GCSE Maths: Dividing fractions (Foundation)
Part of the GCSE Maths fractions chapter (Foundation). Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Dividing fractions
Dividing asks “how many fit into?” How many halves are in 2? There are 4, so 2 ÷ 12 = 4.
Key ideas
- The reciprocal of a fraction is found by turning it upside down. The reciprocal of 23 is 32. The reciprocal of 5 is 15.
- To divide by a fraction, multiply by its reciprocal. (Keep, change, flip.)
- To divide by a whole number, treat it as a fraction over 1, or divide the numerator if it goes in.
- Change mixed numbers to improper fractions first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.34 ÷ 25
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.34 ÷ 25 = 34 × 52
- Look for common factors to cancel. A top number and a bottom number from different fractions can be cancelled if they share a factor.There is nothing to cancel here.
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.3 × 5 = 15
4 × 2 = 8
158 - Write the improper fraction as a mixed number. The top is bigger than the bottom, so the answer is more than 1. Divide to find the whole number and the remainder.15 ÷ 8 = 1 remainder 7
158 = 178
Check. Multiply back: 178 × 25 should give 34. 1.875 × 0.4 = 0.75 and 34 = 0.75. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.214 ÷ 112
- Change the mixed numbers to improper fractions. A mixed number is a whole number plus a fraction. Multiply the whole number by the denominator, add the numerator, and keep the denominator. This lets us calculate with a single fraction.214 = (2 × 4 + 1) over 4 = 94
112 = (1 × 2 + 1) over 2 = 32 - Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.94 ÷ 32 = 94 × 23
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.94 × 23 = 32 × 11
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.3 × 1 = 3
2 × 1 = 2
32 - Write the improper fraction as a mixed number. The top is bigger than the bottom, so the answer is more than 1. Divide to find the whole number and the remainder.3 ÷ 2 = 1 remainder 1
32 = 112
Check. Multiply back: 112 × 112 should give 214. 1.5 × 1.5 = 2.25 and 214 = 2.25. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.6 ÷ 34
- Write the whole number as a fraction over 1. Any whole number can be written over 1 without changing its value. This lets us use the fraction rules.6 = 61
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.61 ÷ 34 = 61 × 43
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.61 × 43 = 21 × 41
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.2 × 4 = 8
1 × 1 = 1
81 - Write the answer. The bottom number is 1, so the answer is a whole number.81 = 8
Check. Multiply back: 8 × 34 should give 6. 8 × 0.75 = 6 and 6 = 6. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.56 ÷ 4
- Write the whole number as a fraction over 1. Any whole number can be written over 1 without changing its value. This lets us use the fraction rules.4 = 41
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.56 ÷ 41 = 56 × 14
- Look for common factors to cancel. A top number and a bottom number from different fractions can be cancelled if they share a factor.There is nothing to cancel here.
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.5 × 1 = 5
6 × 4 = 24
524 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.5 and 24 have no common factor except 1, so 524 is the answer.
Check. Multiply back: 524 × 4 should give 56. 0.2083 × 4 = 0.8333 and 56 = 0.8333. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Decide the operation. “How many bottles fit” asks how many times the bottle size goes into the total, so we divide the total by the size of one bottle.513 ÷ 23
- Change the mixed number to improper fraction. A mixed number is a whole number plus a fraction. Multiply the whole number by the denominator, add the numerator, and keep the denominator. This lets us calculate with a single fraction.513 = (5 × 3 + 1) over 3 = 163
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.163 ÷ 23 = 163 × 32
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.163 × 32 = 81 × 11
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.8 × 1 = 8
1 × 1 = 1
81 - Write the answer. The bottom number is 1, so the answer is a whole number.81 = 8
- Answer the question in words. The answer is a whole number of bottles.8 bottles
Check. Multiply back: 8 × 23 should give 513. 8 × 0.6667 = 5.3333 and 513 = 5.3333. ✓
Watch out. Divide the total by the size of one bottle, not the other way round.
