How to divide fractions: keep, change, flip, and why it works
To divide fractions, multiply the first fraction by the reciprocal of the second: keep the first fraction, change ÷ to ×, and flip the second one. For example, 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8. Multiplying is simpler still: multiply the numerators together and the denominators together.
Dividing fractions looks odd at first, because the rule says to multiply. This guide starts with multiplying fractions, then shows keep, change, flip step by step, explains why multiplying by the reciprocal gives the right answer, and covers mixed numbers and whole numbers, with worked examples throughout. At the end, a free calculator checks each answer and writes out the steps.
Key ideas
- To multiply fractions, multiply the numerators and multiply the denominators: 2/3 × 3/4 = 6/12 = 1/2. No common denominator is needed.
- To divide, keep the first fraction, change ÷ to × and flip the second fraction into its reciprocal: 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8.
- It works because dividing by a number is the same as multiplying by its reciprocal; 3 ÷ 1/2 = 6 because three wholes hold six halves.
- Turn mixed numbers into improper fractions before you flip anything, write a whole number n as n/1, and remember that you can never divide by zero.
How do you multiply fractions?
Multiply the numerators together and the denominators together, then simplify. Unlike addition, multiplication needs no common denominator. For example, 2/3 × 3/4 = (2 × 3)/(3 × 4) = 6/12, and 6/12 simplifies to 1/2.
You can cancel common factors before multiplying to keep the numbers small. In 2/3 × 3/4 the 3 on top and the 3 below cancel, and the 2 and the 4 share a factor of 2, which leaves 1/1 × 1/2 = 1/2 at once. A whole number becomes a fraction over 1: 3 × 2/5 = 3/1 × 2/5 = 6/5, or 1 1/5.
The word “of” in a problem means multiply: half of three quarters is 1/2 × 3/4 = 3/8. That example also shows something worth remembering: multiplying by a fraction smaller than 1 makes a number smaller, because 3/8 is less than 3/4.
How do you divide fractions? Keep, change, flip
Multiply the first fraction by the reciprocal of the second. The reciprocal of a fraction a/b is b/a, the number that multiplied by a/b gives exactly 1: 2/5 × 5/2 = 10/10 = 1. Zero is the one number with no reciprocal, since nothing multiplied by 0 gives 1, and that is why division by zero has no answer.
The steps carry the name keep, change, flip. For 3/4 ÷ 2/5: first keep 3/4 as it is, then change the division sign to multiplication, then flip 2/5 to 5/2. Now multiply: 3/4 × 5/2 = 15/8, which as a mixed number is 1 7/8. Another example: 1/2 ÷ 3/4 = 1/2 × 4/3 = 4/6 = 2/3.
Only the divisor, the second fraction, is flipped. Flipping the first one instead gives 4/3 × 2/5 = 8/15, which is the reciprocal of the right answer, 15/8. To check a result, multiply it by the divisor and you should get back the first fraction: 15/8 × 2/5 = 30/40 = 3/4.
Why does multiplying by the reciprocal work?
Dividing by a number gives the same result as multiplying by its reciprocal. Division asks how many times the divisor fits into the first number, or what number multiplied by the divisor gives the first number back, and multiplying by the reciprocal answers exactly that question.
Start with something you can picture. How many halves are in 3 wholes? Six, so 3 ÷ 1/2 = 6, which is the same as 3 × 2. Measuring cups work the same way: a 1/2 cup holds two 1/4-cup scoops, so 1/2 ÷ 1/4 = 1/2 × 4/1 = 2, and 3/4 of a cup fills the 1/4 cup three times, because 3/4 × 4/1 = 12/4 = 3.
In general, if a/b ÷ c/d = x, then x × c/d = a/b. Multiply both sides by d/c: since c/d × d/c = 1, x = a/b × d/c. There is also a route without flipping: put both fractions over a common denominator and divide the numerators. 3/4 ÷ 2/5 = 15/20 ÷ 8/20 = 15 ÷ 8 = 15/8, the same answer as before.
How do you divide mixed numbers and whole numbers, and check the answer?
Turn mixed numbers into improper fractions first; keep, change, flip only works on plain fractions. For 2 1/2 ÷ 1 1/4: 2 1/2 = 5/2 and 1 1/4 = 5/4, so 5/2 ÷ 5/4 = 5/2 × 4/5 = 20/10 = 2. That makes sense, because one and a quarter fits into two and a half exactly twice.
With a whole number, write it over 1. A fraction divided by a whole number: 10/3 ÷ 5 = 10/3 × 1/5 = 10/15 = 2/3; when the numerator divides evenly you can also divide it directly, 10 ÷ 5 = 2, which gives 2/3 again. A whole number divided by a fraction: 4 ÷ 2/3 = 4/1 × 3/2 = 12/2 = 6, so two thirds fit into four exactly six times.
To check your work, open the free fraction calculator at VITON13 SCHOOL and put ÷ between two fractions, or type 2 1/2 ÷ 1 1/4 in the Expression mode. The steps show the mixed numbers turned into improper fractions, the divisor flipped into its reciprocal, the numerators and denominators multiplied and the answer simplified by the GCD; the result appears as a fraction, a mixed number, a decimal and a percent. If a divisor is zero, the calculator stops and says that division by zero has no value. In an expression, × and ÷ are done before + and −, and brackets first.
In everyday language
Dividing by a fraction asks how many of those pieces fit. Three pizzas hold six halves, so 3 ÷ 1/2 = 6, and flipping 1/2 into 2 and multiplying gives the same six.
Try it yourself
Work out 5/6 ÷ 2/3 on paper with keep, change, flip, then enter it in the fraction calculator and compare the steps.
Expected result
5/6 × 3/2 = 15/12, simplified to 5/4: the mixed number 1 1/4, the decimal 1.25 and 125%.
Check your answer: How many 1/4-cup scoops fill 1 1/2 cups?
Six. 1 1/2 = 3/2, and 3/2 ÷ 1/4 = 3/2 × 4/1 = 12/2 = 6.
Questions
Which fraction do you flip when dividing fractions?
Only the second one, the divisor. The first fraction stays as it is; flipping it instead gives the reciprocal of the correct answer.
Can you divide fractions without flipping?
Yes. Rewrite both fractions over a common denominator and divide the numerators: 3/4 ÷ 2/5 = 15/20 ÷ 8/20 = 15/8. The answer matches keep, change, flip.
Multiply straight across; to divide, keep the first fraction, change ÷ to × and flip the second. Convert mixed numbers first, write whole numbers over 1, and check each answer in the free VITON13 SCHOOL fraction calculator, which shows every step.
Sources and further reading
- Wikipedia: Fraction (multiplication and division of fractions) ↗Sources checked:
- Wikipedia: Multiplicative inverse (reciprocal) ↗Sources checked:
- Mathwords: Keep Change Flip ↗Sources checked: