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Constant Rate
Problems

Master problems involving constant speed and constant density by using unit rates to connect distance, time, mass, and volume.

Pre-Algebra

What Is a Rate?

A rate is a ratio that compares two quantities measured in different units. Whenever you say "miles per hour" or "grams per cubic centimeter," you are describing a rate.

Speed
Distance per unit of time: \(\frac{\text{miles}}{\text{hour}}\)
Density
Mass per unit of volume: \(\frac{\text{grams}}{\text{cm}^3}\)

What Makes a Rate Constant?

A rate is constant when it stays the same no matter how much of each quantity you consider. A car travelling at a constant speed of 60 mph covers the same distance every hour: no speeding up, no slowing down.

Time (hours)
Distance (miles)
0
60
120
180
240
1
2
3
4
Constant: 60 mph

Why Do Constant Rates Matter?

Constant rates appear everywhere in daily life. When a rate is constant, knowing any two quantities lets you find the third: this is what makes these problems solvable.

Road Trips
If a car holds steady at 55 mph, you can predict arrival time for any distance.
Recipes
Doubling a recipe works because ingredient ratios stay constant.
Science
Density lets scientists identify materials: gold is always \(19.3\;\text{g/cm}^3\).

The Speed Formula

For constant speed, the relationship among distance, rate, and time is captured by one formula you can rearrange three ways:

\(d\)
\(r\)
\(\times\)
\(t\)
\(d = r \times t \qquad r = \frac{d}{t} \qquad t = \frac{d}{r}\)

The Density Formula

Density works the same way as speed: it is a constant rate that connects mass and volume.

\(m\)
\(\rho\)
\(\times\)
\(V\)
\(m = \rho \times V \qquad \rho = \frac{m}{V} \qquad V = \frac{m}{\rho}\)

Speed vs. Density

Both are constant rates: they link two quantities through a fixed ratio. The table below shows how they map onto each other.

Constant Speed Constant Density
Rate Speed (r) Density (ρ)
Numerator Distance (d) Mass (m)
Denominator Time (t) Volume (V)
Formula \(d = r \times t\) \(m = \rho \times V\)
Units mph, km/h, m/s \(\text{g/cm}^3\), \(\text{kg/m}^3\)

How to Solve Any Constant Rate Problem

1
Identify the rate and the two quantities. Is it speed (distance, time) or density (mass, volume)?
2
Write the formula. Use \(d = r \times t\) or \(m = \rho \times V\).
3
Plug in the known values and solve for the unknown.
4
Check your units. Make sure numerator and denominator units match and cancel correctly.
5
Verify with estimation. Does the answer make sense in context?

Example 1: Finding Distance

A train travels at a constant speed of 80 mph. How far does it go in 3.5 hours?

Start
End
\(d = ?\)
\(t = 3.5\) h
\(r = 80\) mph
1
Formula: \(d = r \times t\)
2
Substitute: \(d = 80 \times 3.5\)
3
Solve: \(d = \mathbf{280}\) miles

Example 2: Finding Time

A cyclist rides at a constant speed of 12 mph. How long will it take to cover 45 miles?

1
Identify: \(r = 12\) mph, \(d = 45\) miles, \(t = ?\)
2
Rearrange: \(t = \dfrac{d}{r}\)
3
Substitute: \(t = \dfrac{45}{12}\)
4
Solve: \(t = 3.75\) hours = 3 hours 45 minutes

Example 3: Density Problem

A block of aluminum has a density of \(2.7\;\text{g/cm}^3\). If the block's volume is \(150\;\text{cm}^3\), what is its mass?

\(V = 150\;\text{cm}^3\)
\(\rho = 2.7\;\text{g/cm}^3\)
\(m = ?\)
1
Formula: \(m = \rho \times V\)
2
Substitute: \(m = 2.7 \times 150\)
3
Solve: \(m = \mathbf{405}\) grams

Example 4: Two-Part Journey

Mia drives 120 miles at 60 mph, then another 120 miles at 40 mph. What is her total time?

1
Part 1: \(t_1 = \dfrac{120}{60} = 2\) hours
2
Part 2: \(t_2 = \dfrac{120}{40} = 3\) hours
3
Total: \(t = t_1 + t_2 = 2 + 3 = \mathbf{5}\) hours

Watch Your Units

When units in the problem don't match the rate, convert first before plugging in. A mismatch is the most common source of wrong answers.

Wrong
Speed is in mph but time is in minutes: you plug in 30 instead of 0.5
Right
Convert 30 min to \(\frac{30}{60} = 0.5\) hours first, then multiply by the speed in mph

Common Mistakes

✕ Averaging Speeds
For a two-part trip, the average speed is not the mean of the two speeds. You must calculate total distance ÷ total time.
✕ Mismatched Units
Mixing minutes with hours or cm³ with liters without converting leads to answers that are off by large factors.
✕ Dividing Backwards
Students sometimes compute \(r \div d\) instead of \(d \div r\) for time. Always check: does the formula give reasonable units?

Summary Cheat Sheet

Find... Speed Problem Density Problem
The rate \(r = \dfrac{d}{t}\) \(\rho = \dfrac{m}{V}\)
The "total" quantity \(d = r \times t\) \(m = \rho \times V\)
The "per" quantity \(t = \dfrac{d}{r}\) \(V = \dfrac{m}{\rho}\)
Key check Units must cancel to give the target unit Units must cancel to give the target unit

Question 1 of 5

A car travels at a constant speed of 65 mph. How far does it travel in 4 hours?

Question 2 of 5

A runner covers 10 km in 50 minutes. What is her speed in km per hour?

Question 3 of 5

Iron has a density of \(7.87\;\text{g/cm}^3\). What is the mass of a block with volume \(20\;\text{cm}^3\)?

Question 4 of 5

A bus drives 180 miles at 60 mph, then 80 miles at 40 mph. What is the total travel time?

Question 5 of 5

A sample has a mass of 540 g and a volume of \(200\;\text{cm}^3\). What is its density?