Dividing fractions can look confusing because both numbers already have a top and a bottom. The Keep Change Flip method gives you a clear way to rewrite the problem so it becomes multiplication. For example, 3/4 ÷ 2/5 becomes 3/4 × 5/2, which equals 15/8.
This shortcut matters because multiplication of fractions is usually easier and more direct than division.
Understanding Math: How to divide fractions (Keep Change Flip)
Division asks how many groups of one size fit into another amount. This meaning still works when the amounts are fractions. Suppose there is three quarters of a litre of juice, and each serving is two fifths of a litre.
The calculation finds the number of servings. Since one serving is less than a whole litre, the answer can be greater than one.
This is an important check. Students sometimes expect every division answer to become smaller, but division by a fraction smaller than one makes the result larger.
The flipped fraction is called a reciprocal. A fraction multiplied by its reciprocal equals one whole. For example, two fifths multiplied by five halves equals one.
This fact explains the rule rather than making it a trick to memorize. Dividing by two fifths means finding a number that, when multiplied by two fifths, gives the starting amount.
Multiplying by five halves undoes multiplication by two fifths. Division can therefore be replaced with multiplication by the reciprocal because the reciprocal cancels the size of each group.
Before multiplying, simplify when possible. This is often called cancelling common factors. For instance, when multiplying six sevenths by fourteen fifteenths, six and fifteen share a factor of three, while fourteen and seven share a factor of seven.
Reducing first keeps the numbers small and lowers the chance of an arithmetic mistake. It is safe to cancel only a factor from one numerator with a factor from one denominator.
Do not cancel numbers that are merely next to each other across an addition or subtraction sign. Simplify the final answer too, then change an improper fraction into a mixed number when the situation calls for it.
Whole numbers and mixed numbers need preparation before fraction division. Write a whole number with a denominator of one. Change a mixed number into an improper fraction by multiplying the whole number by the denominator, then adding the numerator.
For two and one third, multiply two by three and add one, giving seven thirds. Estimation helps you check the result. If a value near one half is divided by a value near one quarter, the answer should be near two.
A negative answer, zero, or a very tiny answer can reveal an incorrect flip or multiplication. Division by zero is never allowed, and a fraction with zero on the bottom has no value.
Key Facts
- Keep Change Flip means keep the first fraction, change ÷ to ×, and flip the second fraction.
- a/b ÷ c/d = a/b × d/c, as long as b, c, and d are not 0.
- 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8.
- To multiply fractions, multiply the numerators and multiply the denominators: a/b × c/d = ac/bd.
- The reciprocal of c/d is d/c.
- 15/8 can also be written as the mixed number 1 7/8.
Vocabulary
- Dividend
- The dividend is the number being divided, which is the first fraction in a division problem.
- Divisor
- The divisor is the number you are dividing by, which is the second fraction in a division problem.
- Reciprocal
- The reciprocal of a fraction is made by switching its numerator and denominator.
- Numerator
- The numerator is the top number of a fraction and tells how many parts are being counted.
- Denominator
- The denominator is the bottom number of a fraction and tells how many equal parts make one whole.
Common Mistakes to Avoid
- Flipping the first fraction, not the second. This is wrong because Keep Change Flip means the first fraction stays exactly the same and only the divisor becomes its reciprocal.
- Changing the division sign but forgetting to flip the divisor. This is wrong because a/b ÷ c/d must become a/b × d/c, not a/b × c/d.
- Flipping both fractions. This changes the value of the original problem because only dividing by a fraction is rewritten as multiplying by its reciprocal.
- Adding numerators and denominators when multiplying. This is wrong because fraction multiplication uses a/b × c/d = ac/bd, so you multiply across.
Practice Questions
- 1 Solve using Keep Change Flip: 2/3 ÷ 4/5.
- 2 Solve and write your answer as an improper fraction and a mixed number: 5/6 ÷ 1/3.
- 3 Explain why 7/8 ÷ 3/4 is rewritten as 7/8 × 4/3 instead of 8/7 × 4/3.