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Measurement & Geometry · Topic 9 of 9

Gradient

Video coming soon3 worked examples

Theory

The Golden Rule: the two most common ways to lose marks in gradient questions are failing to ensure the dimensions are in consistent units, and failing to express the final gradient as a fraction in its simplest form. Always check your units before doing any calculation!

1. The Gradient Formula

The gradient of a slope is a measure of its steepness. You must use the formula:

Gradient=Vertical HeightHorizontal Distance\text{Gradient} = \frac{\text{Vertical Height}}{\text{Horizontal Distance}}

This formula is provided on your formula sheet, so you do not need to memorise it, but you must know how to apply it correctly.

2. Calculating the True Vertical Height

In real-life contexts (like hills or roads), you are rarely given the vertical height directly. Instead, you are given the starting height above sea level and the ending height above sea level. You must subtract the starting height from the ending height to find the actual vertical height of the triangle before using the formula.

3. Format of the Final Answer

You must read the bold text in the question carefully to see how the examiner wants the final answer presented:

  • Fractions: If asked for a fraction in its simplest form, you must use your Numeracy skills to divide the top and bottom by the highest common factor. A decimal answer will score zero marks for that step.
  • Percentages: To convert a decimal gradient into a percentage (e.g., a 20% slope), simply multiply your decimal answer by 100.

4. Comparing Gradients

When asked to determine which slope is steeper, calculate the gradient for both. The slope with the larger numerical value is the steeper slope.

⚠️ Common Examiner Traps

  • The "Unit Mismatch" Trap: This is the most prevalent trap in the entire course. A question will give the vertical height in millimetres (e.g., 850 mm) and the horizontal base in centimetres (e.g., 165 cm). If you divide 850 by 165 without converting the units to match first, you will lose the majority of the marks.
  • The "Sloping Edge" Trap: a question will occasionally label the diagonal, slanted length of the slope on the diagram instead of the horizontal base. You must use Pythagoras' Theorem first to calculate the horizontal distance across the bottom before you can calculate the gradient.
  • The "Unsimplified Fraction" Trap: If the question states "Give your answer as a fraction in its simplest form", you will lose the final communication mark if you leave your answer as 85165\frac{85}{165} rather than simplifying it down to 1733\frac{17}{33}.

Worked examples

Example 1

Stephen has built a new ramp. The horizontal distance of the ramp is 165 cm and the vertical height is 850 mm.

Calculate the gradient of the ramp. Give your answer as a fraction in its simplest form. (2 marks)

Step 1: Ensure units match. Convert the height from millimetres into centimetres (divide by 10).

850 mm÷10=85 cm850 \text{ mm} \div 10 = 85 \text{ cm}

Step 2: Substitute into the gradient formula.

Gradient=Vertical HeightHorizontal Distance=85165\text{Gradient} = \frac{\text{Vertical Height}}{\text{Horizontal Distance}} = \frac{85}{165}

Step 3: Simplify the fraction. Both numbers end in 5, so divide the top and bottom by 5.

85÷5=1785 \div 5 = 17
165÷5=33165 \div 5 = 33

Final Answer: The gradient is 1733\frac{17}{33}.

Example 2

Tracy decides to walk to the top of Dumyat Hill from Blairlogie car park.

  • The horizontal distance between these two places is 3 kilometres.
  • Blairlogie car park is 21 metres above sea level.
  • The top of Dumyat Hill is 420 metres above sea level.

Calculate the average gradient between the car park and the top of the hill. Give your answer as a fraction in its simplest form. (3 marks)

Step 1: Calculate the true vertical height by subtracting the elevations.

420−21=399 metres420 - 21 = 399 \text{ metres}

Step 2: Ensure units match. Convert the horizontal distance from kilometres into metres (multiply by 1000).

3×1000=3000 metres3 \times 1000 = 3000 \text{ metres}

Step 3: Substitute into the formula.

Gradient=3993000\text{Gradient} = \frac{399}{3000}

Step 4: Simplify the fraction (divide top and bottom by 3).

Final Answer: The gradient is 1331000\frac{133}{1000}.

Example 3

A design for a skatepark ramp has a vertical height of 70 cm and a horizontal base of 2.1 m.

To be suitable, the ramp must have a gradient of 0.35±0.010.35 \pm 0.01.

Determine whether the ramp is suitable. Use your working to justify your answer. (3 marks)

Step 1: Ensure units match. Convert the base from metres to centimetres.

2.1 m×100=210 cm2.1 \text{ m} \times 100 = 210 \text{ cm}

Step 2: Calculate the gradient as a decimal.

Gradient=70210=0.333...\text{Gradient} = \frac{70}{210} = 0.333...

Step 3: Calculate the tolerance limits.

Maximum: 0.35+0.01=0.36\text{Maximum: } 0.35 + 0.01 = 0.36
Minimum: 0.35−0.01=0.34\text{Minimum: } 0.35 - 0.01 = 0.34

Step 4: Make a concluding statement comparing the calculated gradient to the limits.

Final Answer: The ramp is not suitable, because the gradient (0.33) is less than the minimum allowed limit (0.34).

Ready to practise?

9 Gradient questions with answers and video solutions