1. Adding and Subtracting
Fractions
Lesson 2.3.6

2. Adding and Subtracting Fractions
Lesson
2.3.6
Adding and Subtracting Fractions
California Standards:
Number Sense 1.2
Add, subtract, multiply, and divide rational numbers (integers, fractions, and terminating decimals) and take positive rational numbers to whole-number powers.
Number Sense 2.2
Add and subtract fractions by using factoring to find common denominators.
What it means for you:
You’ll see how to add and subtract fractions.
Key words:
denominator
numerator
common denominator

3. Adding and Subtracting Fractions
Lesson
2.3.6
Adding and Subtracting Fractions
Adding and subtracting fractions can be quick, or it can be quite a long process — it all depends on whether the fractions already have a common denominator, or whether you have to find it first.
+ = 2 6 3 5
These fractions have a common denominator, so they can be easily added together. 2 6
+ = = 1 3 5
These fractions have different denominators, which means you’ve got to do a bit of work before you can add them together.

4. Adding and Subtracting Fractions
Lesson
2.3.6
Adding and Subtracting Fractions
You Can Add Fractions With a Common Denominator
If fractions have a common denominator (their denominators are the same), adding them is fairly straightforward. + 2 6 3
To find the numerator of the sum, you add the numerators of the individual fractions. The denominator stays the same. 5 6 =
2 + 3 – 5 6 3 2 =
5 – 3
You subtract fractions with a common denominator in exactly the same way.

5. Adding and Subtracting Fractions
Lesson
2.3.6
Adding and Subtracting Fractions
Example 1
Find 2 7 3
Solution
These two fractions have a common denominator, 7.
So 7 will also be the denominator of the sum.
The numerator of the sum will be = 5.
So = . 2 7 3 5
Solution follows…

6. Adding and Subtracting Fractions
Lesson
2.3.6
Adding and Subtracting Fractions
Example 2
Find – . 7 9 2
Solution
The denominator of the result will be 9 (the fractions’ common denominator).
The numerator of the result will be 7 – 2 = 5.
So – = . 7 9 2 5
Solution follows…

7. Adding and Subtracting Fractions
Lesson
2.3.6
Adding and Subtracting Fractions
Guided Practice
Find the sums and differences in Exercises 1–8. + 1 5 2 – 4 11 3 21 10 17 23 50 19 7 25 9 3 5 7 11 3 5 8 21 + 4 15 7 11 15 42 50 21 25 or 2 25 6 17
Solution follows…

8. Adding and Subtracting Fractions
Lesson
2.3.6
Adding and Subtracting Fractions
You May Need to Find a Common Denominator First
Fractions with unlike denominators cannot be directly added or subtracted. You must first find equivalent fractions with a common denominator. 2 6
+ = = = 1 3
1 • 2
3 • 2 5
1 • 3
2 • 3

9. Adding and Subtracting Fractions
Lesson
2.3.6
Adding and Subtracting Fractions
Example 3
Find 7 9 5 6
Solution
The denominators are different here. This means you need to find two fractions equivalent to them, but with a common denominator.
The common denominator can be any common multiple of 9 and 6. For example you could use 9 × 6 = 54.
Using 54 as a common denominator, the equivalent fractions are: and 7 9 6 42 54
× = =
7 × 6
9 × 6 5 45
5 × 9
6 × 9
Solution continues…
Solution follows…

10. Adding and Subtracting Fractions
Lesson
2.3.6
Adding and Subtracting Fractions
Example 3
Find 7 9 5 6
Solution (continues)
Now you can add these fractions: = , which you can simplify to by dividing the numerator and denominator by 3. 42 54 45 87 29 18
Or you could find the LCM (least common multiple) using prime factorizations and use this as a common denominator.
Since 9 = 32 and 6 = 2 × 3, the LCM is 2 × 3 × 3 = 18.
Solution continues…

11. Adding and Subtracting Fractions
Lesson
2.3.6
Adding and Subtracting Fractions
Example 3
Find 7 9 5 6
Solution (continues)
Using a common denominator of 18, the equivalent fractions
are: and 7 9 2 14 18
× = =
7 × 2
9 × 2 5 6 3 15
5 × 3
6 × 3
Now you can add these fractions: = . 14 18 15 29

12. Adding and Subtracting Fractions
Lesson
2.3.6
Adding and Subtracting Fractions
You can use the LCM or any other common multiple as your common denominator — you’ll end up with the same answer.
But using the LCM means that the numbers in your fractions are smaller and easier to use.

