HSPT: Fractions: divide by multiplying by the reciprocal
Learn to divide fractions with keep, change, flip, and to check your answer with a quick size test.
How do you divide one fraction by another?
Short answer
Keep the first fraction, change division to multiplication, and flip the second fraction. Then multiply across and simplify.
Know first: Multiplying fractions, Simplifying fractions
Step 1 of 5
To divide by a fraction, multiply by its flip
Dividing by a fraction asks how many of these fit? How many quarters fit in one half? Two. And 1/2 × 4/1 = 2, which is the rule in action.
So to divide fractions, keep the first fraction, change ÷ to ×, and flip the second fraction to its reciprocal. Then multiply straight across and simplify.
Step 2 of 5
Flip the second fraction only
Worked example: Sixths divided by fourths
What is (5/6) ÷ (3/4)?
- A10/9
- B5/8
- C9/10
- D8/5
- 1
Keep, change, flip
5/6 × 4/3.
- 2
Multiply
(5 × 4)/(6 × 3) = 20/18.
- 3
Simplify
Divide the top and bottom by 2: 10/9.
- 4
Size check
10/9 is bigger than 5/6, as it should be when you divide by 3/4.
- 5
See why the other choices are there
5/8 multiplies without flipping. 9/10 flips the first fraction instead of the second. 8/5 flips both.
Answer
10/9
Wrong: (5/6) ÷ (3/4) = 6/5 × 3/4 = 9/10
Right: (5/6) ÷ (3/4) = 5/6 × 4/3 = 10/9
Flip the fraction you are dividing by, the second one. Flipping the first gives the reciprocal of the right answer.
Step 3 of 5
Keep, change, flip, then simplify
Check yourself · Question 1
What is (2/9) ÷ (4/5)?
Answer: B
Keep, change, flip: 2/9 × 5/4 = 10/36, which simplifies to 5/18.
Trap answer: 8/45
Forgetting to flip
- Why it looks right
- Multiplying straight across is exactly how fraction multiplication works, so it feels right.
- Why it's wrong
- The question divides. Dividing by 4/5 means multiplying by 5/4, so the 4 belongs on the bottom.
- The right answer
- 5/18, from 2/9 × 5/4 = 10/36.
Check yourself · Question 2
What is 4 ÷ (2/9)?
Answer: C
Write 4 as 4/1. Keep, change, flip: 4/1 × 9/2 = 36/2 = 18. Check the size: 2/9 is small, so many of them fit in 4.
Common mistake
"I flip the first fraction, since it's the one I see first."
- Why it's tempting
- Both fractions are right there, and the rule just says flip.
- Do this instead
- Flip only the divisor, the fraction after the ÷ sign. Then check the size: dividing by a fraction less than 1 should make the answer grow.
Step 4 of 5
What to remember
Remember
- Keep the first fraction, change ÷ to ×, and flip the second.
- Write a whole number as a fraction over 1 before you divide.
- Dividing by a fraction less than 1 gives a bigger answer, which is a fast check.
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
Which fraction did you flip in the practice question, and what wrong choice comes from flipping the other one?