Math Snap

STEP 1

What is this asking?
We need to add two fractions with different denominators!
Watch out!
Don't just add the numerators and denominators directly.
We need a common denominator!

STEP 2

1. Find the Least Common Denominator (LCD)
2. Express each fraction with the LCD
3. Add the fractions

STEP 3

Since $5$ and $11$ are prime numbers and share no common factors other than $1$, their least common multiple, and therefore our least common denominator, is simply their product: $5 \cdot 11 = \mathbf{55}$.

STEP 4

To express $\frac{4}{5}$ with a denominator of $55$, we need to multiply the denominator by $11$.
To keep the fraction's value the same, we must also multiply the numerator by $11$.
This is equivalent to multiplying by $\frac{11}{11}$, which is equal to $1$, and multiplying by $1$ doesn't change the value!
So, we have $\frac{4}{5} \cdot \frac{11}{11} = \frac{4 \cdot 11}{5 \cdot 11} = \frac{\mathbf{44}}{\mathbf{55}}$.

STEP 5

Similarly, for $\frac{9}{11}$, we multiply both the numerator and denominator by $5$: $\frac{9}{11} \cdot \frac{5}{5} = \frac{9 \cdot 5}{11 \cdot 5} = \frac{\mathbf{45}}{\mathbf{55}}$.
We're multiplying by $\frac{5}{5}$ which is just $1$, so the value stays the same!

STEP 6

Now that both fractions have the same denominator, we can add them by adding their numerators and keeping the common denominator: $\frac{44}{55} + \frac{45}{55} = \frac{44 + 45}{55} = \frac{\mathbf{89}}{\mathbf{55}}$.

Our final answer is $\frac{89}{55}$!