Anna and Hannah have $80 each. Their friend offered to invest their money, promising to return a sum r times as great as what they invested. Anna was suspicious, so she invested $20 only, but Hannah invested her entire $80. Fortunately, the friend did indeed return a sum r times as great to each. They decided to make another investment. This time, Hannah invested all of the money returned to her, and Anna invested the money returned to her and the remaining $60. Again, they got a sum r times as great as what they invested. In the end, Hannah had twice the amount Anna had. Write an equation in terms of r for the situation

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Anna and Hannah each start with 80 dollars. Their friend offers an investment opportunity, promising to return r times whatever amount they invest. Anna is suspicious and only invests 20 dollars, while Hannah is more trusting and invests her entire 80 dollars. After the first investment, Anna receives 20r dollars from her 20-dollar investment, giving her a total of 60 plus 20r dollars. Hannah receives 80r dollars from her 80-dollar investment, so she has 80r dollars total. For the second investment, Anna invests all her money - 60 plus 20r dollars - while Hannah invests all her money - 80r dollars. Again, their friend promises to return r times whatever they invest. After the second investment, Anna has 60r plus 20r squared dollars, and Hannah has 80r squared dollars. According to the problem, Hannah has twice what Anna has. This gives us the equation: 80r squared equals 2 times the quantity 60r plus 20r squared, which simplifies to 40r squared equals 120r. Solving the equation 40r squared equals 120r, we factor out 40r to get 40r times the quantity r minus 3 equals zero. This gives us r equals 0 or r equals 3. Since r equals 0 doesn't make sense in this investment context, the answer is r equals 3. The friend returns 3 times whatever amount is invested. Let's analyze the first investment round. Anna invested 20 dollars and received 20r dollars back. Since she kept 60 dollars uninvested, her total after the first round is 60 plus 20r dollars. Hannah invested all 80 dollars and received 80r dollars back, so her total is 80r dollars. For the second investment round, both Anna and Hannah decide to invest all their money. Anna invests 60 plus 20r dollars and will receive r times this amount, which equals 60r plus 20r squared dollars. Hannah invests 80r dollars and will receive r times this amount, which equals 80r squared dollars. After the second investment, Anna has 60r plus 20r squared dollars, while Hannah has 80r squared dollars. The key condition states that Hannah has twice what Anna has. This gives us the equation: 80r squared equals 2 times the quantity 60r plus 20r squared, which simplifies to 40r squared equals 120r. To solve 40r squared equals 120r, we first move all terms to one side to get 40r squared minus 120r equals zero. Factoring out 40r gives us 40r times the quantity r minus 3 equals zero. This means either 40r equals zero or r minus 3 equals zero, giving us r equals zero or r equals 3. Since r equals zero doesn't make sense in this investment context, our answer is r equals 3. We can verify: Anna ends with 360 dollars and Hannah with 720 dollars, and indeed 720 equals 2 times 360.