To solve proportion problems using cross-multiplication, it’s important to understand some basic ideas about ratios and proportions, especially in Year 8 Mathematics.

Proportions are equations that show two ratios are equal. For example, when you see $\frac{a}{b} = \frac{c}{d}$, it means that the products of $a \times d$ and $b \times c$ will be the same if the ratios are equal.

Here's how to use cross-multiplication step by step:

1. Identify the Proportion: Look at the two ratios in your equation. For example, with $\frac{3}{4} = \frac{x}{16}$, the first ratio is $\frac{3}{4}$ and the second is $\frac{x}{16}$.

2. Cross-Multiply: Multiply the top number of one fraction by the bottom number of the other fraction. In the example, you calculate $3 \times 16$ and $4 \times x$:

   $$3 \times 16 = 48$$ $$4 \times x = 4x$$

3. Set Up the Equation: Now create an equation from your cross-multiplication results:

   $$48 = 4x$$

4. Solve for the Variable: To find $x$, divide both sides of the equation by 4:

   $$x = \frac{48}{4}$$ $$x = 12$$

5. Interpret the Result: Here, $x$ equals 12, which fits perfectly into the original proportion.

Let’s go through a couple of examples to clarify this:

Example 1: Imagine a recipe that needs a ratio of 2 cups of flour to 5 cups of sugar. If you want to know how much sugar you need for 8 cups of flour, you can write it as:

$$\frac{2}{5} = \frac{8}{x}$$

Using cross-multiplication, we get:

$$2x = 5 \times 8$$ $$2x = 40$$

Now, divide by 2:

$$x = 20$$

So, you need 20 cups of sugar for 8 cups of flour.

Example 2: In a class, if the ratio of boys to girls is 3 to 2 and there are 18 boys, how many girls are there? We can set it up like this:

$$\frac{3}{2} = \frac{18}{y}$$

Cross-multiplying gives:

$$3y = 2 \times 18$$ $$3y = 36$$

Now, solve for $y$:

$$y = 12$$

So, there are 12 girls in the classroom.

Summary of Steps

• Identify the two ratios.
• Cross-multiply to create a simple equation.
• Set up the equation from the cross-multiplication.
• Solve for the variable, putting it on one side.
• Interpret the solution in the context of the problem.

Using cross-multiplication makes calculations easier and helps you see how different amounts relate to each other. The more you practice with different problems, the better you will get at understanding proportions, which will help you become stronger in math.

In the end, learning to use cross-multiplication will be very helpful. It will not only help you solve proportion problems but also improve your overall problem-solving skills in math. Just remember to see the equal nature of the proportions, follow the steps for cross-multiplication, and develop a clear method for solving ratio questions you might face!