Direct and Inverse Proportion
Proportions are an essential concept in GCSE Maths, forming the foundation of many real-world applications. Understanding direct and inverse proportion is crucial when dealing with problems involving scaling, relationships, and changes in variables. This guide will explain these concepts, including how to calculate amounts, lengths, areas, and volumes using proportions, as well as dealing with exponential growth and decay.
What is Direct Proportion?
Two quantities are said to be in direct proportion if, as one quantity increases, the other increases in the same ratio, or as one decreases, the other decreases in the same ratio. This means that the two quantities always stay in the same proportion to each other.
Mathematically, if $y$ is directly proportional to $x$, we can write this as:
$$y \propto x$$
or equivalently:
$$y = kx$$
where $k$ is the constant of proportionality.
Example:
If 5 apples cost £2, the cost ($y$) is directly proportional to the number of apples ($x$). The constant of proportionality $k$ is $\frac{2}{5} = 0.4$, so the relationship is:
$$\text{Cost} = 0.4 \times \text{Number of Apples}$$
What is Inverse Proportion?
Two quantities are said to be in inverse proportion if, as one quantity increases, the other decreases in such a way that the product of the two quantities remains constant.
Mathematically, if $y$ is inversely proportional to $x$, we can write this as:
$$y \propto \frac{1}{x}$$
or equivalently:
$$y = \frac{k}{x}$$
where $k$ is the constant of proportionality.
Example:
If the time taken ($y$) to travel a fixed distance is inversely proportional to the speed ($x$), the relationship can be expressed as:
$$\text{Time} = \frac{k}{\text{Speed}}$$
If at a speed of 60 mph the time is 2 hours, then $k = 60 \times 2 = 120$. The equation for this relationship is:
$$\text{Time} = \frac{120}{\text{Speed}}$$
Calculating Amounts Using Proportions
Proportions are often used to calculate unknown amounts in various situations, such as scaling up or down based on known quantities.
Example 1: Direct Proportion
If 3 litres of paint cover an area of 12 square metres, how many square metres will 5 litres of paint cover?
Since the coverage is directly proportional to the amount of paint, we set up a proportion:
$$\frac{3 \text{ litres}}{12 \text{ m}^2} = \frac{5 \text{ litres}}{x \text{ m}^2}$$
Solving for $x$:
$$x = \frac{5 \times 12}{3} = 20 \text{ m}^2$$
So, 5 litres of paint will cover 20 square metres.
Example 2: Inverse Proportion
If a machine takes 6 hours to produce 200 units of a product, how long will it take to produce 500 units, assuming the production rate is inversely proportional to the time?
We set up the inverse proportion:
$$\frac{6 \text{ hours}}{200 \text{ units}} = \frac{x \text{ hours}}{500 \text{ units}}$$
Solving for $x$:
$$x = \frac{6 \times 500}{200} = 15 \text{ hours}$$
So, it will take 15 hours to produce 500 units.
Calculating Lengths, Areas, and Volumes Using Proportions
Proportions are frequently used to calculate changes in lengths, areas, and volumes when one of the dimensions is scaled.
Length:
If a figure is enlarged or reduced, the lengths of corresponding sides will change in direct proportion. For example, if the length of a rectangle is doubled, then all corresponding lengths of similar shapes will also double.
Area:
The area of similar figures changes in direct proportion to the square of the linear dimension. If the length of a rectangle is multiplied by a factor of $n$, then the area is multiplied by a factor of $n^2$.
Volume:
The volume of similar figures changes in direct proportion to the cube of the linear dimension. If the length of a cube is multiplied by a factor of $n$, then the volume is multiplied by $n^3$.
Exponential Growth and Decay
Exponential growth and exponential decay refer to situations where quantities increase or decrease at a rate proportional to their current value.
Exponential Growth:
Exponential growth occurs when a quantity increases by a fixed percentage over a regular time period. This type of growth is common in population growth, bacteria growth, and compound interest.
The general formula for exponential growth is:
$$y = a(1 + r)^t$$
Where:
- $y$ is the final amount
- $a$ is the initial amount
- $r$ is the rate of growth (expressed as a decimal)
- $t$ is the time
Example:
If a population of bacteria doubles every 3 hours, and there are initially 100 bacteria, the population after 6 hours will be:
$$y = 100(1 + 1)^2 = 100 \times 4 = 400$$
So, after 6 hours, there will be 400 bacteria.
Exponential Decay:
Exponential decay occurs when a quantity decreases by a fixed percentage over a regular time period. This type of decay is common in radioactive decay, depreciation of assets, and cooling of objects.
The general formula for exponential decay is:
$$y = a(1 - r)^t$$
Where:
- $y$ is the final amount
- $a$ is the initial amount
- $r$ is the rate of decay (expressed as a decimal)
- $t$ is the time
Example:
If the half-life of a substance is 5 years and there are initially 200 grams of it, after 10 years, the amount remaining will be:
$$y = 200(1 - 0.5)^{\frac{10}{5}} = 200 \times 0.5^2 = 200 \times 0.25 = 50 \text{ grams}$$
So, after 10 years, 50 grams of the substance will remain.
Summary
- Direct Proportion: One quantity increases or decreases in the same ratio as another quantity. The formula is $y=kx$.
- Inverse Proportion: One quantity increases while the other decreases in such a way that their product remains constant. The formula is $y = \frac{k}{x}$.
- Exponential Growth and Decay: Quantities increase or decrease at a rate proportional to their current value. Use the formulas $y = a(1 + r)^t$ for growth and $y = a(1 - r)^t$ for decay.
- Scaling in Geometry: Lengths, areas, and volumes of similar shapes change in direct proportion to $n$, $n^2$, and $n^3$, respectively.
Understanding these concepts will help you apply proportion in a wide variety of problems, from scaling geometric shapes to calculating exponential changes over time.
