Luke and Owen have $100 each. Their friend offered to invest their money, promising to return a sum r times as great as what they invested. Luke was suspicious, so he invested $10 only, but Owen invested his entire $100. Fortunately, the friend did indeed return a sum r times as great to each. They decided to make another investment. This time, Owen invested all of the money returned to him, and Luke invested the money returned to him and the remaining $90. Again, they got a sum r times as great as what they invested. In the end, Owen had $337.50 more than Luke. Write an equation in terms of r that models the situation.

Question

Luke and Owen have  $100 each. Their friend offered to invest their money, promising to return a sum  r times as great as what they invested. Luke was suspicious, so he invested  $10 only, but Owen invested his entire  $100. Fortunately, the friend did indeed return a sum  r times as great to each. They decided to make another investment. This time, Owen invested all of the money returned to him, and Luke invested the money returned to him and the remaining  $90. Again, they got a sum  r times as great as what they invested. In the end, Owen had  $337.50 more than Luke. Write an equation in terms of  r that models the situation.

Bytelearn · Accepted Answer

Initial amounts of money: Let's denote the initial amount of money both Luke and Owen have as L and O respectively. We know that L = O = $100. Luke invests $10 and gets back $10r, while Owen invests $100 and gets back $100r after the first round of investment.First round of investment: After the first investment, Luke has $10r from the investment and $90 that he did not invest, so he has a total of $10r + $90. Owen has $100r from his investment.Total amount after first investment: In the second round of investment, Luke invests the $10r he got from the first investment plus the $90 he didn't invest, so he invests a total of $10r + $90. Owen invests the entire $100r he got from the first investment.Second round of investment: After the second investment, Luke has $(10r + 90)r from the investment, and Owen has $(100r)r from his investment.Total amount after second investment: We are given that Owen has $337.50 more than Luke after the second investment. So, the equation that models the situation is: (100r)r = (10r + 90)r + 337.50Equation modeling the situation: To simplify the equation, we distribute the r on both sides: 100r^2 = 10r^2 + 90r + 337.50Simplifying the equation: Now, we subtract 10r^2 + 90r from both sides to get the equation in terms of r: 100r^2 - 10r^2 - 90r = 337.50Combining like terms: Simplify the equation further by combining like terms: 90r^2 - 90r = 337.50Final equation in terms of r: We can divide the entire equation by 90 to simplify it: r^2 - r = 3.75This is the final equation in terms of r that models the situation. We can write it as: r^2 - r - 3.75 = 0

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