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Solving Distance, Rate, and Time Word Problems
Distance, rate, and time problems are a common type of word problem in algebra. These problems involve objects moving at a constant speed, and we need to figure out how far they travel, how long it takes them to travel a certain distance, or how fast they are traveling.
The key to solving these problems is to understand the relationship between distance, rate, and time. The formula that connects these three is:
Distance = Rate × Time
This formula can be rearranged to solve for any of the three variables:
- Rate = Distance / Time
- Time = Distance / Rate
Setting Up the Equation
When solving distance, rate, and time word problems, it’s helpful to set up a table to organize the information. The table should have columns for distance, rate, and time, and rows for each object or situation involved in the problem.
For example, consider the following problem:
Two cars leave from the same point at the same time. Car A travels east at 60 miles per hour, and Car B travels west at 70 miles per hour. How long will it take for the cars to be 260 miles apart?
Here’s how we can set up a table to organize the information:
| Distance | Rate | Time | |
|---|---|---|---|
| Car A | d | 60 | t |
| Car B | 260 – d | 70 | t |
We know the rate of each car and the total distance they need to be apart. We are trying to find the time it takes for them to reach that distance. We can use the formula Distance = Rate × Time to create equations for each car:
- Car A: d = 60t
- Car B: 260 – d = 70t
We now have a system of two equations with two unknowns. We can solve for d and t using substitution or elimination.
Solving the Equation
In this case, we’ll use substitution. From the first equation, we know that d = 60t. We can substitute this value of d into the second equation:
260 – 60t = 70t
Simplifying the equation:
260 = 130t
Solving for t:
t = 260 / 130 = 2 hours
Therefore, it will take 2 hours for the cars to be 260 miles apart.
Key Points to Remember
- Make sure you understand the relationship between distance, rate, and time.
- Set up a table to organize the information in the problem.
- Use the formula Distance = Rate × Time to create equations.
- Solve the system of equations using substitution or elimination.
By following these steps, you can solve any distance, rate, and time word problem.