HSPT: Fractions: add with a common denominator
Learn to add fractions with different denominators by rewriting them over a common denominator, and to spot the wrong answers that skip that step.
How do you add two fractions that have different denominators?
Short answer
Rewrite both fractions over a common denominator by multiplying each top and bottom by the same number. Then add the numerators and simplify.
Know first: Equivalent fractions, Multiples and factors
Step 1 of 5
Add fractions only after the denominators match
A denominator names the size of the pieces. You can add 2 sixths and 3 sixths to get 5 sixths, because the pieces are the same size. You can't add fourths and sixths directly, any more than you can add 3 feet and 2 inches and call it 5.
So first rewrite both fractions with a common denominator. The least common denominator, the smallest number both denominators divide into, keeps the numbers small.
| Denominators | Multiples to check | LCD |
|---|---|---|
| 4 and 6 | 6, 12 | 12 |
| 4 and 5 | 5, 10, 15, 20 | 20 |
| 12 and 8 | 12, 24 | 24 |
List multiples of the larger denominator until the smaller one divides in. Multiplying the two denominators always works too, but can give bigger numbers.
Step 2 of 5
Rewrite, then add the numerators
Worked example: Fourths and sixths
What is 3/4 + 1/6? Give the answer in simplest form.
- A11/12
- B2/5
- C1/3
- D1/6
- 1
Find the LCD
Multiples of 6: 6, 12. Since 4 divides 12, the LCD is 12.
- 2
Rewrite
3/4 = 9/12 (multiply top and bottom by 3). 1/6 = 2/12 (multiply top and bottom by 2).
- 3
Add
9/12 + 2/12 = 11/12. It is already in simplest form.
- 4
See why the other choices are there
2/5 adds the tops and the bottoms: (3 + 1)/(4 + 6). 1/3 is 4/12, which adds the numerators without rewriting them. 1/6 is 4/24, the same mistake over 4 × 6.
Answer
11/12
Find the wrong step
A student adds 5/6 + 4/9.
- 1
Finds the LCD: 18.
- 2
Writes 5/18 + 4/18 = 9/18 = 1/2.
What went wrong
The denominators changed but the numerators didn't. Whatever you multiply the bottom by, you must multiply the top by too.
- 3
Answers 1/2.
The fix
Rewrite fully: 5/6 = 15/18 and 4/9 = 8/18. The sum is 23/18.
Step 3 of 5
Match the pieces, then add
Check yourself · Question 1
What is 1/4 + 2/5? Give the answer in simplest form.
Answer: B
The LCD is 20. 1/4 = 5/20 and 2/5 = 8/20. The sum is 13/20.
Trap answer: 1/3
Adding straight across
- Why it looks right
- Adding top to top and bottom to bottom looks just like multiplying fractions straight across, which is allowed.
- Why it's wrong
- 1/3 is less than 2/5, one of the fractions being added. A sum of two positive fractions must be bigger than each one.
- The right answer
- 13/20, from 5/20 + 8/20.
Check yourself · Question 2
What is 5/12 + 1/8? Give the answer in simplest form.
Answer: C
The LCD is 24. 5/12 = 10/24 and 1/8 = 3/24. The sum is 13/24.
Common mistake
"I add the numerators and add the denominators."
- Why it's tempting
- It is the fastest move and the one that looks most like adding.
- Do this instead
- Check with a sense of size: 1/4 + 2/5 can't be 1/3, because 1/3 is smaller than 2/5. Rewrite with a common denominator first.
Step 4 of 5
What to remember
Remember
- Fractions can be added only when their denominators match.
- Multiply the top and bottom of each fraction by the same number to rewrite it.
- Add the numerators, keep the denominator, and simplify at the end.
Step 5 of 5
Practice on a real question
Use what you just learned on this question, then check the explanation.
In your own words
What common denominator did you use in the practice question, and which wrong choice comes from adding straight across?