Direct and inverse proportion

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课程笔记

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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练习题

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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.

复习笔记

## 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.

闪卡

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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