Learn on PengiAoPS: Introduction to Algebra (AMC 8 & 10)Chapter 7: Proportion

Lesson 4: Rate Problems

In this Grade 4 AoPS Introduction to Algebra lesson, students apply the rate-time-distance relationship (rate × time = distance) and its work equivalent (rate of work × time worked = amount of work done) to solve real-world proportion problems. Learners practice converting units, using inverse proportionality when distance or total work is constant, and accounting for relative motion and moving mediums. The lesson draws on Chapter 7's study of joint proportions to build problem-solving strategies for AMC 8 and AMC 10 style rate problems.

Section 1

Solve Uniform Motion Applications

Property

When planning a road trip, it often helps to know how long it will take to reach the destination or how far to travel each day. We would use the distance, rate, and time formula, $D = rt$ . When the speed of each vehicle is constant, we call applications like this uniform motion problems.

Problem-Solving Strategy:

Read and Understand: Draw a diagram and create a table with columns for Rate, Time, and Distance for each scenario.
Identify Goal: Determine what you need to find.
Name and Assign Variables: Represent unknown quantities with variables and complete the table.
Translate to an Equation: Relate the distances based on the problem (e.g., they are equal, or their sum is a total).
Solve: Use algebra to solve the equation.
Check: Ensure the answer is reasonable.
Answer: State the answer in a complete sentence.

Examples

Two trains travel the same 300-mile route. The express train is 20 mph faster and arrives 1 hour sooner. Set up an equation based on their travel times to find their speeds.
A car and a truck leave the same city, traveling in opposite directions. The car travels at 65 mph and the truck at 55 mph. Their combined distance is $D_{car} + D_{truck}$ . To find when they are 360 miles apart, solve $65t + 55t = 360$ .
A cyclist rides uphill at 8 mph and downhill at 16 mph. If the trip to the top of the hill and back covers the same path, the distance is equal: $8t_{up} = 16t_{down}$ .

Section 2

Uniform Motion: Equal times

Property

When solving uniform motion problems where the time is the same for two different trips, we use the formula $t = \frac{D}{r}$ for each trip and set them equal. Let $t_1$ and $t_2$ be the times for two trips. If $t_1 = t_2$ , then:

\frac{D_1}{r_1} = \frac{D_2}{r_2}

This is common in problems involving a headwind (speed is $r-w$ ) and a tailwind (speed is $r+w$ ), where $r$ is the vehicle's speed and $w$ is the wind or current speed.

Examples

An airplane flies 480 miles with a 40 mph tailwind in the same time it flies 320 miles against it. Let $r$ be the plane's speed. The equation is $\frac{480}{r+40} = \frac{320}{r-40}$ . Solving gives $160r = 32000$ , so $r=200$ mph.

A boat travels 45 miles downstream with a 4 mph current in the same time it travels 27 miles upstream. Let $b$ be the boat's speed. The equation is $\frac{45}{b+4} = \frac{27}{b-4}$ . Solving gives $18b = 288$ , so $b=16$ mph.

Section 3

Uniform Motion: Sum of times

Property

For journeys with multiple parts where the total time is known, the sum of the times for each part equals the total time. Using the formula $t = \frac{D}{r}$ , the equation is:

\frac{D_1}{r_1} + \frac{D_2}{r_2} = T_{\text{total}}

This allows you to find an unknown rate when given the distances of each part and the total duration of the trip.

Examples

Maria trained for 4 hours. She ran 10 miles and biked 30 miles. Her biking speed is 10 mph faster than her running speed, $r$ . The equation is $\frac{10}{r} + \frac{30}{r+10} = 4$ . Solving gives $4r^2 = 100$ , so $r=5$ mph.

A truck drives for 6 hours, covering 120 miles on a highway and 90 miles on a side road. Its highway speed, $r$ , was 25 mph faster than on the side road. The equation is $\frac{120}{r} + \frac{90}{r-25} = 6$ .

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Solve Uniform Motion Applications

Property

Problem-Solving Strategy:

Read and Understand: Draw a diagram and create a table with columns for Rate, Time, and Distance for each scenario.
Identify Goal: Determine what you need to find.
Name and Assign Variables: Represent unknown quantities with variables and complete the table.
Translate to an Equation: Relate the distances based on the problem (e.g., they are equal, or their sum is a total).
Solve: Use algebra to solve the equation.
Check: Ensure the answer is reasonable.
Answer: State the answer in a complete sentence.

Examples

Two trains travel the same 300-mile route. The express train is 20 mph faster and arrives 1 hour sooner. Set up an equation based on their travel times to find their speeds.
A car and a truck leave the same city, traveling in opposite directions. The car travels at 65 mph and the truck at 55 mph. Their combined distance is $D_{car} + D_{truck}$ . To find when they are 360 miles apart, solve $65t + 55t = 360$ .
A cyclist rides uphill at 8 mph and downhill at 16 mph. If the trip to the top of the hill and back covers the same path, the distance is equal: $8t_{up} = 16t_{down}$ .

