Two quantities are in inverse proportion when one becomes 2 times, 3 times, and so on, while the other becomes 1/2 times, 1/3 times, and so on. Written as an equation, this is y = a/x (a ≠ 0). Unlike direct proportion, its graph is not a straight line.
Equation of Inverse Proportion
When y is inversely proportional to x,
y = a/x (a is the constant of proportionality. a ≠ 0, x ≠ 0)
This can also be written as xy = a. It represents a relationship in which “the product of the two quantities is constant.”
- Example: For a rectangle with area 12 cm², length x cm and width y cm → xy = 12 → y = 12/x
- Example: When traveling 60 km at x km per hour, the time y hours → y = 60/x
Example ①
y is inversely proportional to x, and y = 3 when x = 4.
(1) Express y in terms of x.
(2) Find y when x = −6.
Solution:
(1) Substitute x=4 and y=3 into y = a/x: 3 = a/4 → a = 12. Therefore, y = 12/x
(2) y = 12/(−6) = −2
y is inversely proportional to x, and y = 3 when x = 4.
(1) Express y in terms of x.
(2) Find y when x = −6.
Solution:
(1) Substitute x=4 and y=3 into y = a/x: 3 = a/4 → a = 12. Therefore, y = 12/x
(2) y = 12/(−6) = −2
Properties of Inverse Proportion
- x y ＝ a (constant): The product of x and y is always the constant of proportionality a.
- When x becomes 2 or 3 times as large, y becomes 1/2 or 1/3 times as large.
- x = 0 cannot be used (the denominator would be 0).
- Relationship between the sign of a and the quadrants: if a＞0, xy＞0 and the graph lies in the first and third quadrants; if a＜0, it lies in the second and fourth quadrants.
Example ②
Choose the equations in which y is inversely proportional to x, and state the constants of proportionality.
A) y = 3x B) y = 6/x C) y = x + 2 D) xy = −10
Solution:
An inverse proportion has the form y = a/x (or xy = a).
B) y = 6/x (constant of proportionality 6)
D) xy = −10 → y = −10/x (constant of proportionality −10)
A) is a direct proportion, and C) is neither.
Choose the equations in which y is inversely proportional to x, and state the constants of proportionality.
A) y = 3x B) y = 6/x C) y = x + 2 D) xy = −10
Solution:
An inverse proportion has the form y = a/x (or xy = a).
B) y = 6/x (constant of proportionality 6)
D) xy = −10 → y = −10/x (constant of proportionality −10)
A) is a direct proportion, and C) is neither.
Key Points
- Equation of inverse proportion: y = a/x (or xy = a). a is the constant of proportionality.
- The product of x and y is always a and remains constant.
- x = 0 cannot be used (division by zero is not allowed).
- When x becomes 2 times as large, y becomes 1/2 as large (the product remains constant).
Practice Problems
- y is inversely proportional to x, and y = −8 when x = 3. Express y in terms of x.
- For y = 24/x, find y when x = −6 and x when y = 4.
- Express the relationship between the length x cm and width y cm of a rectangle with area 36 cm². Also find y when x = 9.
Answers and Explanations
- Answer: y = −24/x
Explanation: Substitute into y=a/x: −8 = a/3 → a = −24. - Answer: When x = −6, y = −4; when y = 4, x = 6
Explanation: y = 24/(−6) = −4. 4 = 24/x → x = 6. - Answer: y = 36/x; when x=9, y = 4
Explanation: xy = 36 → y = 36/x. 9 × y = 36 → y = 4.