Adding and subtracting algebraic fractions can be tough for Year 11 students, especially when the fractions have different denominators. A lot of learners have a hard time finding a common denominator. This step is really important because it helps us do the math correctly.

What Are Denominators?

First, let’s understand what denominators are.

If we take two fractions, like $\frac{a}{b}$ and $\frac{c}{d}$, the denominators are $b$ and $d$.

Finding the least common denominator (LCD) can be tricky. The LCD is the smallest number that both denominators can divide into evenly. This can get hard, especially if the numbers are big or complicated.

How to Find the Least Common Denominator

Here’s how you can find the LCD, step by step:

1. Break down each denominator into prime factors.
2. Look for the highest powers of these factors.
3. Multiply these together to find the LCD.

For example, if you want to add $\frac{3x}{4}$ and $\frac{5}{6x}$, the denominators are $4$ and $6x$.

Breaking them down:

• The prime factors of $4$ are $2^2$.
• The prime factors of $6x$ are $2 \times 3 \times x$.

So, the LCD for these fractions is $12x$.

Adjusting the Fractions

After you find the LCD, the next step is to adjust each fraction so they both have the same denominator.

This means you will need to multiply the top (numerator) and the bottom (denominator) of each fraction by the number that will change it to the LCD.

For example:

• To change $\frac{3x}{4}$ to have a denominator of $12x$, you would multiply both the top and the bottom by $3x$: $\frac{3x \cdot 3x}{4 \cdot 3x} = \frac{9x^2}{12x}$
• To change $\frac{5}{6x}$, you multiply both the top and bottom by $2$: $\frac{5 \cdot 2}{6x \cdot 2} = \frac{10}{12x}$

Adding or Subtracting the Fractions

Now that both fractions have the same denominator, you can add or subtract their tops (numerators).

Continuing with our example: $\frac{9x^2}{12x} + \frac{10}{12x} = \frac{9x^2 + 10}{12x}$

Simplifying the Result

Finally, remember that you often need to simplify the fraction. This means you might have to factor the top and cancel out any common factors with the bottom.

Simplifying can be tricky, which is why many students find this part frustrating. But with some practice and a clear plan, you can get better at it!