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The time needed to paint a fence varies directly with the length of the fence and inversely with the number of painters. If it takes five hours to paint 200 feet of fence with three painters, how long will it take five painters to paint 500 feet of fence?

The time needed to paint a fence varies directly with the length of the fence and inversely with the number of painters. If it takes 55 hours to paint 200200 feet of fence with 33 painters, how long will it take 55 painters to paint 500500 feet of fence?

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Q. The time needed to paint a fence varies directly with the length of the fence and inversely with the number of painters. If it takes 55 hours to paint 200200 feet of fence with 33 painters, how long will it take 55 painters to paint 500500 feet of fence?
  1. Calculate Rate: First, let's find the rate at which one painter can paint the fence. We use the formula: Time=Length of fence×Number of paintersRate\text{Time} = \frac{\text{Length of fence} \times \text{Number of painters}}{\text{Rate}}. Here, we know the time (55 hours), the length (200200 feet), and the number of painters (33). So, we solve for Rate.\newlineRate = 200 feet×3 painters5 hours=120 feet per hour per painter\frac{200 \text{ feet} \times 3 \text{ painters}}{5 \text{ hours}} = 120 \text{ feet per hour per painter}.
  2. Calculate Time: Now, we need to calculate the time it will take for five painters to paint 500500 feet of fence using the rate we found. We rearrange the formula to solve for Time: Time=(Length of fence×Number of painters)/Rate\text{Time} = (\text{Length of fence} \times \text{Number of painters}) / \text{Rate}.\newlineTime=(500 feet×5 painters)/120 feet per hour per painter=20.83 hours.\text{Time} = (500 \text{ feet} \times 5 \text{ painters}) / 120 \text{ feet per hour per painter} = 20.83 \text{ hours}.

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