A motorist was to travel from town A to town B, a distance of 80 miles. He traveled the first 24 minutes at a certain rate; traffic then increased and for the next 6 minutes he averaged 10 miles per hour less than his original speed; then traffic eased up and he traveled the remaining distance at a rate 50% greater than his original rate. He arrived 22 earlier than he would had he traveled the whole distance at his original rate. Find his original rate.
PLEASE PROVIDE SOME EXPLANATIONS
let x be the original rate (mi/min)
Let y be the time traveled at 50% faster speed.
Since distance = time*speed,
24x + 6(x-1/6) + y(1.5x) = 80
24+6+y = 80/x - 22
x = 13/16
y = 604/13
original speed = 13/16 mi/min = 48.75 mi/hr
check:
24 min @ 13/16 mi/min = 39/2 mi
6 min @ 31/48 mi/min = 31/8 mi
remainder 453/8 mi @ 39/32 = 604/13 min
total time: 994/13 min
80/(13/16) = 1280/13
1280/13 - 994/13 = 286/13 = 22 min
strange answer, but it works.
To solve this problem, let's break it down into smaller steps.
Step 1: Identify the given information
- Total distance: 80 miles
- Time traveled at the original rate: 24 minutes
- Time traveled at a reduced rate: 6 minutes
- Time saved: 22 minutes
Step 2: Convert the given times to hours
- 24 minutes is 24/60 = 0.4 hours
- 6 minutes is 6/60 = 0.1 hours
- 22 minutes is 22/60 = 0.3667 hours
Step 3: Calculate the time traveled at the original rate
- Let's assume the original rate is 'r'.
- The time traveled at the original rate is given by: Distance / Rate
- For the first part of the journey, the distance traveled is 0.4r.
Step 4: Calculate the time traveled at the reduced rate
- The reduced rate is 10 mph less than the original rate, so it is (r - 10).
- The time traveled at the reduced rate is given by: Distance / Rate
- For the second part of the journey, the distance traveled is 0.1(r - 10).
Step 5: Calculate the time traveled at the increased rate
- The increased rate is 50% greater than the original rate, so it is (1.5r).
- The time traveled at the increased rate is given by: Distance / Rate
- For the third part of the journey, the distance traveled is 80 - (0.4r + 0.1(r - 10)).
Step 6: Set up the equation using the time saved
- The time saved is 0.3667 hours.
- Equation: Time traveled at original rate - (Time traveled at reduced rate + Time traveled at increased rate) = Time saved
Step 7: Solve the equation to find the original rate, 'r'
- Substitute the values from steps 3, 4, and 5 into the equation.
- Solve the equation for 'r'.
After following these steps, you should be able to find the original rate of the motorist.