12. Maria, Carolina and Pedro receive \( \$ 800 \) from their grandmother in the ratio Maria: Carolina: Pedro \( =7: 5: 4 \). (a) Calculate how much money each receives. (b) Maria spends \( \frac{2}{7} \) of her money and then invests the rest for two years at \( 5 \% \) per year simple interest. How much money does Maria have at the end of the two years? (c) Carolina spends all of.her money on a hi-fi set and two years later sells it at a loss of \( 20 \% \). How much money does Carolina have at the end of the two years (d) Pedro spends some of his money and at the end of the two years he has \( \$ 100 \). Write down and simplify the ratio of the amounts of money Maria, Carolina and Pedro have at the end of the two years.

To solve the problem, we will go through each part step by step. ### (a) Calculate how much money each receives. The total ratio of Maria, Carolina, and Pedro is \( 7 + 5 + 4 = 16 \) parts. Now, we can find out how much each person receives: - **Maria's share**: \[ \text{Maria's share} = \frac{7}{16} \times 800 = 350 \] - **Carolina's share**: \[ \text{Carolina's share} = \frac{5}{16} \times 800 = 250 \] - **Pedro's share**: \[ \text{Pedro's share} = \frac{4}{16} \times 800 = 200 \] So, the amounts received are: - Maria: \( \$350 \) - Carolina: \( \$250 \) - Pedro: \( \$200 \) ### (b) Maria spends \( \frac{2}{7} \) of her money and then invests the rest for two years at \( 5\% \) per year simple interest. First, we calculate how much Maria spends: \[ \text{Amount spent by Maria} = \frac{2}{7} \times 350 = 100 \] Now, we find out how much money she has left after spending: \[ \text{Amount left} = 350 - 100 = 250 \] Next, we calculate the simple interest for two years: \[ \text{Simple Interest} = \text{Principal} \times \text{Rate} \times \text{Time} = 250 \times 0.05 \times 2 = 25 \] Finally, we find out how much money Maria has at the end of the two years: \[ \text{Total amount} = \text{Principal} + \text{Interest} = 250 + 25 = 275 \] ### (c) Carolina spends all of her money on a hi-fi set and two years later sells it at a loss of \( 20\% \). Carolina spends all of her money, which is \( \$250 \). Now, we calculate how much she gets back after selling it at a loss of \( 20\% \): \[ \text{Loss} = 20\% \text{ of } 250 = 0.20 \times 250 = 50 \] \[ \text{Amount after selling} = 250 - 50 = 200 \] ### (d) Pedro spends some of his money and at the end of the two years he has \( \$100 \). Now we have the amounts at the end of two years: - Maria: \( \$275 \) - Carolina: \( \$200 \) - Pedro: \( \$100 \) Now, we write down and simplify the ratio of the amounts of money Maria, Carolina, and Pedro have at the end of the two years: \[ \text{Ratio} = 275 : 200 : 100 \] To simplify this ratio, we can divide each term by \( 25 \): \[ \frac{275}{25} : \frac{200}{25} : \frac{100}{25} = 11 : 8 : 4 \] Thus, the simplified ratio of the amounts of money Maria, Carolina, and Pedro have at the end of the two years is: \[ \boxed{11 : 8 : 4} \]