Hello, Super Mathematicians!
Welcome to the wonderful world of Fractions! Have you ever tried to share a pizza that was cut into 6 slices with a friend whose pizza was cut into 8 slices? It's tricky to know who ate more! In these notes, we're going to learn how to solve problems just like that.
We'll become experts at adding, subtracting, and comparing fractions, even when their bottom numbers (the denominators) are different. It might sound a bit tricky, but don't worry! We'll go step-by-step, and you'll be a fraction master in no time. Let's get started!
Quick Review: What We Already Know!
Before we jump into new things, let's refresh our memories on some key fraction facts.
What is a Fraction?
A fraction represents a part of a whole. It has two main parts:
Numerator: The top number. It tells us how many parts we have.
Denominator: The bottom number. It tells us how many equal parts the whole is divided into.
Example: In the fraction $$ \frac{3}{4} $$, we have 3 parts out of a total of 4 equal parts.
Types of Fractions
Proper Fractions: The numerator is smaller than the denominator. (Example: $$ \frac{1}{2} $$, $$ \frac{5}{8} $$)
Improper Fractions: The numerator is bigger than or equal to the denominator. (Example: $$ \frac{5}{4} $$, $$ \frac{3}{3} $$)
Mixed Numbers: A whole number and a proper fraction together. (Example: $$ 1\frac{1}{4} $$)
Adding & Subtracting with the SAME Denominator
This is the easy part! When the denominators are the same, you just add or subtract the numerators. The denominator stays the same!
Example: $$ \frac{2}{7} + \frac{3}{7} = \frac{2+3}{7} = \frac{5}{7} $$
The Big Challenge: Fractions with Different Denominators
What happens when we want to add $$ \frac{1}{2} $$ and $$ \frac{1}{4} $$? We can't just add the top numbers. Think about it: a pizza slice from a pizza cut in half is much bigger than a slice from a pizza cut into four pieces. It's not fair to add them directly!
The Golden Rule: To add or subtract fractions, they must have the same denominator.
So, our mission is to make the denominators the same. How do we do that? We give them a "makeover" to create equivalent fractions!
Our Superpower: Finding a Common Denominator
A common denominator is a number that both original denominators can divide into. The best one to use is the Least Common Multiple (LCM). This just means the smallest number that is in both of their times tables.
How to find the LCM (Least Common Multiple)
Let's find the LCM for the fractions $$ \frac{1}{4} $$ and $$ \frac{1}{6} $$.
- List the multiples (times tables) of the first denominator (4): 4, 8, 12, 16, 20...
- List the multiples of the second denominator (6): 6, 12, 18, 24...
- The first number you find in both lists is the LCM! Here, it's 12.
So, 12 is our new, super, common denominator!
Memory Aid: Think of it as finding a "Common Friend" for the two bottom numbers!
Adding Fractions with Different Denominators
Let's solve $$ \frac{1}{4} + \frac{1}{6} $$ step-by-step.
Step 1: Find the LCM
We already did this! The LCM of 4 and 6 is 12. This will be our new denominator.
Step 2: Make Equivalent Fractions
We need to change both fractions so they have a denominator of 12. Remember this important trick: Whatever you do to the bottom, you must do to the top!
For $$ \frac{1}{4} $$: To get from 4 to 12, we multiply by 3. So, we must also multiply the numerator by 3.
$$ \frac{1 \times 3}{4 \times 3} = \frac{3}{12} $$
For $$ \frac{1}{6} $$: To get from 6 to 12, we multiply by 2. So, we must also multiply the numerator by 2.
$$ \frac{1 \times 2}{6 \times 2} = \frac{2}{12} $$
Step 3: Add the New Fractions
Now that the denominators are the same, we can just add the numerators!
$$ \frac{3}{12} + \frac{2}{12} = \frac{3+2}{12} = \frac{5}{12} $$
Our answer is $$ \frac{5}{12} $$. Great job!
Key Takeaway
To add fractions with different denominators:
1. Find LCM -> 2. Make Equivalent Fractions -> 3. Add Numerators
Subtracting Fractions with Different Denominators
Guess what? Subtracting uses the exact same steps! The only difference is that you subtract the numerators in the end. Let's try $$ \frac{2}{3} - \frac{1}{4} $$.
Step 1: Find the LCM
Multiples of 3: 3, 6, 9, 12, 15...
Multiples of 4: 4, 8, 12, 16...
The LCM is 12.
Step 2: Make Equivalent Fractions
For $$ \frac{2}{3} $$: 3 x 4 = 12. So, $$ \frac{2 \times 4}{3 \times 4} = \frac{8}{12} $$
For $$ \frac{1}{4} $$: 4 x 3 = 12. So, $$ \frac{1 \times 3}{4 \times 3} = \frac{3}{12} $$
Step 3: Subtract the New Fractions
$$ \frac{8}{12} - \frac{3}{12} = \frac{8-3}{12} = \frac{5}{12} $$
See? You've got this!
Watch Out! Common Mistakes
A very common mistake is to add or subtract the denominators. Don't do it! The denominator tells you the size of the slice. The slice size doesn't change when you add or subtract.
Example of what NOT to do: $$ \frac{1}{2} + \frac{1}{3} \neq \frac{2}{5} $$
Mixed Operations (Adding and Subtracting Together)
Sometimes a problem has both addition and subtraction. No problem! We just work from left to right after we find a common denominator for ALL the fractions.
Let's solve: $$ \frac{1}{2} + \frac{3}{4} - \frac{5}{8} $$
Step 1: Find the LCM for all three denominators (2, 4, and 8)
Multiples of 2: 2, 4, 6, 8, 10...
Multiples of 4: 4, 8, 12...
Multiples of 8: 8, 16...
The LCM for all three is 8.
Step 2: Make Equivalent Fractions
$$ \frac{1}{2} = \frac{1 \times 4}{2 \times 4} = \frac{4}{8} $$
$$ \frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8} $$
$$ \frac{5}{8} $$ (This one already has the right denominator! Lucky us!)
Step 3: Solve from Left to Right
Our new problem is: $$ \frac{4}{8} + \frac{6}{8} - \frac{5}{8} $$
First, do the addition: $$ \frac{4}{8} + \frac{6}{8} = \frac{10}{8} $$
Now, do the subtraction: $$ \frac{10}{8} - \frac{5}{8} = \frac{5}{8} $$
The final answer is $$ \frac{5}{8} $$.
Key Takeaway
For mixed operations, find a common denominator for all fractions, then solve from left to right, one step at a time.
Comparing Fractions
Which is bigger, $$ \frac{2}{3} $$ or $$ \frac{3}{4} $$? It's hard to tell just by looking. But if we give them a common denominator, it becomes super easy!
- Find the LCM of the denominators (3 and 4). The LCM is 12.
- Make equivalent fractions:
$$ \frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12} $$
$$ \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} $$
- Compare the numerators. Now we compare $$ \frac{8}{12} $$ and $$ \frac{9}{12} $$. Since 9 is bigger than 8, we know that:
$$ \frac{9}{12} $$ is bigger than $$ \frac{8}{12} $$, which means $$ \frac{3}{4} $$ is bigger than $$ \frac{2}{3} $$.
Did you know?
The line in a fraction that separates the numerator and the denominator is called a vinculum. It's a Latin word that means 'bond' or 'link'.