13. Adding and Subtracting Fractions
Lesson
2.3.6
Adding and Subtracting Fractions
Example 4
By putting both fractions over a common denominator,
find: – . 3 20 11 15
Solution 15 20 5 3 2
LCM
Use a table to find the LCM of 15 and 20 — this is 5 × 3 × 2 × 2 = 60.
Find equivalent fractions with denominator 60: and 44 60
= =
11 × 4
15 × 4 11 15 3 20 9
3 × 3
20 × 3
Solution continues…
Solution follows…

14. Adding and Subtracting Fractions
Lesson
2.3.6
Adding and Subtracting Fractions
Example 4
By putting both fractions over a common denominator,
find: – . 3 20 11 15
Solution (continued) 44 60 = 11 15 3 20 9
and
So rewriting the subtraction gives: – = – = 44 60 11 15 3 20 9 35
This can be simplified to = . 35 60 7 12

15. Adding and Subtracting Fractions
Lesson
2.3.6
Adding and Subtracting Fractions
Guided Practice
Calculate the answers in Exercises 9–11. 9.
10.
11. – – 2 5 14 15 14 15 6 8
– = + 2 3 7 10 21 30 20 41
+ = + 2 3 4 7 14 21 12 2
– = –
Solution follows…

16. Adding and Subtracting Fractions
Lesson
2.3.6
Adding and Subtracting Fractions
Be Extra Careful if There are Negative Signs
As always in math, if there are negative numbers around, you have to be extra careful.
Remember — subtracting a negative number is exactly the same as adding a positive number.

17. Adding and Subtracting Fractions
Lesson
2.3.6
Adding and Subtracting Fractions
Example 5
Find – 3 5 –2 7
Solution
This looks tricky because of all the negative signs. So take things slowly and carefully. –3 5 2 7 +
This sum can be rewritten as — it means the same.
Now you can find a common denominator and solve the problem in the same way as before.
Solution continues…
Solution follows…

18. Adding and Subtracting Fractions
Lesson
2.3.6
Adding and Subtracting Fractions
Example 5
Find – 3 5 –2 7
Solution (continued)
The LCM of 5 and 7 is 5 × 7 = Put both these fractions over a common denominator of 35. –3
–3 × 7
5 × 7 5
–21 35 = 2
2 × 5
7 × 5 7 10 35 =
Now you can add the two fractions in the same way as before. – 3 5 –2 7
–21 35 = 10 + –
–11 11
or –

19. Adding and Subtracting Fractions
Lesson
2.3.6
Adding and Subtracting Fractions
Guided Practice
Calculate the answers in Exercises 12–14. Simplify your answers.
12.
13.
14. – 2 3 8 2 3 16 24
– = – = – 8 9 7 7 6 – 4 9 7 6 21 18
+ = = 4 9 8 29 – 25 48 7 16 25 48
– = – = – = – 7 16 21 4 1 12
Solution follows…

20. Adding and Subtracting Fractions
Lesson
2.3.6
Adding and Subtracting Fractions
Independent Practice
Calculate the following. Give all your answers in their simplest form. + 2 7 8 – 5 19 30 20 3 16 9 17 21 10 7 6 5 5 9 1 16 59 60 75 56
131
272 – 1
420 –
Solution follows…

21. Adding and Subtracting Fractions
Lesson
2.3.6
Adding and Subtracting Fractions
Round Up
Keep practicing this until it becomes routine... (i) find a common denominator, (ii) put both fractions over this denominator, (iii) do the addition or subtraction, (iv) simplify your answer if possible.
You’ll see this again next Lesson.