Compound Measures

\( \text{Speed} = \frac{\text{Distance}}{\text{Time}},\quad \text{Density} = \frac{\text{Mass}}{\text{Volume}},\quad \text{Pressure} = \frac{\text{Force}}{\text{Area}} \)
Measures GCSE
Question 4 of 20
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\( A bus travels at 50 km/h for 2 hours. Distance=\,? \)

Hint (H)
\( Distance=Speed×Time. \)

Explanation

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Statement

Compound measures link three quantities together. They are widely used in physics and everyday life. The three most common compound measures are:

\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}}, \quad \text{Density} = \frac{\text{Mass}}{\text{Volume}}, \quad \text{Pressure} = \frac{\text{Force}}{\text{Area}} \]

Why it’s true

  • Speed: Measures how far something travels in a given time, so speed = distance ÷ time.
  • Density: Measures how much matter is packed into a space, so density = mass ÷ volume.
  • Pressure: Measures how much force is spread across a surface, so pressure = force ÷ area.

Recipe (how to use it)

  1. Identify which two quantities are given and which one you need to find.
  2. Rearrange the formula if needed (e.g. Distance = Speed × Time, Mass = Density × Volume).
  3. Substitute the values into the formula.
  4. Check units carefully (e.g. m/s, g/cm³, N/m²).

Spotting it

These formulas appear whenever questions involve “per unit” ideas: kilometres per hour, kilograms per cubic metre, newtons per square metre.

Common pairings

  • Speed-time-distance triangle.
  • Density-mass-volume triangle.
  • Pressure-force-area triangle.

Mini examples

  1. A car travels 150 km in 3 hours. Speed = 150 ÷ 3 = 50 km/h.
  2. A block has mass 200 g and volume 50 cm³. Density = 200 ÷ 50 = 4 g/cm³.
  3. A force of 200 N is applied over area 25 m². Pressure = 200 ÷ 25 = 8 N/m².

Pitfalls

  • Wrong units: Always check consistency (e.g. convert minutes to seconds).
  • Mixing formulas: Don’t confuse speed with acceleration, or pressure with stress in physics.
  • Wrong rearrangement: Be careful when solving for different variables.

Exam strategy

  • Memorise the three formulas and the triangle versions.
  • Always write down the rearranged formula before substituting values.
  • Check the final units match what is expected.

Summary

Speed, density, and pressure are examples of compound measures that connect three physical quantities. Rearranging the formulas allows you to solve for the missing variable, and careful use of units is essential. These appear regularly in GCSE maths and science contexts.

Worked examples

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  1. A cyclist travels 90 km in 3 hours. Find speed.
    1. \( Speed=Distance/Time \)
    2. \( 90/3=30 km/h \)
    Answer: 30 km/h
  2. A car travels at 60 km/h for 2.5 hours. Find distance.
    1. \( Distance=Speed×Time \)
    2. \( 60×2.5=150 km \)
    Answer: 150 km
  3. A train travels 240 km at 80 km/h. Find time taken.
    1. \( Time=Distance/Speed \)
    2. \( 240/80=3 h \)
    Answer: 3 hours
  4. \( Mass=600 g, Volume=200 cm³. Find density. \)
    1. \( Density=Mass/Volume \)
    2. \( 600/200=3 g/cm³ \)
    Answer: 3 g/cm³
  5. A cube has volume 100 cm³ and mass 250 g. Find density.
    1. \( Density=250/100=2.5 g/cm³ \)
    Answer: 2.5 g/cm³
  6. \( Force=300 N, Area=50 m². Find pressure. \)
    1. \( Pressure=Force/Area \)
    2. \( 300/50=6 N/m² \)
    Answer: 6 N/m²
  7. A runner covers 21 km in 105 minutes. Find average speed in km/h.
    1. \( 105 min=1.75 h \)
    2. \( Speed=21/1.75=12 km/h \)
    Answer: 12 km/h
  8. A metal has mass 7.8 g and volume 1 cm³. Find density.
    1. \( Density=7.8/1=7.8 g/cm³ \)
    Answer: 7.8 g/cm³
  9. \( Force=120 N applied on area=0.4 m². Find pressure. \)
    1. \( Pressure=120/0.4=300 N/m² \)
    Answer: 300 N/m²
  10. A car travels 150 miles in 2.5 hours. Find average speed in mph.
    1. \( Speed=150/2.5=60 mph \)
    Answer: 60 mph