Lesson 19b · Quick Check

Slope, Rate, or Percent?

One calculation can lead to different descriptions. Check each step separately.
Three-step habit: calculate $m=\frac{\Delta y}{\Delta x}$, read the axes to name the rate and its units, then use $m\times100\%$ only when the question asks for an equal-unit physical gradient.
  1. Problem 1 · Distance and time
    A horse travels from $10$ km to $30$ km between hour $2$ and hour $6$.
    1. Calculate the numerical slope.
    2. State the rate of change with units.
    3. Should this be converted to a percentage? Explain briefly.
    calculate → read axes → apply the percent gate →
  2. Problem 2 · Trail gradient
    A trail rises $6$ m over a horizontal run of $15$ m.
    1. Calculate the slope.
    2. Write the percentage gradient.
    3. Interpret the percentage in words.
    rise/run → decimal → percentage → meaning →
  3. Problem 3 · Battery graph
    A phone battery drops from $96\%$ to $72\%$ over $3$ hours.
    1. Calculate the slope, including its sign.
    2. State the rate precisely, using “percentage points”.
    3. Explain why multiplying the answer by $100\%$ would be wrong.
    change in displayed percentage ÷ time → precise units → explain →
  4. Problem 4 · Repair the rule
    A student says, “A slope of $4$ always means a rate of $400\%$.”
    1. Rewrite the statement so it is true.
    2. Give one example where slope $4$ means $400\%$ and one where it does not.
    state when percent applies → contrast two contexts →
Show answers
Problem 1a
$m=\frac{30-10}{6-2}=\frac{20}{4}=5$.
Problem 1b
The rate of change is $5$ km/h.
Problem 1c
No. Kilometres and hours are different kinds of units, so $5$ km/h does not become $500\%$.
Problem 2a
$m=\frac{6}{15}=0.4$.
Problem 2b
Percentage gradient $=0.4\times100\%=40\%$.
Problem 2c
The trail rises $40$ m for every $100$ m travelled horizontally.
Problem 3a
$m=\frac{72-96}{3-0}=\frac{-24}{3}=-8$.
Problem 3b
The battery decreases by $8$ percentage points per hour.
Problem 3c
The $y$-axis already measures percentage points. The slope is a change in those points per hour, not a physical rise/run gradient.
Problem 4a
A slope of $4$ means a $400\%$ gradient only when rise and run are compatible lengths and the gradient is requested as a percentage.
Problem 4b
A rise of $4$ m over $1$ m across is a $400\%$ gradient. An income increase of €4 per hour is €4/hour, not $400\%$.
Ready for Lesson 20Keep answering in three layers: slope number, rate with units, percentage only when appropriate.
← All lessons
25:00

Ask your tutor