Direct and inverse proportion

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Apuntes de la lección

Direct Proportion

Two quantities are in direct proportion if their ratio is constant.
If y is directly proportional to x, we write y ∝ x.
The equation is y = kx, where k is the constant of proportionality (k > 0).
k can be found using k = y / x for any non-zero x.
When one quantity increases, the other increases in the same ratio; when one decreases, the other decreases.
The graph of y = kx is a straight line through the origin (0, 0).
Example: distance travelled at constant speed is directly proportional to time, with speed as k.

Solving Direct Proportion Problems

Use the unitary method: find the value for one unit, then multiply.
Step 1: Identify the two quantities and confirm direct proportion.
Step 2: Find k by dividing one quantity by the other.
Step 3: Use y = kx to find the unknown value.
Example: if 3 pens cost £1.50, one pen costs £0.50, so 7 pens cost 7 × 0.50 = £3.50.

Exchange Rates and Currency Conversion

Currency conversion is a direct proportion problem.
The exchange rate is the constant of proportionality k.
To convert from pounds to another currency, multiply by the rate.
To convert back, divide by the rate.
Example: if £1 = $1.25, then £40 = 40 × 1.25 = $50; $75 = 75 ÷ 1.25 = £60.

Inverse Proportion

Two quantities are in inverse proportion if their product is constant.
If y is inversely proportional to x, we write y ∝ 1/x.
The equation is y = k / x, which rearranges to xy = k.
k is the constant of proportionality (k ≠ 0).
As one quantity increases, the other decreases in the same ratio.
Example: for a fixed journey, speed and time are inversely proportional (s × t = d).

Solving Inverse Proportion Problems

Step 1: Confirm inverse proportion (product constant).
Step 2: Find k by multiplying a known pair: k = x × y.
Step 3: Use y = k / x to find the unknown.
Example: if 4 workers take 6 hours, k = 24 worker-hours; 3 workers take 24 ÷ 3 = 8 hours.
More workers mean less time, so the answer should be smaller than the original time.

Graphs of Proportion

Direct proportion: straight line through the origin, gradient k.
Inverse proportion: a curve called a rectangular hyperbola.
The graph of inverse proportion never crosses either axis.
For inverse proportion, the product of x and y is the same at every point on the curve.

Direct vs Inverse Proportion

Direct: variables change together; ratio y/x is constant.
Inverse: one increases as the other decreases; product xy is constant.
Direct equation: y = kx.
Inverse equation: y = k / x.
Direct graph: straight line through origin.
Inverse graph: hyperbolic curve.

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Preguntas de práctica

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1.Which statement correctly describes direct proportion between two variables x and y?
Easy
- Ay = kx for some positive constant k
- By = k / x for some non-zero constant k
- Cx + y = k for some constant k
- Dxy = k for some non-zero constant k
2.Which statement correctly describes inverse proportion between two variables x and y?
Easy
- Ay = kx for some positive constant k
- By = k / x for some non-zero constant k
- Cx + y = k for some constant k
- Dy = x + k for some constant k
3.The graph of two variables that are directly proportional is a straight line passing through the origin.
Easy
True or false?
4.The circumference C of a circle is directly proportional to its diameter d. What is the constant of proportionality?
Medium
- Aπ
- B2π
- C1/π
- Dπ/2
5.If y is inversely proportional to x and y = 8 when x = 5, what is the value of y when x = 10?
Medium
- A4
- B16
- C2
- D40
6.Which of the following situations describe inverse proportion? (select all that apply)
Medium
- AThe time taken to complete a job is inversely proportional to the number of workers.
- BThe distance travelled at a constant speed is directly proportional to the time.
- CThe time taken for a fixed journey is inversely proportional to the speed.
- DThe area of a square is directly proportional to the square of its side length.
- EThe pressure of a fixed mass of gas at constant temperature is inversely proportional to its volume.
7.Match each proportionality statement to its equation form.
Medium
- y is directly proportional to x
- y is inversely proportional to x
- y is directly proportional to the square of x
- y = kx
- y = k / x
- y = kx²
8.Order these steps to solve a direct proportion problem using the unitary method.
Medium
- Find the value of the quantity for one unit.
- Write down the known proportional relationship.
- Multiply the unit value by the required number of units.
- State the final answer with units.

