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ScotlandApplications of MathematicsSyllabus dot point

How do you investigate a situation involving gradient, calculating the steepness of a slope or ramp as a ratio and relating it to vertical and horizontal distance?

Investigating a situation involving gradient, calculating gradient as vertical height divided by horizontal distance, and using it to find an unknown height or distance, including ramps and slopes.

A focused answer to the SQA National 5 Applications of Mathematics geometry content on gradient, covering calculating gradient as vertical height divided by horizontal distance, interpreting it as the steepness of a slope or ramp, and using it to find an unknown height or horizontal distance.

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  1. What this dot point is asking
  2. What gradient measures
  3. Finding an unknown height or distance
  4. Comparing with a guideline
  5. Examples in context
  6. Try this

What this dot point is asking

The SQA wants you to investigate a situation involving gradient: calculate gradient as the vertical height divided by the horizontal distance, interpret it as the steepness of a slope or ramp, and use it to find an unknown height or horizontal distance.

What gradient measures

Gradient describes how steep a slope, ramp or hill is. It compares how far something rises vertically with how far it travels horizontally. Because it is a comparison of two lengths, the gradient itself has no units; it is just a number (or a fraction or ratio) that captures the steepness, so two slopes can be compared directly by their gradients regardless of their actual sizes.

A gradient can be written as a fraction, a decimal or a ratio, and the SQA often asks you to compare it with a recommended value. Gradients are sometimes given as ratios such as 1 in 121 \text{ in } 12 on road signs, which means the same as the fraction 112\tfrac{1}{12}: a rise of 11 for every 1212 travelled horizontally. Converting between the ratio, fraction and decimal forms lets you compare a measured slope with a stated limit.

Finding an unknown height or distance

Because gradient links three quantities, knowing any two lets you find the third by rearranging.

Comparing with a guideline

Gradient questions in Applications of Mathematics are usually about a real decision: is a ramp gentle enough, is a road too steep, does a path meet a standard? You calculate the gradient, compare it with the guideline, and justify whether it passes. When comparing, put both gradients in the same form first; a decimal is often easiest, so a guideline of 112≈0.083\tfrac{1}{12} \approx 0.083 can be compared directly with a measured gradient such as 0.090.09, which is steeper and so fails.

Examples in context

Gradient appears in accessibility (wheelchair ramps), construction (roof pitch and road slopes), and the outdoors (the steepness of a hill on a map). Each rests on dividing vertical height by horizontal distance, or rearranging to find a missing length, then judging the result against a standard, the skills here. A steeper ramp may save space but be harder to use, which is the kind of trade-off the SQA asks you to weigh.

Try this

Q1. A slope rises 44 metres over 5050 metres horizontally. Find the gradient. [2 marks]

  • Cue. 450=0.08\dfrac{4}{50} = 0.08.

Q2. A ramp has gradient 0.20.2 and a horizontal distance of 1515 metres. Find the rise. [2 marks]

  • Cue. 0.2×15=30.2 \times 15 = 3 metres.

Q3. A path rises 22 metres at a gradient of 0.050.05. Find the horizontal distance. [2 marks]

  • Cue. 20.05=40\dfrac{2}{0.05} = 40 metres.

Exam-style practice questions

Practice questions written in the style of SQA exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

SQA N5 Apps style3 marksA wheelchair ramp rises 0.60.6 metres over a horizontal distance of 7.27.2 metres. Calculate its gradient. A guideline recommends a gradient no steeper than 112\tfrac{1}{12}. Does this ramp meet the guideline?
Show worked answer →

Gradient is vertical height divided by horizontal distance: 0.67.2\dfrac{0.6}{7.2} (1 mark). Evaluate: 0.67.2=112≈0.083\dfrac{0.6}{7.2} = \dfrac{1}{12} \approx 0.083 (1 mark). Compare with the guideline: the ramp's gradient is exactly 112\tfrac{1}{12}, so it just meets the recommendation of no steeper than 112\tfrac{1}{12} (1 mark). Markers reward the gradient calculation, the value, and a justified comparison. A steeper ramp would have a larger gradient.

SQA N5 Apps style3 marksA path has a gradient of 0.150.15. Over a horizontal distance of 4040 metres, how much does it rise?
Show worked answer →

Rearrange the gradient relationship: gradient == vertical height ÷\div horizontal distance, so vertical height == gradient ×\times horizontal distance (1 mark). Substitute: 0.15×400.15 \times 40 (1 mark). Evaluate: 0.15×40=60.15 \times 40 = 6 metres of rise (1 mark). Markers reward rearranging for the height, the substitution, and the answer with units. Multiplying the gradient by the horizontal distance gives the vertical rise.

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