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

슬라이드

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연습 문제

무료 미리 보기 — 56개 중 8개 문제. 가입하면 전부 볼 수 있어요.

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

더 많은 문제

9.y is directly proportional to (x - 1)². When x = 5, y = 4. Find y when x = 7.
Medium
- A36
- B16
- C9
- D24
10.y is inversely proportional to the square root of (x+1). When x = 8, y = 2. Find y when x = 99.
Medium
- A0.6
- B0.5
- C0.4
- D0.8
11.p is directly proportional to (q+2)². When q = 1, p = 1. Find p when q = 10.
Medium
- A16
- B9
- C25
- D36
12.y is inversely proportional to (x+3)². When x = 2, y = 8. Find y when x = 7.
Medium
- A2
- B4
- C1
- D8
13.y is directly proportional to the cube root of (x+3). When x = 5, y = 2/3. Find y when x = 24.
Hard
- A1
- B4/3
- C2/3
- D8/3
14.y ∝ 1/x. When y = 8, x = 4. Find y when x = 49.
Medium
- A32/49
- B49/32
- C2/7
- D7/2
15.y is inversely proportional to the cube of x. If y = 4 when x = 2, find y when x = 1/2.
Hard
- A256
- B32
- C64
- D128
16.y is directly proportional to (x - 4). When x = 16, y = 3. Find y in terms of x.
Medium
- Ay = (1/4)(x-4)
- By = 3(x-4)
- Cy = 4(x-4)
- Dy = (x-4)/3
17.y is directly proportional to x. When x = 5, y = 20. Find y when x = 8.
Easy
- A32
- B25
- C28
- D40
18.y is inversely proportional to x. When x = 2, y = 9. Find y when x = 3.
Easy
- A6
- B13.5
- C4.5
- D6.75
19.y is directly proportional to the square of x. When x = 2, y = 12. Find y when x = 5.
Medium
- A75
- B30
- C60
- D150
20.y is inversely proportional to the square of x. When x = 3, y = 4. Find y when x = 6.
Medium
- A1
- B8
- C2
- D0.5
21.y is directly proportional to the square root of x. When x = 16, y = 8. Find y when x = 25.
Medium
- A10
- B12.5
- C12
- D6.4
22.y is inversely proportional to the cube of x. When x = 2, y = 4. Find y when x = 1/2.
Hard
- A256
- B64
- C16
- D0.5
23.y is directly proportional to (x - 1)². When x = 3, y = 24. Find y when x = 6.
Medium
- A150
- B96
- C54
- D75
24.y is inversely proportional to the square root of x. When y = 7, x = 2.25. Write y in terms of x.
Medium
- Ay = 10.5/√x
- By = 15.75/√x
- Cy = 7/√x
- Dy = 10.5/x
25.t is inversely proportional to the square of (x+1). When x = 2, t = 5. Write t in terms of x.
Medium
- At = 45/(x+1)²
- Bt = 5/(x+1)²
- Ct = 25/(x+1)²
- Dt = 15/(x+1)²
26.m is inversely proportional to the square of (p-1). When p = 4, m = 5. Find m when p = 6.
Medium
- A1.25
- B3.2
- C2
- D0.8
27.y is inversely proportional to the square of (x+2). When x = 3, y = 2. Find y when x = 8.
Medium
- A0.5
- B1
- C0.25
- D4
28.The distance d metres a car travels is directly proportional to the square of the time t seconds. When t = 2, d = 20. Find d when t = 5.
Medium
- A125
- B50
- C100
- D250
29.The current I amperes in a circuit is inversely proportional to the resistance R ohms. When R = 5, I = 2. Find I when R = 8.
Medium
- A1.25
- B3.2
- C0.8
- D1.6
30.Which of the following equations represents direct proportion between y and x?
Easy
- Ay = kx
- By = k/x
- Cy = kx²
- Dy = k√x
31.If y is inversely proportional to x, which of the following is true?
Easy
- Axy = k
- Bx/y = k
- Cy = kx
- Dx = ky
32.The symbol ∝ means
Easy
- Aproportional to
- Bequal to
- Capproximately equal to
- Dgreater than
33.In the equation y = kx, k is called the
Easy
- Aconstant of proportionality
- Bvariable
- Ccoefficient
- Dexponent
34.If y is directly proportional to x, the graph of y against x is
Easy
- Aa straight line through the origin
- Ba curve
- Ca straight line not through the origin
- Da horizontal line
35.y is directly proportional to the square of x. When x = 2, y = 20. Find y when x = 5.
Medium
- A125
- B50
- C100
- D200
36.y is directly proportional to the square of x. When x = 3, y = 36. Find y when x = 5.
Medium
- A100
- B60
- C30
- D45
37.y is inversely proportional to x. When x = 5, y = 4. Find y when x = 10.
Easy
- A2
- B8
- C20
- D0.5
38.y is directly proportional to the cube of x. When x = 2, y = 16. Find y when x = 3.
Medium
- A54
- B24
- C72
- D48
39.y is inversely proportional to the square root of x. When x = 4, y = 6. Find y when x = 9.
Medium
- A4
- B9
- C8
- D12
40.y is directly proportional to (x + 2). When x = 4, y = 18. Find y when x = 7.
Medium
- A27
- B21
- C24
- D30
41.Which symbol is used to show that one quantity is directly proportional to another?
Easy
- A∝
- B∞
- C≈
- D=
42.If y is directly proportional to x, which of the following equations represents the relationship?
Easy
- Ay = k/x
- By = kx
- Cy = kx²
- Dy = k + x
43.In the equation y = kx, what is the name of k?
Easy
- Aconstant of proportionality
- Bcoefficient
- Cvariable
- Dgradient
44.If y is directly proportional to x, the graph of y against x is a straight line through the origin.
Easy
True or false?
45.If y is directly proportional to the cube of x, which equation is correct?
Easy
- Ay = kx³
- By = k/x³
- Cy = kx
- Dy = kx²
46.Match each equation to the correct type of proportion.
Medium
- y = kx
- y = k/x
- y = kx²
- y = k/x²
- Direct proportion
- Inverse proportion
- Direct proportion to square
- Inverse proportion to square
47.Arrange the steps for solving a direct proportion problem in the correct order.
Easy
- Identify variables and write formula y = kx
- Find k using given values
- Rewrite formula with k
- Use formula to find unknown
48.Which of the following statements about inverse proportion are true? (Select all that apply)
Medium
- AAs one variable increases, the other decreases.
- BThe graph is a straight line through the origin.
- CThe product of the two variables is constant.
- Dy ∝ 1/x means y = k/x.
49.If y is directly proportional to x, then the graph of y against x is a straight line through the origin.
Easy
True or false?
50.y is inversely proportional to the square of (x+1). When x = 2, y = 9. Find y when x = 5.
Medium
- A4
- B1
- C9
- D2.25
51.Given that y is directly proportional to x and y = 12 when x = 4, find y when x = 7.
Easy
52.Complete the sentence.
Easy
If y is inversely proportional to x, then y = k / ____
53.y is directly proportional to the square of x. When x = 3, y = 27. Find y when x = 4.
Medium
54.Write an equation for 'y is inversely proportional to the cube root of x'.
Medium
55.Explain how to determine whether two quantities are directly proportional from a table of values.
Medium
56.y is directly proportional to the square of x. When x = 3, y = 27. Find y when x = 4.
Medium
- A58
- B53
- C43
- D48

복습 노트

## 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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