Proportion

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Direct Proportion

Direct proportion means as one variable increases, the other increases by the same factor; the ratio stays constant.
Symbol: ∝, e.g. y ∝ x means y is directly proportional to x.
General form: y = kx, where k is the constant of proportionality.
The graph of y = kx is a straight line through the origin with gradient k.
For powers/roots: y ∝ x² ⇒ y = kx²; y ∝ √x ⇒ y = k√x; y ∝ x³ ⇒ y = kx³; y ∝ ∛x ⇒ y = k∛x.
Steps: (1) Write formula in terms of k, (2) Substitute given values to find k, (3) Rewrite equation with k, (4) Use equation to find unknown.

Direct proportion: y = kx

Inverse Proportion

Inverse proportion means as one variable increases, the other decreases by the same factor.
y is inversely proportional to x means y ∝ 1/x, so y = k/x.
The graph of y = k/x is a hyperbola (curved, never touching axes).
For powers/roots: y ∝ 1/x² ⇒ y = k/x²; y ∝ 1/√x ⇒ y = k/√x; y ∝ 1/x³ ⇒ y = k/x³; y ∝ 1/∛x ⇒ y = k/∛x.
Steps: (1) Write formula in terms of k, (2) Substitute given values to find k, (3) Rewrite equation with k, (4) Use equation to find unknown.

Inverse proportion: y = k/x

Finding the Constant of Proportionality

For direct proportion: substitute given (x, y) into y = kxⁿ (or root) and solve for k.
For inverse proportion: substitute given (x, y) into y = k/xⁿ (or root) and solve for k.
Always rewrite the full equation after finding k before using it for other values.
In harder questions, you must set up the proportion yourself from the wording.

Direct Proportion with Powers and Roots

y ∝ x²: y = kx². Example: if y=18 when x=3, then k=2, so y=2x².
y ∝ √x: y = k√x. Example: if y=6 when x=9, then k=2, so y=2√x.
y ∝ (x−1)²: y = k(x−1)². Example: if y=4 when x=5, then k=1, so y=(x−1)².
y ∝ ∛(x+3): y = k∛(x+3). Example: if y=2/3 when x=5, then k=1/3, so y=(1/3)∛(x+3).

Inverse Proportion with Powers and Roots

y ∝ 1/x²: y = k/x². Example: if y=7.5 when x=4, then k=120, so y=120/x².
y ∝ 1/√x: y = k/√x. Example: if y=7 when x=2.25, then k=10.5, so y=10.5/√x.
y ∝ 1/(x+1)²: y = k/(x+1)². Example: if t=5 when x=2, then k=45, so t=45/(x+1)².
y ∝ 1/(x+3)²: y = k/(x+3)². Example: if y=8 when x=2, then k=200, so y=200/(x+3)².

Solving Proportion Problems Step by Step

Step 1: Identify variables and write proportion statement (e.g., y ∝ x²).
Step 2: Convert to equation using k (e.g., y = kx²).
Step 3: Substitute given pair to find k.
Step 4: Write final equation (e.g., y = 2x²).
Step 5: Use equation to find required value or solve for variable.

Common Exam Question Types

Given a direct or inverse proportion statement and one pair of values, find another value.
Find the formula connecting two variables (e.g., d in terms of t).
Percentage change problems: e.g., if y is increased by 21%, find % increase in x when x ∝ √y.
Inverse proportion with linear expressions: e.g., y ∝ 1/(x+2)².
Finding minimum whole number (e.g., number of people) from inverse proportion.

Graphs of Proportion

Direct proportion y = kx: straight line through origin, gradient k.
Direct proportion y = kx²: parabola through origin.
Direct proportion y = k√x: increasing curve, concave down.
Inverse proportion y = k/x: hyperbola, decreasing, asymptotic to axes.
Inverse proportion y = k/x²: similar hyperbola but steeper decrease.

Shapes of direct and inverse proportion

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練習題

免費預覽——56 題中的 8 題。註冊即可查看全部。

1.Which symbol is used to show that one quantity is proportional to another?
Easy
- A∝
- B≈
- C=
- D≡
2.If y is directly proportional to x, which equation represents this relationship?
Easy
- Ay = kx
- By = k/x
- Cy = x + k
- Dy = x - k
3.If y is inversely proportional to x, which equation represents this relationship?
Easy
- Ay = k/x
- By = kx
- Cy = x + k
- Dy = x - k
4.Given that y ∝ x and y = 12 when x = 4, find y when x = 7.
Medium
- A21
- B28
- C3
- D48
5.y is inversely proportional to x. When x = 9, y = 8. Find y when x = 6.
Medium
- A12
- B10.67
- C6
- D4.5
6.y is directly proportional to the square root of x. When x = 9, y = 6. Find y when x = 25.
Medium
- A10
- B8
- C15
- D12
7.y is inversely proportional to x². When x = 4, y = 7.5. Find y when x = 5.
Medium
- A4.8
- B6
- C9.375
- D3.84
8.A ball falls d metres in t seconds. d is directly proportional to the square of t. The ball falls 44.1 m in 3 seconds. Find d when t = 2.
Medium
- A19.6
- B29.4
- C9.8
- D14.7

複習筆記

## Proportion Revision Notes ### Direct Proportion - **Definition**: As one variable increases, the other increases by the same factor. Ratio stays constant. - **Symbol**: ∝ (e.g., y ∝ x) - **Equation**: y = kx, where k is the constant of proportionality. - **With powers/roots**: - y ∝ x² → y = kx² - y ∝ √x → y = k√x - y ∝ x³ → y = kx³ - y ∝ ∛x → y = k∛x - **Steps**: 1. Write formula: y = k × (expression) 2. Substitute given values to find k. 3. Rewrite equation with k. 4. Use equation to find unknown. ### Inverse Proportion - **Definition**: As one variable increases, the other decreases by the same factor. - **Symbol**: y ∝ 1/x - **Equation**: y = k/x - **With powers/roots**: - y ∝ 1/x² → y = k/x² - y ∝ 1/√x → y = k/√x - y ∝ 1/x³ → y = k/x³ - y ∝ 1/∛x → y = k/∛x - **Steps**: Same as direct proportion but with reciprocal. ### Key Points - Always find k first using given pair. - For expressions like (x+1)², treat as single variable. - In exam, you may need to derive equation without being told. - Graphs: direct proportion gives straight line through origin; inverse gives hyperbola.

Direct proportion: Two variables are directly proportional if one is a constant multiple of the other: y = kx.
Inverse proportion: Two variables are inversely proportional if one is proportional to the reciprocal of the other: y = k/x.
Constant of proportionality (k): The constant value that relates two proportional variables, found by substituting known values.
Proportional symbol (∝): Symbol used to indicate that one variable is proportional to another.

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