Solving an Equation for a Rate Problem
| | Rate (km/h) | Time (h) | Distance (km) |
| :---- | :---------- | :------- | :------------ |
| Trip1 | 40 | $0.9-x$ | 40(0.9-x) |
| Trip2 | 5 | $x$ | $5x$ |
Talia took the bus from her home to the bank and then walked back to her home along the same route. The bus traveled at an average speed of [tex]$40 km / h$[/tex] and she walked at an average speed of [tex]$5 km / h$[/tex]. To determine the time, [tex]$x$[/tex], that it took Talia to walk home, she used the equation [tex]$40(0.9-x)=5 x$[/tex].
The time of Trip 2 is hours.
The distance of Trip 1 is km.
Answer (1)
Solve the equation 40 ( 0.9 − x ) = 5 x for x to find the time of Trip 2.
Simplify the equation to 36 − 40 x = 5 x , then combine like terms to get 36 = 45 x .
Divide both sides by 45 to find x = 0.8 hours.
Substitute x = 0.8 into the expression 40 ( 0.9 − x ) to find the distance of Trip 1, which is 40 ( 0.9 − 0.8 ) = 4 km.
The time of Trip 2 is 0.8 hours and the distance of Trip 1 is 4 km.
Explanation
Understanding the Problem We are given a rate problem where Talia takes a bus to the bank and walks back home. We need to find the time it took her to walk home (Trip 2) and the distance of her bus trip to the bank (Trip 1). We are given the equation 40 ( 0.9 − x ) = 5 x , which relates the speeds, times, and distances of the two trips.
Solving for Time (x) First, we need to solve the equation 40 ( 0.9 − x ) = 5 x for x . This equation represents the fact that the distance to the bank is the same as the distance back home. Let's solve for x :
Expand the left side of the equation: 40 ( 0.9 − x ) = 36 − 40 x
So the equation becomes: 36 − 40 x = 5 x
Add 40 x to both sides: 36 = 45 x
Divide both sides by 45: x = 45 36 = 5 × 9 4 × 9 = 5 4 = 0.8
So, x = 0.8 hours.
Calculating the Distance of Trip 1 Now that we have the value of x , which represents the time of Trip 2 (walking back home), we can find the distance of Trip 1 (bus trip to the bank). The distance of Trip 1 is given by 40 ( 0.9 − x ) . Substitute x = 0.8 into this expression:
D i s t an ce = 40 ( 0.9 − 0.8 ) = 40 ( 0.1 ) = 4
So, the distance of Trip 1 is 4 km.
Final Answer Therefore, the time of Trip 2 is 0.8 hours, and the distance of Trip 1 is 4 km.
Examples
Understanding rate problems like this can help in everyday scenarios such as planning trips or calculating travel times. For instance, if you know the distance to a destination and the speed at which you're traveling, you can calculate the time it will take to get there. Similarly, if you know the time and speed, you can determine the distance. This is useful for estimating arrival times, comparing different routes, or optimizing travel plans.
