When you're working on equations with mixed fractions in GCSE Mathematics, it's really important to understand fractions and how to work with them. Mixed fractions, like $1 \frac{1}{2}$, can look tricky at first. But don't worry! With a few tips, you'll find them much easier to deal with.

Step 1: Change Mixed Fractions to Improper Fractions

The first thing you should do when you see a mixed fraction is to change it into an improper fraction.

For example, let's take $1 \frac{1}{2}$. Here’s how to convert it:

$$1 \frac{1}{2} = \frac{2 \times 1 + 1}{2} = \frac{3}{2}$$

Making this change helps a lot. So, if you have an equation like:

$$x + 1 \frac{1}{2} = 3$$

You can change $1 \frac{1}{2}$ to $\frac{3}{2}$:

$$x + \frac{3}{2} = 3$$

Step 2: Get the Variable Alone

Now that you’ve changed all the mixed fractions, the next step is to get the variable alone. You can do this by removing the fraction. Subtract the fraction from both sides of the equation:

$$x = 3 - \frac{3}{2}$$

To make this subtraction easier, you can turn 3 into a fraction, like this:

$$3 = \frac{6}{2}$$

So now you have:

$$x = \frac{6}{2} - \frac{3}{2} = \frac{6 - 3}{2} = \frac{3}{2}$$

This means $x = \frac{3}{2}$.

Step 3: Getting Rid of Fractions

Sometimes, it might be easier to just get rid of all the fractions. You can do this by multiplying everything in the equation by the least common multiple (LCM) of the denominators.

For example, for the previous equations, the LCM is 2. If you multiply the entire equation by 2, it looks like this:

$$2(x) + 2(1 \frac{1}{2}) = 2(3)$$

This gives you:

$$2x + 3 = 6$$

Now, you can solve for $x$ easily:

1. First, subtract 3 from both sides:

   $$2x = 3$$

2. Then, divide by 2:

   $$x = \frac{3}{2}$$

Example Problem

Let’s practice with an example. Look at the equation:

$$2x + 3 \frac{1}{4} = 5$$

1. First, change $3 \frac{1}{4}$ to an improper fraction, which is $\frac{13}{4}$.

2. Now the equation looks like this:

   $$2x + \frac{13}{4} = 5$$

3. Change 5 into fourths: $5 = \frac{20}{4}$.

4. Rearranging gives you:

   $$2x = \frac{20}{4} - \frac{13}{4} = \frac{7}{4}$$

5. Now divide by 2:

   $$x = \frac{7}{8}$$

Final Thoughts

With practice, solving equations with mixed fractions will get a lot easier. Just remember these key steps: convert mixed fractions to improper fractions, isolate the variable, and if you need to, eliminate the fractions by multiplying by the LCM of the denominators. Keep trying different examples, and soon you’ll feel great about tackling any equation you see!