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Math Help - Distance Problem

  1. #1
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    Distance Problem

    Two different routes between two cities differ by 21 mi. Two people made the trip between the cities in exactly the same time. One traveled the shorter route at 50 mi/hr. and the other traveled the longer route at 55mi/hr. Find the length of each route.
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  2. #2
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    Hello reiward
    Quote Originally Posted by reiward View Post
    Two different routes between two cities differ by 21 mi. Two people made the trip between the cities in exactly the same time. One traveled the shorter route at 50 mi/hr. and the other traveled the longer route at 55mi/hr. Find the length of each route.
    Use ratios. In one hour travelling at 50 and 55 mph, the distances travelled are (of course!) 50 and 55 miles, a ratio of 50:55 = 10:11.

    So the two routes between the cities are in the ratio 10:11. In other words, the longer route is \tfrac{1}{10} longer than the shorter route. This is 21 miles. The shorter route is therefore 10\times21=210 miles, and the longer is \tfrac{11}{10}\times210 = 231 miles.

    Grandad
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  3. #3
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    Hello, reiward!

    Another approach . . .

    We use: . \text{Distance} \:=\:\text{Speed} \times \text{Time} \quad\rightarrow\quad T \:=\:\frac{D}{S}


    Two different routes between two cities differ by 21 miles.
    Two people made the trip between the cities in exactly the same time.
    One traveled the shorter route at 50 mi/hr. and the other traveled the longer route at 55mi/hr.
    Find the length of each route.

    Let D = length of the longer route.
    Then D-21 = length of the shorter route.

    One person drove the shorter route at 50 mph.
    . . This took: . \frac{D-21}{50} hours.

    The other drove the longer route at 55 mph.
    . . This took: . \frac{D}{55} hours.

    The two times are equal: . \frac{D-21}{50} \:=\:\frac{D}{55}

    Cross-multiply: . 55(D-21) \:=\:50D \quad\Rightarrow\quad 55D - 1155 \:=\:50D \quad\Rightarrow\quad 5D \:=\:1155


    Therefore: . \begin{array}{cccc}D \;=\; 231\text{ miles} &&\text{longer route} \\ D-21 \;=\; 210\text{ miles} && \text{shorter route} \end{array}

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