### Instruction: Darren has borrowed $100$ clams from Ethan at a $10\%$ simple daily interest. Meanwhile, Fergie has borrowed $150$ clams from Gertie at a $5\%$ simple daily interest. In how many days will Darren and Fergie owe the same amounts, assuming that they will not make any repayments in that time period?### Response: To solve this problem, I need to find the number of days, let's call it $x$, when the debts of Darren and Fergie are equal.
I know that simple interest means that the interest is calculated only on the original principal, not on the accumulated interest.
So, for Darren, the interest he owes after $x$ days is $0.1x \times 100 = 10x$ clams, and his total debt is $100 + 10x$ clams.
For Fergie, the interest she owes after $x$ days is $0.05x \times 150 = 7.5x$ clams, and her total debt is $150 + 7.5x$ clams.
To find $x$, I need to set these two expressions equal to each other and solve for $x$.
So I have $100 + 10x = 150 + 7.5x$.
Subtracting $100$ from both sides, I get $10x = 50 + 7.5x$.
Subtracting $7.5x$ from both sides, I get $2.5x = 50$.
Dividing both sides by $2.5$, I get $x = 20$.
This means that after $20$ days, Darren and Fergie will owe the same amount of money to their creditors.