Más preguntas

9.Which of the following relationships are examples of direct proportion? (select all that apply)
Medium
- AThe distance travelled by a car at constant speed and the time taken
- BThe time taken to complete a job and the number of workers (for a fixed amount of work)
- CThe circumference of a circle and its diameter
- DThe force acting on an object and its acceleration (for a fixed mass)
- EThe speed of a car and the time taken for a fixed journey
10.Which of the following relationships are examples of inverse proportion? (select all that apply)
Medium
- AThe number of workers and the time taken to build a wall (for a fixed amount of work)
- BThe speed of a car and the time taken for a fixed journey
- CThe distance travelled by a car at constant speed and the time taken
- DThe number of days a fixed supply of food lasts and the number of people sharing it
- EThe circumference of a circle and its diameter
11.Match each proportionality statement to the correct equation form. (k is a non-zero constant.)
Medium
- y is directly proportional to x
- y is inversely proportional to x
- y is directly proportional to the square of x
- y is inversely proportional to the square root of x
- y = kx
- y = k/x
- y = kx²
- y = k/√(x)
12.Order these steps to solve an inverse proportion problem using the unitary method.
Medium
- Write down the known pair of values and their product.
- Calculate the constant k by multiplying the known pair.
- Use k to find the missing value by dividing k by the new value.
- Check that the answer makes sense (e.g., more workers means less time).
13.The exchange rate is £1 = $1.25. How many dollars do you get for £80?
Medium
- A$100
- B$64
- C$80
- D$125
14.If y is inversely proportional to x and y = 6 when x = 2, what is the value of y when x = 3?
Medium
- A4
- B9
- C1
- D12
15.The graph of two variables that are inversely proportional is a straight line through the origin.
Easy
True or false?
16.If y is directly proportional to x and y = 24 when x = 6, what is the value of y when x = 9?
Easy
- A36
- B30
- C32
- D27
17.If y is inversely proportional to x and y = 10 when x = 4, what is the value of y when x = 8?
Medium
- A5
- B20
- C2.5
- D8
18.A car travels at a constant speed. The distance travelled is directly proportional to the time taken. If the car travels 180 km in 3 hours, how far does it travel in 5 hours?
Medium
- A300 km
- B270 km
- C360 km
- D240 km
19.A rectangle has a fixed area of 36 cm². Which statement about its length l and width w is true?
Medium
- Al and w are inversely proportional because lw = 36.
- Bl and w are directly proportional because l/w is constant.
- Cl and w are directly proportional because l + w is constant.
- Dl and w are neither directly nor inversely proportional.
20.The graph of y against x for two variables in direct proportion is a straight line. Which point must this line always pass through?
Medium
- A(0, 0)
- B(1, 1)
- C(0, 1)
- D(1, 0)
21.Which of the following pairs of variables are inversely proportional? (select all that apply)
Medium
- AThe speed of a car and the time taken for a fixed journey.
- BThe number of workers and the time taken to complete a fixed job (assuming equal rates).
- CThe side length of a square and its perimeter.
- DThe number of people sharing a prize equally and the amount each receives.
- EThe radius of a circle and its circumference.
22.In direct proportion, as one variable increases, the other variable also increases.
Easy
True or false?
23.Match each proportionality statement to its equation form, where k is a non-zero constant.
Medium
- y is directly proportional to x
- y is inversely proportional to x
- y is directly proportional to the square of x
- y = kx
- y = k / x
- y = kx²
24.A map has a scale of 1 : 25000. Two towns are 8 cm apart on the map. What is the actual distance between them, in kilometres?
Medium
- A0.2 km
- B2 km
- C20 km
- D200 km
25.A car travels 150 km in 2 hours. At the same constant speed, how long will it take to travel 375 km?
Hard
- A4 hours
- B5 hours
- C6 hours
- D7.5 hours
26.The force F newtons acting on an object is directly proportional to its acceleration a m/s². When a = 4, F = 20. Find the force when a = 7.
Medium
- A28 N
- B35 N
- C40 N
- D56 N
27.The cost C pounds of hiring a hall is directly proportional to the number of hours h it is used. If C = 45 when h = 3, find C when h = 5.
Medium
- A£60
- B£75
- C£90
- D£135
28.The time t hours taken to build a wall is inversely proportional to the number of workers n. If 4 workers take 12 hours, how long will 6 workers take?
Hard
- A6 hours
- B8 hours
- C9 hours
- D18 hours
29.The pressure P of a fixed mass of gas is inversely proportional to its volume V. When P = 120 kPa, V = 3 m³. Find P when V = 4 m³.
Medium
- A80 kPa
- B90 kPa
- C120 kPa
- D160 kPa
30.The volume V cm³ of a metal cube is directly proportional to the cube of its side length s cm. When s = 2, V = 8. Find V when s = 3.
Hard
- A12 cm³
- B18 cm³
- C27 cm³
