Final answer:
Gardeners A and B work together for 4 days and complete 2/3 of the garden. Gardener B then needs an additional 5 days to complete the remaining 1/3 of the garden alone. Hence, the entire garden is landscaped in a total of 9 days.
Step-by-step explanation:
The problem at hand is a work-rate problem involving two gardeners, A and B. Gardener A can complete the garden in 10 days, and Gardener B can complete it in 15 days. Therefore, A's work rate is 1/10 of the garden per day, and B's work rate is 1/15 of the garden per day. When they work together for 4 days, they complete a fraction of the garden together: 4 days * (1/10 + 1/15) of the garden.
First, we need to determine the least common denominator for the fractions, which is 30. So, we can rewrite this as: 4 days * (3/30 + 2/30) of the garden, simplifying to 4 days * 5/30, or simply 20/30 of the garden. This means after 4 days, A and B have completed 20/30 (or 2/3) of the garden together.
This leaves 1/3 of the garden to be completed by B alone. Since B's rate is 1/15 per day, we can determine how many days it will take for B to finish the remaining garden by solving for x in 1/3 = x/15. Multiplying both sides by 15 gives us 5 = x, meaning it will take B exactly 5 more days.
Therefore, the total days taken to landscape the entire garden is the 4 days of combined work plus the additional 5 days of B working alone, which gives us 9 days in total.