Compound Measures - Speed, Density, Pressure | Mathematics for GCSE/IGCSE - Year 11 PDF Download

Compound Measures

What is a compound measure?

• Compound measures involve calculations that incorporate multiple measurements.
• They are commonly employed to quantify rates or ratios.
  • This concept measures how much one quantity changes when another is increased by 1.
  • Examples of such measurements include:
    • Speed: Speed refers to how much the distance changes for each unit of time.
    • Density: Density signifies how heavy an object is for each unit of its area or volume.
    • Pressure: Pressure indicates the amount of force applied to an object for each unit of its area.
    • Flow Rate: Flow rate represents how much the volume changes for each unit of time.
    • Population Density: Population density reveals how many people there are for each unit of area.

How do I find the units for a compound measure?

• Employ the formula for a compound measure to deduce its units.
• Utilize the units associated with the quantities in the formula to determine the units of the compound measure.
• Represent a division as a/b or  ab^-1 and pronounce it as "a per b."
• Examples include:
  • 
    If the distance is measured in km and the time is measured in mins then the speed is measured in km/min.
  • 
    If the force is measured in N and the area is measured in cm2 then the pressure is measured in N/cm².

How do I find the formula for a compound measure?

• The units associated with a compound measure can aid in recalling its formula.
• Simply remember the purpose of each unit of measurement.
• For a unit expressed as a/b, the corresponding formula will involve the quantity measured by a divided by the quantity measured by b.
• Examples include:
  • kg is a measure of mass and cm³ is a measure of volume
  • Therefore
  • Density can be measured in kg/cm³

Can I use a formula triangle?

• Formula triangles can be used based on personal preference.
• Follow these steps if you choose to use them:
• Steps:
  1. Identify the units provided in the question or information to formulate an initial equation.
  2. Construct a formula triangle, like for Speed, Distance, and Time, as needed.
     
  3. Break down the question into known and unknown components.
  4. Rearrange the formula using the triangle if necessary.
  5. Substitute the known values and compute the solution.

Speed, Density & Pressure

 What are speed, density and pressure? 

• Speed, density, and pressure are combined measurements that are derived from other fundamental measures.
  • Speed is connected to distance and time.
  • Density is linked to mass and volume.
  • Pressure is associated with force and area.
• The relationship between each of these sets of measures follows the same pattern
  • Sometimes referred to as “formula triangles”

• It's important to recall the formula triangle for speed, distance, and time.
• In exams, formulas for other rates will typically be provided, allowing you to construct the corresponding formula triangle.
• If you don't remember the formula triangles, don't fret. You can often deduce them from the information provided in the question or from the units involved.

Problems involving speed, distance & time

• Speed is typically quantified in units such as meters per second (m/s) or miles per hour (mph), with alternatives like kilometers per hour (km/h) also prevalent. 
• These units signify that speed denotes the distance covered per unit of time, as expressed by the formula speed = distance ÷ time. 
• In this context, "speed" generally refers to average speed, especially in more challenging problems featuring multiple journeys or segments within a longer journey.

Problems involving density, mass & volume

• Density is commonly expressed in units such as grams per cubic centimeter (g/cm³) or kilograms per cubic meter (kg/m³). 
• These units signify that density represents the mass contained within a given volume, as denoted by the formula density = mass ÷ volume. 
• In more challenging problems involving alloys, liquids, or gases, it's not uncommon to encounter scenarios where two or more substances have been combined. In such cases, it may be necessary to employ a volume formula to ascertain the volume of an object beforehand.

Problems involving pressure, force & area

• Pressure is commonly quantified in units such as Newtons per square meter (N/m²).
• These units of pressure are frequently referred to as Pascals (Pa) rather than N/m².
• The units of pressure signify that pressure denotes the force exerted over a given area, expressed by the formula pressure = force ÷ area.
• It's important to recognize that weight represents a force; it should not be confused with mass.