Constant Rate Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Constant Rate.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A constant rate of change means the output increases (or decreases) by the same fixed amount for every unit increase in the input — the hallmark of a linear function.

Constant rate means steady, uniform progress — like a car traveling at a fixed speed: every hour, the same number of miles is added to the total.

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How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A linear function f(x) = mx + b has constant rate m — the slope is the same everywhere on the graph, giving a perfectly straight line.

Common stuck point: Constant rate \neq constant value. Rate is the CHANGE per unit.

Sense of Study hint: Compare the change in y between consecutive x-values. If the change is the same every time, the rate is constant.

Worked Examples

Example 1

easy
A car travels at a constant speed of 60 km/h. Write the distance function d(t), find d(2.5), and compute the rate of change from t=1 to t=3.

Solution

  1. 1
    Constant rate 60 km/h means distance = rate \times time: d(t) = 60t.
  2. 2
    Evaluate: d(2.5) = 60 \times 2.5 = 150 km.
  3. 3
    Average rate of change from t=1 to t=3: \frac{d(3)-d(1)}{3-1} = \frac{180-60}{2} = \frac{120}{2} = 60 km/h. (Constant rate means average rate equals instantaneous rate.)

Answer

d(t)=60t; d(2.5)=150 km; rate =60 km/h
A constant rate of change produces a linear function. For any linear function f(x)=mx+b, the average rate of change between any two points always equals the slope m.

Example 2

medium
Find the equation of the line passing through (2, 5) and (6, 13), interpret the slope, and write it in slope-intercept form y = mx + b.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A plumber charges a \50 flat fee plus \75 per hour. Write C(h) and find the cost for 3 hours. At what hour does the cost reach \275$?

Example 2

medium
Determine whether the data is consistent with a constant rate of change: x: 0, 2, 4, 6 and y: 3, 7, 12, 15. If not, find where it fails.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

rate of changelinear functions