How Can We Tell the Difference Between Direct and Inverse Proportions in Math?

Learning about ratios and proportions is really important in Year 10 math, especially when we look at direct and inverse proportions. Even though they sound similar, they mean different things and are used in different ways.

Direct Proportion

Let’s start with direct proportion. When we say that two quantities are in direct proportion, it means that when one quantity goes up, the other one goes up too, at a steady rate. We can write this as:

$y \propto x$

This shows that $y$ is directly proportional to $x$. We can express this with the formula:

$y = kx$

Here, $k$ is a constant number that stays the same.

Example of Direct Proportion:

Think about when you're buying apples. If one kilogram of apples costs £2, we can show the cost ($C$) like this:

$C = 2a$

In this case, $a$ is the number of kilograms you buy. So, if you buy 2 kilograms, the cost would be:

$C = 2 \times 2 = £4$

As you can see, the cost goes up as the amount of apples you buy increases. This is direct proportion.

Inverse Proportion

Now, let’s talk about inverse proportion. When two quantities are inversely proportional, it means that when one goes up, the other goes down, but their product stays the same. We can show this as:

$y \propto \frac{1}{x}$

In simpler terms, it becomes:

$y = \frac{k}{x}$

Again, $k$ is a constant.

Example of Inverse Proportion:

Think about how the number of workers affects the time it takes to finish a job. If you have more workers, the time it takes to complete the job gets shorter. If we let $T$ stand for time and $N$ stand for the number of workers, we can write:

$T = \frac{k}{N}$

For example, if 2 workers finish a project in 12 hours, we can find $k$:

$k = T \cdot N = 12 \times 2 = 24$

Now, if 3 workers are working on the same project, the time taken ($T$) would be:

$T = \frac{24}{3} = 8 \text{ hours}$

Key Differences

To help you remember, here’s a quick summary of the differences between direct and inverse proportions:

• Direct Proportion:

  • When one quantity increases, the other also increases.
  • Formula: $y = kx$.
  • Example: Cost of apples compared to quantity.

• Inverse Proportion:

  • When one quantity increases, the other decreases.
  • Formula: $y = \frac{k}{x}$.
  • Example: Time to complete a job compared to number of workers.

By understanding these definitions and their differences, you’ll be on your way to mastering ratios and proportions in your math studies!