Interpret density per area or volume and preserve its units
Divide a quantity by its supporting area or volume · Use density to estimate a total under a stated uniform model · Convert density units consistently
Section Paper 4 · Historical Extended · calculator
Exam scope and assessment notes
Extended content; not a Core requirement. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Metric area and volume conversions
- Ratios and division
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Divide a quantity by its supporting area or volume
- Use density to estimate a total under a stated uniform model
- Convert density units consistently
Read the denominator
An area density is quantity/area, such as seedlings per m² or persons per m². A volume density is quantity/volume, such as mass per cm³ or energy per m³. The denominator is part of the meaning. An average density does not imply each small part has that same density.
Use one consistent unit system
Mass density ρ=m/V gives m=ρV and V=m/ρ for a uniform model. With grams and cm³, density is g/cm³; with kilograms and m³, it is kg/m³. Since 1 g=0.001 kg and 1 cm³=0.000001 m³, 1 g/cm³=1000 kg/m³.
For a planting model of 4 seedlings per m² over 60 m², estimate 240 seedlings. A stated density is a mathematical assumption here, not evidence that every patch has that spacing.
Sources for this lesson
Pause and explain
What does 3 seedlings per m² mean in a uniform planting model?
Worked example
A sample has mass 540 g and volume 200 cm³. Find its average mass density.
Show the worked solution
- Density=mass/volume=540/200.
- Keep the mass and volume units together.
Answer 2.7 g/cm³.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A model places 180 people uniformly across 60 m². Find the area density.
Give me a hint
Divide count by area.
Compare my reasoning
- 180/60=3.
- The denominator is area, so the unit is people per m².
3 people per m².
Look for these in your work
- Use the stated area
- Preserve the density unit
A uniform material sample has mass 420 g and volume 150 cm³. Find its density and the model mass of 400 cm³ of that material.
Give me a hint
Find density before scaling volume.
Compare my reasoning
- Density=420/150=2.8 g/cm³.
- Model mass=2.8×400=1120 g.
2.8 g/cm³; 1120 g.
Look for these in your work
- Use the uniform-material condition
- Check units cancel correctly
A uniform material has density 800 kg/m³. Find the mass of 2000 cm³, and express the density in g/cm³.
Give me a hint
Convert 2000 cm³ to 0.002 m³.
Compare my reasoning
- Mass=800×0.002=1.6 kg.
- 800 kg/m³=0.8 g/cm³, giving the same mass 1600 g.
1.6 kg; density 0.8 g/cm³.
Look for these in your work
- Convert volume cubically
- Cross-check another unit system
Common mistakes
- Per square metre and per cubic metre measure different densities.
- Convert both numerator and denominator when changing density units.
What can you explain now?
What does 3 seedlings per m² mean in a uniform planting model?
Compare with the explanation
An estimated 3 seedlings for each square metre.
Multiply the area in m² by 3 for the model total.
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Make it stick
What does 3 seedlings per m² mean in a uniform planting model?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.