**Example 1**: Add the fractions .

The denominators of the two fractions are both 7. By having the same denominators, we can easily add these fractions by adding their numerators and copying the common denominator which is 7.

We can also show the addition process using circles.

- The first fraction can be represented by a circle divided equally in seven parts with three pieces shaded in red.

**Observe**: The **numerator** tells us how many areas are shaded while the **denominator** tells us how many equal parts the circle is divided.

- In the same manner, the second fraction looks like this.

- Since the two circles are both divided in seven (7) equal parts, we should be able to overlap them. The new circle after addition has five (5) shaded regions which is the
**accumulation**of both red and blue pieces.

**Example 2**: Add the fractions .

Let’s combine these fractions using the addition rule. Again, add the numerators and copy the common denominator.

After you add fractions, always find the opportunity to simplify the added fractions by reducing it to the lowest term. We can do so by dividing both the numerator and denominator by their greatest common divisor.

**Common divisor**is a nonzero whole number that can evenly divide two or more numbers.

**Greatest Common Divisor**(GCD) is the largest number among the common divisors of two or more numbers.

Obviously, the numerator and denominator have a common divisor of 2. However, is there a number larger than 2 that can also evenly divide both of them?

Yes, there is! The number 4 is the greatest common divisor of 12 and 16. Therefore, we will use this number to reduce the fraction to its lowest term.

Divide the top and bottom by the **GCD = 4** to get the final answer.

**Example 3:** Add the fractions .

**Solution**:

Since the denominators of the two fractions are equal, add the numerators and copy the common denominator.

The top and bottom numbers of the fraction are divisible by 2 and 6. However, we always want the largest common divisor to reduce the fraction to its lowest term. Thus, the **GCD = 6**.

- Divide the top and bottom numbers by 6.

**Example 4:** Add the fractions .

**Solution**:

All three fractions have the same denominators. The rule in adding fractions with equal denominators still holds!

- Get the sum of the three numerators, and copy the common denominator.

The greatest common division between the numerator and denominator is 5.

- Divide top and bottom by 5.

**Example 5:** Subtract the fractions .

This time around, we are going to subtract the numerators instead of adding them.

Looking at the result after subtraction, the **only** common divisor between the numerator and denominator is **1**. Thus, the final answer remains to be . Think about it, dividing the top and bottom by 1 won’t change the value of the fraction.

How does it look graphically?

Suppose you have a green cake. And you cut it in 5 equal portions. This can be represented in fraction as .

If you ate two slices of the cake () , you should have three leftover pieces ( ).

The plate should look something like this.

**Example 6:** Subtract the fractions .

The two fractions have the same denominators which means we should be able to easily subtract their numerators.

The answer can still be further simplified using a common divisor of 3. So, divide the numerator and denominator by 3 to reduce the fraction to its lowest terms.

**Example 7:** Subtract the fractions .

**Solution**:

Since the denominators of the two fractions are equal, **subtract** their numerators and copy the common denominator.

The numerator and divisor are divisible by 3 and 9. However, we always want the largest common divisor to reduce the fraction to its lowest term. Thus, the **GCD = 9**.

- Divide the top and bottom numbers by 9.

**Example 8:** Subtract the fractions .

**Solution**:

Subtract the numerators, and reduce the resulting fraction to its lowest term using the **GCD = 11**.