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Problem 1172

Two cars traveled the same distance. One car traveled at 50 mph and the other car traveled at \(60 \mathrm{mph}\). It took the slower car 50 minutes longer to make the trip. How long did it take the faster car to make the trip?

Short Answer

Expert verified
The time taken by the faster car to travel the distance is \(4 \) hours and \(10 \) minutes.
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Step by step solution

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Step 1: Assign variables to the given information

Let the time taken by the faster car to travel the distance be \(t\) hours. Then, the time taken by the slower car to travel the same distance would be \((t + \frac{50}{60})\) hours, as it takes 50 minutes more than the faster car.

Step 2: Write the equation for the distance of each car

Since both cars travel the same distance, we can write the equation as: Distance travelled by slower car = Distance travelled by the faster car. So, using the formula distance = speed × time, we get: 50 × (t + \(\frac{50}{60}\)) = 60 × t

Step 3: Simplify the equation

First, modify the equation to have terms without fractions by multiplying the equation by 60: 60 × 50 × (t + \(\frac{50}{60}\)) = 60 × 60 × t This results in: 50 × (60t + 50) = 60 × 60t Now, distribute and simplify: 3000t + 2500 = 3600t

Step 4: Solve for t

Subtract 3000t from both sides: 2500 = 600t Now, divide both sides by 600: t = \(\frac{2500}{600}\) To simplify the fraction, divide both the numerator and the denominator by the greatest common divisor, which is 100. t = \(\frac{25}{6}\)

Step 5: Interpret the result

The time taken by the faster car to travel the distance is \(\frac{25}{6}\) hours. To express this result in hours and minutes, take the integer part as the hours and multiply the decimal part by 60. So, the faster car takes 4 hours and \(\frac{1}{6}\) of an hour or 4 hours and 10 minutes to travel the distance.

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