Ratio, proportion & rates of change

Direct and Inverse Proportion

Year 10 · Year 11

✓By the end of this lesson students will be able to understand and apply the concept of direct proportion, including direct proportion to a power.
✓By the end of this lesson students will be able to understand and apply the concept of inverse proportion, including inverse proportion to a power.
✓By the end of this lesson students will be able to form and solve equations involving direct and inverse proportion to find unknown values.
✓By the end of this lesson students will be able to recognise and interpret the graphs of direct and inverse proportion.

Key concepts

Direct Proportion

Two quantities are in direct proportion if they increase or decrease at the same rate. This means that if one quantity doubles, the other quantity also doubles. If one quantity halves, the other quantity also halves. The ratio between the two quantities remains constant. We use the symbol '∝' to denote proportionality.

y ∝ x (read as 'y is directly proportional to x') This can be written as an equation: y = kx, where k is the constant of proportionality.

Direct Proportion to a Power

A quantity can be directly proportional to the square, cube, square root, or any power of another quantity. For example, the area of a circle is directly proportional to the square of its radius.

y ∝ x^n (read as 'y is directly proportional to x to the power of n') This can be written as an equation: y = kx^n, where k is the constant of proportionality.

Graphs of Direct Proportion

The graph of direct proportion (y = kx) is a straight line that passes through the origin (0,0). The gradient of this line is the constant of proportionality, k. If y is directly proportional to x^2 (y = kx^2), the graph is a parabola passing through the origin.

Inverse Proportion

Two quantities are in inverse proportion if an increase in one quantity leads to a decrease in the other, and vice versa. This means that if one quantity doubles, the other quantity halves. If one quantity halves, the other quantity doubles. The product of the two quantities remains constant.

y ∝ 1/x (read as 'y is inversely proportional to x') This can be written as an equation: y = k/x, where k is the constant of proportionality.

Inverse Proportion to a Power

A quantity can be inversely proportional to the square, cube, square root, or any power of another quantity. For example, the force of gravity is inversely proportional to the square of the distance between two objects.

y ∝ 1/x^n (read as 'y is inversely proportional to x to the power of n') This can be written as an equation: y = k/x^n, where k is the constant of proportionality.

Graphs of Inverse Proportion

The graph of inverse proportion (y = k/x) is a reciprocal curve (a hyperbola). For positive values of k, the curve lies in the first and third quadrants, approaching the x and y axes but never touching them. If y is inversely proportional to x^2 (y = k/x^2), the graph is also a reciprocal curve, but it approaches the axes more steeply.

Key facts to remember

1Direct proportion: y ∝ x, which means y = kx.
2Inverse proportion: y ∝ 1/x, which means y = k/x.
3The symbol 'k' represents the constant of proportionality.
4To find 'k', substitute a known pair of values into the proportionality equation.
5The graph of direct proportion (y = kx) is a straight line through the origin.
6The graph of inverse proportion (y = k/x) is a reciprocal curve (hyperbola) that does not touch the axes.
7Proportionality can involve powers, such as y ∝ x^2, y ∝ 1/√x, etc.
8Once 'k' is found, the full equation can be used to find any other unknown value.

Worked examples

Example 1

y is directly proportional to x. When x = 5, y = 20. Find the value of y when x = 8.

I1. Write the proportionality statement: y ∝ x

II2. Convert to an equation using the constant of proportionality, k: y = kx

III3. Use the given values (x = 5, y = 20) to find k:

IV 20 = k(5)

V k = 20 / 5

VI k = 4

VII4. Write the full equation with the value of k: y = 4x

VIII5. Use this equation to find y when x = 8:

9 y = 4(8)

10 y = 32

Answer

y = 32

Example 2

P is inversely proportional to the square of V. When P = 100, V = 2. Find the value of P when V = 5.

I1. Write the proportionality statement: P ∝ 1/V^2

II2. Convert to an equation using the constant of proportionality, k: P = k/V^2

III3. Use the given values (P = 100, V = 2) to find k:

IV 100 = k/(2^2)

V 100 = k/4

VI k = 100 × 4

VII k = 400

VIII4. Write the full equation with the value of k: P = 400/V^2

95. Use this equation to find P when V = 5:

10 P = 400/(5^2)

11 P = 400/25

12 P = 16

Answer

P = 16

Example 3

A graph shows a curve in the first quadrant that passes through the points (1, 6), (2, 3), and (3, 2). Describe the relationship between the variables x and y.

I1. Examine the given points: (1, 6), (2, 3), (3, 2).

II2. Observe the trend: As x increases, y decreases. This suggests an inverse relationship.

III3. Test for inverse proportion (y = k/x) by calculating the product x × y for each point:

IV For (1, 6): 1 × 6 = 6

V For (2, 3): 2 × 3 = 6

VI For (3, 2): 3 × 2 = 6

VII4. Since the product x × y is constant (equal to 6), the relationship is inverse proportion, and the constant of proportionality k = 6.

VIII5. Formulate the relationship.

Answer

y is inversely proportional to x (or y = 6/x).

If it were direct proportion, the ratio y/x would be constant. For example, 6/1 = 6, but 3/2 ≠ 6, so it's not direct proportion.

Common mistakes

✗Confusing direct and inverse proportion, leading to incorrect initial equations (e.g., using y = kx for inverse proportion).
✗Forgetting to include the constant of proportionality, 'k', or incorrectly calculating its value.
✗Not correctly applying powers (e.g., squaring x when y is proportional to x^2, or squaring the denominator for inverse square proportion).
✗Assuming 'k' must always be positive; 'k' can be any real number (positive or negative, but usually positive in typical GCSE problems).
✗Incorrectly interpreting graphs, especially distinguishing between y = kx and y = kx^2, or y = k/x and y = k/x^2.

Exam tips

★Always start by writing down the correct proportionality statement (e.g., y ∝ x or y ∝ 1/x^2).
★Immediately convert the proportionality statement into an equation using 'k' (e.g., y = kx or y = k/x^2).
★Use the given pair of values to find the constant 'k' first. This is a crucial step and often carries marks.
★Write down the full equation with the calculated 'k' before attempting to find any other unknown values.
★Be careful with calculations, especially when dealing with fractions, powers, or square roots.

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