Proportion

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इन सवालों के जवाब देकर एनर्जी कमाएं, फिर मछली पकड़ें और घूमें। कोई अकाउंट नहीं चाहिए।

टीचर्स के लिए: Proportion (Maths [CIE], Extended) के लिए इस्तेमाल के लिए तैयार लेसन स्लाइड्स, रिवीज़न नोट्स — इन्हें अपने लेसन में इस्तेमाल करें, या टॉपिक को एक इंटरैक्टिव क्लास एक्टिविटी की तरह चलाएं जिसे आपके स्टूडेंट्स लाइव गेम की तरह खेलें।

लेसन नोट्स

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

स्लाइड्स

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साइन अप करें — फ्री लॉग इन

प्रैक्टिस सवाल

फ्री प्रीव्यू — 56 में से 8 सवाल। सभी देखने के लिए साइन अप करें।

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

और सवाल

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

रिवीज़न नोट्स

## 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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