- D64 cm³
31.The force F newtons exerted by a spring is directly proportional to its extension x cm. If F = 12 when x = 4, find the extension when F = 21.
Medium
- A5 cm
- B7 cm
- C9 cm
- D12 cm
32.Which of the following are examples of inverse proportion? (select all that apply)
Medium
- AThe number of workers and the time taken to complete a job, for a fixed amount of work
- BThe speed of a car and the time taken for a fixed journey
- CThe number of identical pipes filling a tank and the time taken to fill it
- DThe side length of a square and its perimeter
- EThe number of people sharing a prize equally and the amount each receives
33.Which of the following statements are true for quantities in direct proportion? (select all that apply)
Medium
- ATheir ratio is constant.
- BTheir product is constant.
- CTheir graph is a straight line through the origin.
- DAs one increases, the other increases.
- EAs one increases, the other decreases.
34.If two quantities are in inverse proportion, their product is constant.
Easy
True or false?
35.The graph of two quantities in inverse proportion is a straight line through the origin.
Medium
True or false?
36.Match each real-life relationship to the type of proportion it shows.
Medium
- The cost of petrol and the number of litres bought
- The time taken to travel a fixed distance and the speed
- The number of workers and the time to build a wall
- The distance travelled at constant speed and the time taken
- Direct proportion
- Inverse proportion
37.Order the steps to solve this problem: 'If 5 pens cost £7.50, how much do 8 pens cost?'
Medium
- Divide £7.50 by 5 to find the cost of one pen: £1.50
- Multiply £1.50 by 8 to find the cost of 8 pens
- State the final answer: £12.00
38.What is the name of the symbol ∝ in a proportionality statement such as y ∝ x?
Easy
- Ais proportional to
- Bis equal to
- Cis greater than
- Dis approximately equal to
39.Which formula correctly states that y is directly proportional to x with constant of proportionality k?
Easy
- Ay = kx
- By = k/x
- Cy = x/k
- Dxy = k
40.
The table shows corresponding values of two variables.
x 2 4 6
y 6 12 18
Are x and y in direct proportion, and if so, what is the constant of proportionality?
Medium
- AYes, k = 3
- BYes, k = 1/3
- CNo, they are not in direct proportion
- DYes, k = 6
41.If y is directly proportional to x and y = 12 when x = 3, find the value of y when x = 7.
Easy
42.Complete the sentence about direct proportion notation.
Easy
The symbol used to mean 'is proportional to' is ____ .
43.In the equation y = kx, what name is given to the constant k?
Medium
44.If y is directly proportional to x and y = 15 when x = 5, find the value of y when x = 8.
Medium
45.Complete the sentence about inverse proportion.
Easy
If y is inversely proportional to x, then the product xy is equal to a constant called the ____ of proportionality.
46.What name is given to the constant k in the equation y = k/x for inverse proportion?
Medium
47.How many euros does the student receive for £200?
Medium
48.The student then changes the 230 euros back to pounds at the rate £1 = €1.20. How many pounds does the student get? Give your answer to the nearest penny.
Hard
49.How much money has the student lost in the two exchanges? Give your answer to the nearest penny.
Hard
50.Complete the sentence about the constant of proportionality.
Easy
For y directly proportional to x, the constant of proportionality k is found by calculating y ____ x.
51.If y is directly proportional to x and y = 42 when x = 7, find y when x = 11.
Medium
52.If y is inversely proportional to x and y = 9 when x = 4, find y when x = 6.
Medium
53.Explain the difference between direct proportion and inverse proportion, and give one example of each.
Hard
54.A car travels 240 km in 3 hours. At the same constant speed, how far will it travel in 5 hours? Show your working.
Hard
55.It takes 6 builders 15 days to build a wall. How many days will it take 10 builders working at the same rate? Show your working.
Hard
56.If y is directly proportional to x and y = 12 when x = 3, find the value of y when x = 7.
Easy
- A34
- B28
- C31
- D25
57.If y is directly proportional to x and y = 15 when x = 5, find the value of y when x = 8.
Medium
- A24
- B28
- C26
- D22
58.How many euros does the student receive for £200?
Medium
- A207 euros
- B230 euros
- C253 euros
- D276 euros
59.The student then changes the 230 euros back to pounds at the rate £1 = €1.20. How many pounds does the student get? Give your answer to the nearest penny.
Hard
- A210.84 pounds
- B172.5 pounds
- C230.01 pounds
- D191.67 pounds
60.How much money has the student lost in the two exchanges? Give your answer to the nearest penny.
Hard
- A7.33 pounds
- B8.33 pounds
- C10.33 pounds
- D9.33 pounds
61.If y is directly proportional to x and y = 42 when x = 7, find y when x = 11.
Medium
- A73
- B80
- C66
- D59
62.If y is inversely proportional to x and y = 9 when x = 4, find y when x = 6.
Medium
- A5
- B7
- C8
- D6

