Distance Problems - 2

Sabtu, 25 April 2026

In the following problems people are making a round trip. The average speed in each direction will be different and the total trip time will be given. The equation to solve is

Time to destination + Time on return trip = Total trip time.

To get the time to and from the destination, use D = RT and solve for T. The equation to solve becomes,

            

 

Example

 

A jogger jogged seven miles to a park then jogged home. He jogged 1 mph faster to the park than he jogged on the way home. The round trip took 2 hours 34 minutes. How fast did he jog to the park?

   Let r represent the jogger’s average speed on the way home. He jogged 1 mph faster to the park, so r + 1 represents his average speed to the park. The distance to the park is 7 miles, so D = 7.

Time to the park + Time home = 2 hours 34 minutes

The time to the park is represented by . The time home is represented by . The round trip is 2 hours 34 minutes . The equation to solve becomes  . The LCD is 30r(r + 1).

The jogger’s average speed to the park was 5 + 1 = 6 mph.

 

Practice

 

1.      A man rode his bike six miles to work. The wind reduced his average speed on the way home by 2 mph. The round trip took 1 hour 21 minutes. How fast was he riding on the way to work?

2.      On a road trip a saleswoman traveled 120 miles to visit a customer. She averaged 15 mph faster to the customer than on the return trip. She spent a total of 4 hours 40 minutes driving. What was her average speed on the return trip?

3.      A couple walked on the beach from their house to a public beach four miles away. They walked 0.2 mph faster on the way home than on the way to the public beach. They walked for a total of 2 hours 35 minutes. How fast did they walk home?

4.      A family drove from Detroit to Buffalo, a distance of 215 miles, for the weekend. They averaged 10 mph faster on the return trip. They spent a total of seven hours on the road. What was their average speed on the trip from Detroit to Buffalo? (Give your solution accurate to one decimal place.)

5.      Boston and New York are 190 miles apart. A professor drove from his home in Boston to a conference in New York. On the return trip, he faced heavy traffic and averaged 17 mph slower than on his way to New York. He spent a total of 8 hours 5 minutes on the road. How long did his trip from Boston to New York last?

 

Solutions

 

1.      Let r represent the man’s average speed on the way to work. Then r – 2 represents the man’s average speed on his way home. The distance each way is 6 miles, so the time he rode to work is  , and the time he rode home is . The total time is 1 hour 21 minutes = hours. The equation to solve is . The LCD is 20r(r – 2).

 

 

The man’s average speed on his way to work was 10 mph.

2.      Let r represent the saleswoman’s average speed on her return trip. Her average speed on the way to the customer is r + 15. The distance each way is 120 miles. She spent a total of 4 hours 40 minutes =  hours driving. The time spent driving to the customer is  The time spent driving on the return trip is .   The equation to solve is  . The LCD is 3r(r + 15).

 

The saleswoman averaged 45 mph on her return trip.

3.      Let r represent the couple’s average rate on their way home, then r – 0.2 represents the couple’s average speed to the public beach. The distance to the public beach is 4 miles. They walked for a total of 2 hours 35 minutes = 2 ³⁵/₆₀ = 2 ⁷/₁₂ = ³¹/₁₂ hours.The time spent walking to the public beach is  . The time spent walking home is .   The equation to solve is. The LCD is 12r(r – 0.2).

(Multiplying by 10 to clear the decimals would result in fairly large numbers for the quadratic formula.)

The couple walked home at the rate of ¹⁶/₅ = 3.2 mph.

4.      Let r represent the average speed from Detroit to Buffalo. The average speed from Buffalo to Detroit is r + 10. The distance from Detroit to Buffalo is 215 miles and the total time the family spent driving is 7 hours. The time spent driving from Detroit to Buffalo is   . The time spent driving from Buffalo to Detroit is  . The equation to solve is  . The LCD is r(r + 10).

The family averaged 56.8 mph from Detroit to Buffalo.

5.      Let r represent the average speed on his trip from Boston to New York. Because his average speed was 17 mph slower on his return trip, r – 17 represents his average speed on his trip from New York to Boston. The distance between Boston and New York is 190 miles. The time on the road from Boston to New York is  and the time on the road from New York to Boston is . The time on the road from Boston to New York plus the time on the road from New York to Boston is 8 hours 5 minutes = 8 ⁵/₆₀ = 8 ¹/₁₂ = ⁹⁷/₁₂ hours. The equation to solve is . The LCD is 12r(r – 17).

The rate cannot be 7 ¹/₉₇ because the total round trip is only 8 hours 5 minutes. The professor’s average speed from Boston to New York is 57 mph. We want his time on the road from Boston to New York. His time on the road from Boston to New York is hours or 3 hours 20 minutes.

 

 

 

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Labels: Mathematician

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