do all the fruits on Angélica's fruit stand weigh?

b) In how many ways can Pedro, Angélica's grandson, choose 2 different fruits for his breakfast, using the fruits from the fruit stand?

c) How many smoothies can be made with these fruits, using $600 \mathrm{~g}$ of fruits? (It is allowed to use repeated fruits, but only whole quantities of fruit).
Solution

a) Let's organize the information:
I) 1 orange + 1 mango + 1 papaya = 1 pineapple $(600 \mathrm{~g})$;

II) 2 bananas = 1 papaya;

III) 1 orange + 1 mango = 4 bananas;

IV) 1 mango = 1 orange + 100 g.

In (I), by substituting the orange, mango, and papaya using (II) and (III), we find that 6 bananas are equivalent to one pineapple, meaning each banana weighs $100 \mathrm{~g}$, and consequently, each papaya weighs $200 \mathrm{~g}$, and an orange and a mango together weigh $400 \mathrm{~g}$. From (IV), it can be concluded that each mango weighs $250 \mathrm{~g}$ and each orange weighs $150 \mathrm{~g}$. Therefore, the total weight of all the fruits is $12 \cdot 100 + 600 + 4 \cdot 150 + 2 \cdot 250 + 3 \cdot 200 = 3,500 \mathrm{~g}$, which is the same as $3.5 \mathrm{~kg}$.

b) There are 5 types of fruits to choose from, and we need to choose two. This means the combinations are: banana and papaya; banana and mango; banana and pineapple; banana and orange; papaya and mango; papaya and pineapple; papaya and orange; mango and pineapple; mango and orange; and finally, pineapple and orange. Thus, Pedro can choose two different fruits in 10 ways. Another way to find this result without listing all possibilities is $\frac{5 \cdot 4}{2}=10$, which means we have 5 options for the first fruit, 4 for the second, and since the order in which we choose the first and second fruits does not matter, we divide the result by 2.
c) Let's list the possibilities:

| Pineapple | Mango | Papaya | Orange | Banana |
| :---: | :---: | :---: | :---: | :---: |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 1 | 0 |
| 0 | 1 | 0 | 1 | 2 |
| 0 | 2 | 0 | 0 | 1 |
| 0 | 0 | 1 | 2 | 1 |
| 0 | 0 | 1 | 0 | 4 |
| 0 | 0 | 2 | 0 | 2 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 2 | 3 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 6 |

Therefore, there are 11 different smoothies.