Section 2

Uniform Motion: Equal times

Property

\frac{D_1}{r_1} = \frac{D_2}{r_2}

This is common in problems involving a headwind (speed is $r-w$ ) and a tailwind (speed is $r+w$ ), where $r$ is the vehicle's speed and $w$ is the wind or current speed.

Examples

An airplane flies 480 miles with a 40 mph tailwind in the same time it flies 320 miles against it. Let $r$ be the plane's speed. The equation is $\frac{480}{r+40} = \frac{320}{r-40}$ . Solving gives $160r = 32000$ , so $r=200$ mph.

A boat travels 45 miles downstream with a 4 mph current in the same time it travels 27 miles upstream. Let $b$ be the boat's speed. The equation is $\frac{45}{b+4} = \frac{27}{b-4}$ . Solving gives $18b = 288$ , so $b=16$ mph.

Section 3

Uniform Motion: Sum of times

Property

For journeys with multiple parts where the total time is known, the sum of the times for each part equals the total time. Using the formula $t = \frac{D}{r}$ , the equation is:

\frac{D_1}{r_1} + \frac{D_2}{r_2} = T_{\text{total}}

This allows you to find an unknown rate when given the distances of each part and the total duration of the trip.

Examples

Maria trained for 4 hours. She ran 10 miles and biked 30 miles. Her biking speed is 10 mph faster than her running speed, $r$ . The equation is $\frac{10}{r} + \frac{30}{r+10} = 4$ . Solving gives $4r^2 = 100$ , so $r=5$ mph.

A truck drives for 6 hours, covering 120 miles on a highway and 90 miles on a side road. Its highway speed, $r$ , was 25 mph faster than on the side road. The equation is $\frac{120}{r} + \frac{90}{r-25} = 6$ .

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Solve Uniform Motion Applications

Property

Problem-Solving Strategy:

Read and Understand: Draw a diagram and create a table with columns for Rate, Time, and Distance for each scenario.
Identify Goal: Determine what you need to find.
Name and Assign Variables: Represent unknown quantities with variables and complete the table.
Translate to an Equation: Relate the distances based on the problem (e.g., they are equal, or their sum is a total).
Solve: Use algebra to solve the equation.
Check: Ensure the answer is reasonable.
Answer: State the answer in a complete sentence.

Examples

Two trains travel the same 300-mile route. The express train is 20 mph faster and arrives 1 hour sooner. Set up an equation based on their travel times to find their speeds.
A car and a truck leave the same city, traveling in opposite directions. The car travels at 65 mph and the truck at 55 mph. Their combined distance is $D_{car} + D_{truck}$ . To find when they are 360 miles apart, solve $65t + 55t = 360$ .
A cyclist rides uphill at 8 mph and downhill at 16 mph. If the trip to the top of the hill and back covers the same path, the distance is equal: $8t_{up} = 16t_{down}$ .

Section 2

Uniform Motion: Equal times

Property

\frac{D_1}{r_1} = \frac{D_2}{r_2}

This is common in problems involving a headwind (speed is $r-w$ ) and a tailwind (speed is $r+w$ ), where $r$ is the vehicle's speed and $w$ is the wind or current speed.

Examples

An airplane flies 480 miles with a 40 mph tailwind in the same time it flies 320 miles against it. Let $r$ be the plane's speed. The equation is $\frac{480}{r+40} = \frac{320}{r-40}$ . Solving gives $160r = 32000$ , so $r=200$ mph.

A boat travels 45 miles downstream with a 4 mph current in the same time it travels 27 miles upstream. Let $b$ be the boat's speed. The equation is $\frac{45}{b+4} = \frac{27}{b-4}$ . Solving gives $18b = 288$ , so $b=16$ mph.

Section 3

Uniform Motion: Sum of times

Property

For journeys with multiple parts where the total time is known, the sum of the times for each part equals the total time. Using the formula $t = \frac{D}{r}$ , the equation is:

\frac{D_1}{r_1} + \frac{D_2}{r_2} = T_{\text{total}}

This allows you to find an unknown rate when given the distances of each part and the total duration of the trip.

Examples

Maria trained for 4 hours. She ran 10 miles and biked 30 miles. Her biking speed is 10 mph faster than her running speed, $r$ . The equation is $\frac{10}{r} + \frac{30}{r+10} = 4$ . Solving gives $4r^2 = 100$ , so $r=5$ mph.

A truck drives for 6 hours, covering 120 miles on a highway and 90 miles on a side road. Its highway speed, $r$ , was 25 mph faster than on the side road. The equation is $\frac{120}{r} + \frac{90}{r-25} = 6$ .