Distance, rate, and time problems are a staple in algebra and have practical applications in everyday life. The fundamental relationship that governs these problems is:
Distance = Rate × Time
Or, more concisely:
[latex]d = r cdot t[/latex]
Where:
- d represents distance
- r represents rate (speed or velocity)
- t represents time
This guide will delve into various scenarios and provide a structured approach to solving them, with a special focus on problems where A Car Traveled At An Average Speed Of 80 (km/h or mph, depending on the context).
Understanding the Basics
Before tackling complex problems, let’s reinforce the basic concept. If a car traveled at an average speed of 80 km/h for 2 hours, the total distance covered would be:
Distance = 80 km/h × 2 h = 160 km
Utilizing Tables for Organization
More intricate problems often involve multiple variables or scenarios. A table is invaluable for organizing the information. Here’s a basic table structure:
| Object/Person | Rate (r) | Time (t) | Distance (d) |
|---|
Remember, the Distance column is always calculated by multiplying the Rate and Time columns.
Distance rate time chart example
Example Problems and Solutions
Let’s explore some common problem types:
1. Meeting in Opposite Directions
Joey and Natasha start at the same location and walk in opposite directions. Joey walks 2 km/h faster than Natasha. After 3 hours, they are 30 kilometers apart. How fast did each walk?
| Person | Rate (r) | Time (t) | Distance (d) |
|---|---|---|---|
| Natasha | r | 3 h | 3r |
| Joey | r + 2 | 3 h | 3(r + 2) |
Equation: 3r + 3(r + 2) = 30
Solving for r:
[latex]begin{array}{rrrrrrl} 3r&+&3(r&+&2)&=&30 \ 3r&+&3r&+&6&=&30 \ &&&-&6&&-6 \ hline &&&&dfrac{6r}{6}&=&dfrac{24}{6} \ \ &&&&r&=&4 text{ km/h} end{array}[/latex]
Natasha walks at 4 km/h, and Joey walks at 6 km/h.
2. Round Trip with Varying Rates
Nick and Chloe paddled downstream at 12 km/h and upstream at 4 km/h. The total trip took 1 hour. How long did they paddle downstream?
| Direction | Rate (r) | Time (t) | Distance (d) |
|---|---|---|---|
| Downstream | 12 km/h | t | 12t |
| Upstream | 4 km/h | 1 – t | 4(1 – t) |
Equation: 12t = 4(1 – t)
Solving for t:
[latex]begin{array}{rrlll} 12(t)&=&4(1&-&t) \ 12t&=&4&-&4t \ +4t&&&&+&4t \ hline dfrac{16t}{16}&=&dfrac{4}{16}&& \ \ t&=&0.25&& end{array}[/latex]
They paddled downstream for 0.25 hours.
3. Catch-Up Scenario
Terry leaves home on a bike at 20 km/h. Sally leaves 6 hours later on a scooter at 80 km/h. How long will it take Sally to catch up?
| Person | Rate (r) | Time (t) | Distance (d) |
|---|---|---|---|
| Terry | 20 km/h | t | 20t |
| Sally | 80 km/h | t – 6 | 80(t – 6) |
Equation: 20t = 80(t – 6)
Solving for t:
[latex]begin{array}{rrrrr} 20(t)&=&80(t&-&6) \ 20t&=&80t&-&480 \ -80t&&-80t&& \ hline dfrac{-60t}{-60}&=&dfrac{-480}{-60}&& \ \ t&=&8&& end{array}[/latex]
Sally catches up after Terry travels for 8 hours (meaning Sally travels for 2 hours).
4. Variable Speed Over a Fixed Distance
On a 130 km trip, a car traveled at an average speed of 55 km/h and then reduced its speed to 40 km/h. The trip took 2.5 hours. How long was the car travelling at 40 km/h?
| Segment | Rate (r) | Time (t) | Distance (d) |
|---|---|---|---|
| 55 km/h | 55 km/h | t | 55t |
| 40 km/h | 40 km/h | 2.5 – t | 40(2.5 – t) |
Car speed changes during a trip
Equation: 55t + 40(2.5 – t) = 130
Solving for t:
[latex]begin{array}{rrrrrrr} 55(t)&+&40(2.5&-&t)&=&130 \ 55t&+&100&-&40t&=&130 \ &-&100&&&&-100 \ hline &&&&dfrac{15t}{15}&=&dfrac{30}{15} \ \ &&&&t&=&2 end{array}[/latex]
The car traveled at 40 km/h for 0.5 hours.
Key Takeaways
- Master the formula: Distance = Rate × Time
- Organize with tables: Clearly define rates, times, and distances for each object or segment of the journey.
- Formulate the equation: Based on the problem’s context, create an equation that relates the distances, rates, and times.
- Solve for the unknown: Use algebraic techniques to solve for the desired variable.
- Interpret the result: Ensure your answer makes sense within the problem’s context.
Distance, rate, and time problems require careful reading, organization, and algebraic manipulation. By understanding the fundamental principles and practicing various problem types, you can confidently solve these challenges. Remember to pay close attention to the units (km/h, mph, hours, minutes) to avoid errors.
