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Identifying
Constant Rate

Learn to determine whether a situation involves a constant rate and justify your reasoning using tables and graphs.

Pre-Algebra

What Is a Rate?

A rate compares two quantities with different units. It tells you how much of one thing changes for each unit of another.

Speed
\( 60 \) miles per \( 1 \) hour
Price
\( \$3 \) per \( 1 \) pound
Growth
\( 2 \) inches per \( 1 \) month

What Makes a Rate Constant?

A rate is constant when the ratio between the two quantities stays the same no matter which data points you compare.

Think of it like a machine: if you always put in the same input change, you always get the exact same output change.

Why Constant Rate Matters

Recognizing a constant rate lets you predict values, write equations, and understand whether a relationship is linear.

Predict
Use the rate to find values you have not measured yet
Model
Write the equation \( y = rx + b \) to describe the pattern
Verify
Check if real-world data truly follows a linear pattern

The Constant-Rate Table Test

For every pair of consecutive rows, compute \( \dfrac{\Delta y}{\Delta x} \). If the result is always the same, the rate is constant.

Time (hours) Distance (miles) 1 50 2 100 3 150 4 200 +50 +50 +50
\( \text{Rate} = 50 \frac{\text{miles}}{\text{hour}} \text{ (constant!)} \)

The Constant-Rate Graph Test

On a graph, a constant rate appears as a perfectly straight line. If the points curve or bend, the rate is not constant.

Time (hours) Distance (mi) 0 1 2 3 4 0 50 100 150 Straight line = constant rate

Constant vs. Changing Rate

Constant Rate Changing Rate
Constant Rate
Straight line, equal \( \Delta y \) for equal \( \Delta x \)
Changing Rate
Curved line, \( \Delta y \) varies even when \( \Delta x \) is the same

How to Identify Constant Rate

1
Find the changes. Compute \( \Delta x \) and \( \Delta y \) between consecutive entries in the table.
2
Divide. Calculate \( \dfrac{\Delta y}{\Delta x} \) for each consecutive pair.
3
Compare. If every ratio is the same, the rate is constant.
4
Graph check. Plot the points. If they form a straight line, that confirms a constant rate.

Example 1: Water Tank

A tank fills at a steady pace. The table shows time and water level.

Minutes 0 5 10 15 Gallons 0 15 30 45
\(\frac{+15}{+5}=3\)
\(\frac{+15}{+5}=3\)
\(\frac{+15}{+5}=3\)
\( \text{Constant rate: } 3 \frac{\text{gallons}}{\text{minute}} \)

Example 2: Not Constant

A car drives through city traffic. Is the rate constant?

Hours 1 2 3 4 Miles 20 50 65 100
\(\frac{+30}{+1}=30\)
\(\frac{+15}{+1}=15\)
\(\frac{+35}{+1}=35\)
\( \text{Not constant: } 30, 15, 35 \text{ differ!} \)

Example 3: Graph Check

A jogger records her distance every 10 minutes. Plot the data and decide.

0 10 20 30 40 0 1 2 3 Minutes Miles Straight line ✓

Interpreting the Unit Rate

Once you confirm a constant rate, express it as a unit rate: the amount of \( y \) per one unit of \( x \).

1
Pick any row pair. Use \( \dfrac{\Delta y}{\Delta x} \) from one consecutive pair.
2
Simplify. Reduce to a fraction with denominator \( 1 \).
3
Attach units. Write the full unit rate: e.g., 3 gallons per minute.

What Non-Constant Looks Like

0 1 2 3 4 0 4 8 12 16 Time Value Curved = not constant +1 +3 +5 +7

Common Mistakes

✕ Checking only one pair
You must check every consecutive pair. One matching ratio does not prove constancy.
✕ Ignoring unequal \( \Delta x \)
If the \( x \)-values are not evenly spaced, you must still divide \( \Delta y \div \Delta x \), not just look at \( \Delta y \) alone.
✕ Eyeballing the graph
A graph that looks "close to straight" may still have a changing rate. Always verify with the table.

Summary Comparison

Constant Rate Non-Constant Rate
Table test \( \dfrac{\Delta y}{\Delta x} \) is always the same \( \dfrac{\Delta y}{\Delta x} \) changes
Graph shape Straight line Curve or broken line
Equation form \( y = mx + b \) (linear) Not linear
Real-world clue "steady," "per," "each" "accelerating," "varies"

Question 1 of 5

A table shows: when \( x \) goes up by 2, \( y \) always goes up by 6. What is the constant rate?

Question 2 of 5

Which graph feature tells you the rate is constant?

Question 3 of 5

A table has these \( (x, y) \) pairs: \( (0, 5) \), \( (2, 11) \), \( (4, 17) \), \( (6, 23) \). Is the rate constant?

Question 4 of 5

A student says "the \( y \)-values go up by 4, 4, 4 so the rate is constant." However the \( x \)-values go up by 1, 2, 1. Is the student correct?

Question 5 of 5

A truck's total distance is recorded at different times: \( 120 \) miles after \( 2 \) hours, \( 180 \) miles after \( 3 \) hours, and \( 240 \) miles after \( 4 \) hours. Does this represent a constant rate?