Louis would take 120/7 hours or approximately 17.14 hours to collect a bucket of blueberries alone, after calculating their combined work rate and solving the equation for Louis' individual rate.
To determine how long it would take Louis to collect a bucket of blueberries alone, we first need to consider the rate at which Ken and Louis combined can fill the bucket. We can express this as the combined rate of work, and then find Louis' individual rate of work. Ken fills a bucket in 15 hours, so his rate is 1 bucket per 15 hours. Together, Ken and Louis fill a bucket in 8 hours, so the combined rate is 1 bucket per 8 hours.
Let's represent Louis' time to fill a bucket alone as L hours. Then, Ken's work rate is 1/15 and Louis' work rate is 1/L. Together, their combined rate of 1 bucket per 8 hours is:
1/15 + 1/L = 1/8
Solving this equation for L involves finding a common denominator (in this case, 120L), multiplying through, and rearranging terms:
- Multiply each term by 120L: 8L + 120 = 15L
- Subtract 8L from both sides: 120 = 7L
- Divide by 7 to find Louis' time: L = 120/7
Finally, calculating 120 divided by 7 gives us Louis' time to fill a bucket alone.
Louis would take 120/7 hours, which is approximately 17.14 hours to collect a bucket of blueberries alone.