John and Bill walk towards each other's houses. If both leave at 10 am, they meet 10 minutes later. If Bill leaves 3 minutes after John, they meet 9 minutes after Bill leaves. How long does it take for Bill to walk to John's house?
Answer (2)
To solve the problem about John and Bill walking towards each other's house and the time it takes for Bill to reach John's house, we need to establish a constant rate of speed for each individual and then create equations based on the two scenarios provided: both leaving at the same time and Bill leaving 3 minutes after John.
Scenario Analysis
In the first scenario, they meet 10 minutes after leaving at 10 am, which means John and Bill combined cover the distance between their houses in 10 minutes. In the second scenario, when Bill leaves 3 minutes after John, they still meet each other, now 9 minutes after Bill leaves. This would be 12 minutes after John leaves since he left 3 minutes before Bill.
Let's denote the total distance between their houses as D and define their walking rates as rates of John (RJ) and Bill (RB). If John walks for 12 minutes and Bill for 9 minutes before they meet, and because distance equals rate times time, then:
12RJ + 9RB = D (1)
But we know from the first scenario, where they walk towards each other and meet in 10 minutes, that:
10(RJ + RB) = D (2)
Solving the Equations
Equation (2) can help us simplify Equation (1) because it gives us the value of RJ + RB. We can divide Equation (1) by the number of minutes, and set them equal to each other, which gives us:
12/10RJ + 9/10RB = RJ + RB
After simplifying, RB = 2/10RJ or RB = 1/5RJ. This means that Bill's rate is one-fifth of John's rate, so it takes Bill 5 times longer to walk a certain distance than it does for John.
So, if they take 10 minutes to meet when they start at the same time, then Bill would take 50 minutes to walk the whole distance to John's house because,
10 minutes (John and Bill's meeting time) x 5 (since Bill walks 5 times slower than John) = 50 minutes (The time it takes for Bill to walk to John's house).
Bill takes 60 minutes to walk to John's house. This conclusion comes from analyzing two meeting scenarios and setting up equations based on their speeds. We determined the relationship between their speeds and used it to find the time it takes Bill to reach John's house.
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