Mathematics (4024)
Topic 15 of 18Cambridge O Levels

Proportionality & Variation

Understanding how one quantity changes in a predictable relationship with another.

Proportionality and variation are fundamental mathematical concepts used to describe how quantities relate to each other. When two quantities are linked in such a way that a change in one causes a predictable change in the other, they are said to be in variation. Understanding these relationships is crucial for solving problems in science, engineering, and economics.


### Direct Variation


This is the simplest form of relationship. We say that y is directly proportional to x if an increase in x causes a proportional increase in y, and a decrease in x causes a proportional decrease in y. For example, the more hours you work at a fixed hourly wage, the more money you earn.


  • Notation: The symbol for 'is proportional to' is ∝. So, we write **y ∝ x**.
  • Equation: To turn the proportionality into an equation, we introduce a **constant of proportionality**, denoted by **k**. The equation becomes:

    • y = kx


    Solving Direct Variation Problems:

    The process involves four key steps:

  • Formulate the relationship: Write the statement using the proportionality symbol (e.g., y ∝ x).
  • Write the equation: Introduce the constant k to form an equation (y = kx).
  • Find the constant, k: Use a given pair of values for x and y to substitute into the equation and solve for k.
  • Solve the problem: Use the equation with the calculated value of k to find any unknown values.

    • Example: If *C* is directly proportional to *r*, and *C* = 20 when *r* = 4, find *C* when *r* = 7.

  • Relationship: C ∝ r
  • Equation: C = kr
  • Find k: 20 = k(4) => k = 20/4 = 5. The formula is C = 5r.
  • Solve: C = 5(7) = 35.

    • ### Inverse Variation


    In this relationship, an increase in one quantity causes a proportional decrease in the other. We say that y is inversely proportional to x. For instance, the faster you travel, the less time a journey takes.


  • Notation: We write this as **y ∝ 1/x**.
  • Equation: The corresponding equation is:

    • y = k/x or xy = k


    The four-step solving process remains the same, but using the inverse variation equation.


    Example: If *t* is inversely proportional to *s*, and *t* = 3 when *s* = 60, find *t* when *s* = 90.

  • Relationship: t ∝ 1/s
  • Equation: t = k/s
  • Find k: 3 = k/60 => k = 3 * 60 = 180. The formula is t = 180/s.
  • Solve: t = 180/90 = 2.

    • ### Variation with Powers and Roots


    Variation is not limited to linear relationships. A quantity can be proportional to the square, cube, square root, or any other power of another variable.


  • Direct Variation with Powers:
  • "y is proportional to the square of x" means y ∝ x², leading to the equation y = kx².
  • "P is proportional to the cube root of V" means P ∝ ³√V, leading to the equation P = k³√V.
  • Inverse Variation with Powers:
  • "F is inversely proportional to the square of d" means F ∝ 1/d², leading to the equation F = k/d².

    • The method for solving these problems is identical: formulate the relationship, write the equation with k, find k, and then solve.


    ### Joint Variation


    Joint variation occurs when a variable depends on two or more other variables. It can be a combination of direct and inverse relationships.


  • Formulation: For example, "*z* varies directly as *x* and inversely as the square of *y*."
  • Notation: z ∝ x/y²
  • Equation: **z = kx/y²**

    • To find the constant k, you will be given a complete set of corresponding values for all variables (z, x, and y).


    Example: *A* varies jointly as *b* and the square of *h*. Given that *A* = 60 when *b* = 5 and *h* = 2, find *A* when *b* = 3 and *h* = 4.

  • Relationship: A ∝ bh²
  • Equation: A = kbh²
  • Find k: 60 = k(5)(2²) => 60 = k(20) => k = 3. The formula is A = 3bh².
  • Solve: A = 3(3)(4²) = 3 * 9 * 16 = 432.

    • Key Points to Remember

    • 1**Direct Variation (y ∝ x)** means y = kx. As one variable increases, the other increases proportionally.
    • 2**Inverse Variation (y ∝ 1/x)** means y = k/x. As one variable increases, the other decreases proportionally.
    • 3The **constant of proportionality (k)** must always be calculated first using a given set of values.
    • 4The 4-step process is: State the relationship (∝), form the equation with k, find k, then solve the problem.
    • 5Variation can involve powers and roots, such as y ∝ x² (direct square) or y ∝ 1/√x (inverse square root).
    • 6**Joint Variation** describes a relationship where a variable depends on two or more other variables (e.g., z = kxy/w).
    • 7Always establish the full formula (with the value of k) before finding unknown quantities.
    • 8The symbol '∝' is shorthand for 'is proportional to'.

    Pakistan Example

    Cost of Electricity Generation at a Dam

    In Pakistan, the power output of a hydroelectric dam like Tarbela or Mangla can be modelled using variation. Let's say the electrical power (P, in megawatts) generated is directly proportional to the flow rate of water (Q, in cubic meters per second) and the square of the dam's height (h, in meters). The relationship is P ∝ Qh². If a dam with a height of 140m and a flow rate of 500 m³/s generates 3500 MW, we can find the constant of proportionality, k. The formula is P = kQh². Substituting the values: 3500 = k(500)(140²). Solving this gives the value of k. This formula can then be used by engineers to predict the power output if the water flow rate changes during different seasons, a common scenario in Pakistan's river systems.

    Quick Revision Infographic

    Mathematics — Quick Revision

    Proportionality & Variation

    Key Concepts

    1**Direct Variation (y ∝ x)** means y = kx. As one variable increases, the other increases proportionally.
    2**Inverse Variation (y ∝ 1/x)** means y = k/x. As one variable increases, the other decreases proportionally.
    3The **constant of proportionality (k)** must always be calculated first using a given set of values.
    4The 4-step process is: State the relationship (∝), form the equation with k, find k, then solve the problem.
    5Variation can involve powers and roots, such as y ∝ x² (direct square) or y ∝ 1/√x (inverse square root).
    6**Joint Variation** describes a relationship where a variable depends on two or more other variables (e.g., z = kxy/w).

    Formulas to Know

    Direct Variation (y ∝ x)** means y = kx. As one variable increases, the other increases proportionally.
    Inverse Variation (y ∝ 1/x)** means y = k/x. As one variable increases, the other decreases proportionally.
    Joint Variation** describes a relationship where a variable depends on two or more other variables (e.g., z = kxy/w).
    Pakistan Example

    Cost of Electricity Generation at a Dam

    In Pakistan, the power output of a hydroelectric dam like Tarbela or Mangla can be modelled using variation. Let's say the electrical power (P, in megawatts) generated is directly proportional to the flow rate of water (Q, in cubic meters per second) and the square of the dam's height (h, in meters). The relationship is P ∝ Qh². If a dam with a height of 140m and a flow rate of 500 m³/s generates 3500 MW, we can find the constant of proportionality, k. The formula is P = kQh². Substituting the values: 3500 = k(500)(140²). Solving this gives the value of k. This formula can then be used by engineers to predict the power output if the water flow rate changes during different seasons, a common scenario in Pakistan's river systems.

    SeekhoAsaan.com — Free RevisionProportionality & Variation Infographic

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