A step-by-step way can help Year 8 students feel more confident when working with proportions. Let's break it down!

Understanding Proportions

First, let's understand what proportions are. A proportion says that two ratios are equal. For example, if we write $\frac{a}{b} = \frac{c}{d}$, we are comparing two ratios: $a:b$ and $c:d$.

Step 1: Identify the Proportions

In a problem, look for the ratios you are given. For example, in the proportion $\frac{3}{4} = \frac{x}{20}$, the numbers $3$ and $4$ are the first group, and $x$ and $20$ are the second group.

Step 2: Cross-Multiplication

Next, we will use cross-multiplication to get rid of the fractions. This means you multiply the top number of one side by the bottom number of the other side:

$3 \cdot 20 = 4 \cdot x$

This gives us:

$60 = 4x$

Step 3: Solve for the Unknown

Now, we need to solve for $x$. To do this, you divide both sides by $4$:

$x = \frac{60}{4} = 15$

Step 4: Check Your Work

It’s important to check your answer. Put $x = 15$ back into the original ratios to see if both sides match:

$\frac{3}{4} = \frac{15}{20}$

Since both fractions reduce to $\frac{3}{4}$, we know our answer is correct!

Example Problem

Let's try another example:

Solve $\frac{5}{x} = \frac{10}{12}$.

1. Identify the Proportions: Here, the ratios are $5:x$ and $10:12$.
2. Cross-Multiply:

$5 \cdot 12 = 10 \cdot x$ $60 = 10x$

3. Solve for $x$:

$x = \frac{60}{10} = 6$

4. Check Your Work:

$\frac{5}{6} \text{ vs } \frac{10}{12} \Rightarrow \frac{5}{6} = \frac{5}{6}$

Conclusion

By following these steps—identify, cross-multiply, solve, and check—students will have a good way to solve proportions easily. Happy calculating!