Apuntes de repaso

## Direct and Inverse Proportion ### Direct proportion Two quantities are **directly proportional** if their ratio is constant. If y is directly proportional to x, then $y = kx$ where k is the **constant of proportionality** (k > 0). Also written y ∝ x. For x ≠ 0, $k = \frac{y}{x}$. **Graph:** a straight line through the origin. **Behaviour:** as one increases, the other increases at the same rate. **Examples** - Distance travelled at constant speed is directly proportional to time; speed is k. - Circumference of a circle is directly proportional to diameter; k = π. - Map distance is directly proportional to real distance; k is the map scale. ### Inverse proportion Two quantities are **inversely proportional** if their product is constant. If y is inversely proportional to x, then $y = \frac{k}{x}$ or equivalently $xy = k$ where k is a non-zero constant. Also written y ∝ 1/x. **Graph:** a rectangular hyperbola; it never crosses either axis. **Behaviour:** as one increases, the other decreases. **Example** For a fixed distance d, speed s and time t satisfy s × t = d, so time is inversely proportional to speed. ### Unitary method Find the value of **one** unit first, then multiply to get the required amount. Example: if 5 pens cost £2, then 1 pen costs £2 ÷ 5 = £0.40, so 8 pens cost 8 × £0.40 = £3.20. ### Exchange rates Currency conversion uses direct proportion. If £1 = $1.25, then £20 = 20 × 1.25 = $25. ### Key points to remember - Direct: $y = kx$, graph is a straight line through (0,0). - Inverse: $y = \frac{k}{x}$, graph is a curve that never touches the axes. - Direct: both quantities change in the same direction. - Inverse: one increases while the other decreases. - Always identify k from the given information.

