Solving Distance, Rate, and Time Word Problems

Distance, rate, and time problems are a common type of word problem in algebra and physics. These problems involve calculating one of these three quantities when the other two are known. The relationship between these quantities is given by the following formula:

Distance = Rate x Time

This formula can be rearranged to solve for any of the three variables. For example, if you know the distance and time, you can solve for the rate by dividing the distance by the time. Similarly, if you know the rate and time, you can solve for the distance by multiplying the rate by the time.

Setting Up an Equation

When solving distance, rate, and time word problems, it is important to identify the given information and what you are trying to find. Then, you can set up an equation that relates the three quantities. Here is a step-by-step process:

1. Read the problem carefully and identify the given information. What are you trying to find?
2. Define variables to represent the unknown quantities. For example, you might use d for distance, r for rate, and t for time.
3. Write down the formula that relates the three quantities.
4. Substitute the known values into the formula. This will give you an equation with one unknown variable.
5. Solve for the unknown variable. This may involve simplifying the equation, combining like terms, or isolating the variable.

Examples of Distance, Rate, and Time Word Problems

Here are a few examples of distance, rate, and time word problems:

1. Example 1: A car travels 100 miles in 2 hours. What is its average speed?

Solution:

• Given: Distance = 100 miles, Time = 2 hours
• To find: Rate (speed)
• Formula: Rate = Distance / Time
• Substitute: Rate = 100 miles / 2 hours
• Solve: Rate = 50 miles per hour
2. Example 2: A train travels at a speed of 60 miles per hour. How long will it take to travel 240 miles?

Solution:

• Given: Rate = 60 miles per hour, Distance = 240 miles
• To find: Time
• Formula: Time = Distance / Rate
• Substitute: Time = 240 miles / 60 miles per hour
• Solve: Time = 4 hours
3. Example 3: A plane flies at a speed of 500 miles per hour for 3 hours. How far does it travel?

Solution:

• Given: Rate = 500 miles per hour, Time = 3 hours
• To find: Distance
• Formula: Distance = Rate x Time
• Substitute: Distance = 500 miles per hour x 3 hours
• Solve: Distance = 1500 miles

Solving Problems with Rational Equations

Some distance, rate, and time problems may involve more complex scenarios, requiring you to use rational equations. A rational equation is an equation that contains fractions with variables in the denominator. Here's a step-by-step process for solving these problems:

1. Identify the given information and the unknown variable you need to find.
2. Set up a rational equation that relates the distance, rate, and time using the formula Distance = Rate x Time.
3. Clear the fractions by multiplying both sides of the equation by the least common denominator (LCD).
4. Solve the resulting equation for the unknown variable. This may involve simplifying the equation, combining like terms, or isolating the variable.

Example of a Problem Using Rational Equations

A boat travels 10 miles upstream in 2 hours and 10 miles downstream in 1 hour. Find the speed of the boat in still water and the speed of the current.

Solution:

• Let x be the speed of the boat in still water and y be the speed of the current.
• Upstream: Speed = x - y, Time = 2 hours, Distance = 10 miles. Equation: 10 = (x - y) x 2
• Downstream: Speed = x + y, Time = 1 hour, Distance = 10 miles. Equation: 10 = (x + y) x 1
• Simplifying the equations: 5 = x - y and 10 = x + y
• Solving the system of equations: Add the two equations to eliminate y, resulting in 15 = 2x. Therefore, x = 7.5 miles per hour.
• Substitute x = 7.5 into either of the original equations to solve for y. Using the equation 10 = x + y, we get 10 = 7.5 + y. Therefore, y = 2.5 miles per hour.

Therefore, the speed of the boat in still water is 7.5 miles per hour, and the speed of the current is 2.5 miles per hour.

Conclusion

Solving distance, rate, and time word problems involves understanding the relationship between these three quantities and setting up an equation that relates them. By following the steps outlined above, you can solve a variety of distance, rate, and time problems, including those that involve rational equations.