Let's denote the speed of the car leaving City B as \( r \) mph. Therefore, the speed of the car leaving City A would be \( r + 20 \) mph since it is traveling 20 mph faster.
Since they meet after 1 hour and 24 minutes, we first need to convert that time into hours: \[ 1 \text{ hour} + 24 \text{ minutes} = 1 + \frac{24}{60} = 1.4 \text{ hours} \]
Next, we know that the distance covered by both cars combined equals the distance between the two cities, which is 112 miles. Thus, the total distance traveled by both cars can be expressed as: \[ \text{Distance} = \text{Speed} \times \text{Time} \] For the two cars, we have: \[ (r + 20) \times 1.4 + r \times 1.4 = 112 \]
This simplifies to: \[ 1.4(r + 20) + 1.4r = 112 \]
Now combine the terms: \[ 1.4r + 28 + 1.4r = 112 \] \[ 2.8r + 28 = 112 \]
Next, isolate \( r \): \[ 2.8r = 112 - 28 \] \[ 2.8r = 84 \] \[ r = \frac{84}{2.8} = 30 \]
Thus, the speed of the car leaving City B is \( 30 \) mph. Now to find the speed of the car leaving City A: \[ r + 20 = 30 + 20 = 50 \text{ mph} \]
In conclusion, the rates of the cars are:
- The speed of the car from City A is 50 mph.
- The speed of the car from City B is 30 mph.