Direct proportion: Two quantities are directly proportional if their ratio is constant. If y is directly proportional to x, then y = kx for some positive constant k.
Inverse proportion: Two quantities are inversely proportional if their product is constant. If y is inversely proportional to x, then y = k/x or xy = k for some non-zero constant k.
Constant of proportionality: The constant k in y = kx (direct) or y = k/x (inverse). For direct proportion, k = y/x; for inverse proportion, k = xy.
Unitary method: A method where you first find the value of one unit, then multiply to find the required value.
Exchange rate: The rate at which one currency can be converted into another; used in direct proportion problems.
Proportion: An equation stating that two ratios are equal, e.g. a/b = x/y = k.

Tarjetas

What does it mean for two variables to be directly proportional?

Their corresponding values have a constant ratio. If y is directly proportional to x, then y = kx for some positive constant k.

What is the symbol for 'is proportional to'?

The symbol is ∝. For example, y ∝ x means y is directly proportional to x.

In y = kx, what is k called?

k is the constant of proportionality (also called the constant of variation). It equals the ratio y/x.

How do you find the constant of proportionality k when y is directly proportional to x?

Divide y by x: k = y/x. Use any corresponding pair of values.

What is the graph of direct proportion?

A straight line through the origin. Its gradient is the constant of proportionality k.

What does it mean for two variables to be inversely proportional?

Their product is constant. If y is inversely proportional to x, then y = k/x, or equivalently xy = k, for some non-zero constant k.

In inverse proportion y = k/x, what is k?

k is the constant of proportionality, equal to the product xy.

How do you find k when y is inversely proportional to x?

Multiply the corresponding values: k = xy. Use any pair of values.

What is the graph of inverse proportion?

A rectangular hyperbola. It never crosses either axis because neither x nor y can be zero.

How do direct and inverse proportion differ as variables change?

In direct proportion, the variables increase or decrease together. In inverse proportion, an increase in one variable causes a decrease in the other.

If an object travels at constant speed, how are distance and time related?

Distance is directly proportional to time. The constant of proportionality is the speed.

For a fixed distance, how are speed and time related?

Speed and time are inversely proportional: s × t = d, where d is the constant distance.

The circumference of a circle is directly proportional to its diameter. What is the constant of proportionality?

The constant is π. So C = πd.

What is the unitary method?

A method where you first find the value of one unit (by dividing), then multiply to find the required value.

How do you use the unitary method to solve a direct proportion problem?

Find the value for one unit by dividing, then multiply by the number of units needed.

How do you use the unitary method to solve an inverse proportion problem?

Find the total amount (the constant product), then divide by the new value to find the other variable.

If 5 pens cost £2, how do you find the cost of 8 pens using the unitary method?

First find the cost of 1 pen: £2 ÷ 5 = £0.40. Then multiply by 8: £0.40 × 8 = £3.20.

If 4 workers take 6 hours to do a job, how long would 3 workers take?

The total work is 4 × 6 = 24 worker-hours. With 3 workers, time = 24 ÷ 3 = 8 hours.

How do you convert between currencies using a given exchange rate?

Multiply the amount by the exchange rate to convert to the other currency. For example, if £1 = $1.30, then £20 = 20 × 1.30 = $26.

If £1 = €1.15, how many euros do you get for £50?

50 × 1.15 = €57.50.

If $1 = £0.80, how many pounds do you get for $200?

200 × 0.80 = £160.

What is the relationship between direct proportion and linear equations?

Direct proportion y = kx is a linear equation with y-intercept 0 and gradient k > 0.

What does the notation y ∝ 1/x mean?

y is inversely proportional to x. This means y = k/x for some non-zero constant k.

If y is directly proportional to x and y = 12 when x = 3, find y when x = 5.

First find k: k = 12/3 = 4. Then y = 4 × 5 = 20.

If y is inversely proportional to x and y = 6 when x = 2, find y when x = 4.

First find k: k = 6 × 2 = 12. Then y = 12/4 = 3.

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