Watch out
Only flip the second fraction (the one you divide by). Also 12 ÷ 14 = 2 but 14 ÷ 12 = 12: order matters when dividing.Exercise 3G Dividing fractions
Non-calculatorMethod. The reciprocal of a number is found by turning the fraction upside down (swap the top and the bottom).
- Say what a reciprocal is. The reciprocal of a number is 1 divided by that number. For a fraction it is the same fraction turned upside down.reciprocal of 23
- Flip the fraction. Swap the numerator and the denominator.23 becomes 32
- Write the answer. If the new fraction has 1 on the bottom, it is a whole number.32
Check. A number times its reciprocal is 1: 23 × 32 = 1. ✓
Watch out. Only flip the fraction. Do not change its sign or add anything. The reciprocal of 5 is 1/5, not −5.
Method. The reciprocal of a number is found by turning the fraction upside down (swap the top and the bottom).
- Say what a reciprocal is. The reciprocal of a number is 1 divided by that number. For a fraction it is the same fraction turned upside down.reciprocal of 58
- Flip the fraction. Swap the numerator and the denominator.58 becomes 85
- Write the answer. If the new fraction has 1 on the bottom, it is a whole number.85
Check. A number times its reciprocal is 1: 58 × 85 = 1. ✓
Watch out. Only flip the fraction. Do not change its sign or add anything. The reciprocal of 5 is 1/5, not −5.
Method. The reciprocal of a number is found by turning the fraction upside down (swap the top and the bottom).
- Say what a reciprocal is. The reciprocal of a number is 1 divided by that number. For a fraction it is the same fraction turned upside down.reciprocal of 4
- Write the whole number as a fraction over 1. Every whole number can be written over 1 without changing its value.4 = 41
- Flip the fraction. Swap the numerator and the denominator.41 becomes 14
- Write the answer. If the new fraction has 1 on the bottom, it is a whole number.14
Check. A number times its reciprocal is 1: 4 × 14 = 1. ✓
Watch out. Only flip the fraction. Do not change its sign or add anything. The reciprocal of 5 is 1/5, not −5.
Method. The reciprocal of a number is found by turning the fraction upside down (swap the top and the bottom).
- Say what a reciprocal is. The reciprocal of a number is 1 divided by that number. For a fraction it is the same fraction turned upside down.reciprocal of 17
- Flip the fraction. Swap the numerator and the denominator.17 becomes 71
- Write the answer. If the new fraction has 1 on the bottom, it is a whole number.7
Check. A number times its reciprocal is 1: 17 × 7 = 1. ✓
Watch out. Only flip the fraction. Do not change its sign or add anything. The reciprocal of 5 is 1/5, not −5.
Method. The reciprocal of a number is found by turning the fraction upside down (swap the top and the bottom).
- Say what a reciprocal is. The reciprocal of a number is 1 divided by that number. For a fraction it is the same fraction turned upside down.reciprocal of 312
- Change the mixed number to an improper fraction. We need a single top and a single bottom before we can flip it.312 = 72
- Flip the fraction. Swap the numerator and the denominator.72 becomes 27
- Write the answer. If the new fraction has 1 on the bottom, it is a whole number.27
Check. A number times its reciprocal is 1: 72 × 27 = 1. ✓
Watch out. Only flip the fraction. Do not change its sign or add anything. The reciprocal of 5 is 1/5, not −5.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.2 ÷ 12
- Write the whole number as a fraction over 1. Any whole number can be written over 1 without changing its value. This lets us use the fraction rules.2 = 21
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.21 ÷ 12 = 21 × 21
- Look for common factors to cancel. A top number and a bottom number from different fractions can be cancelled if they share a factor.There is nothing to cancel here.
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.2 × 2 = 4
1 × 1 = 1
41 - Write the answer. The bottom number is 1, so the answer is a whole number.41 = 4
Check. Multiply back: 4 × 12 should give 2. 4 × 0.5 = 2 and 2 = 2. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.3 ÷ 14
- Write the whole number as a fraction over 1. Any whole number can be written over 1 without changing its value. This lets us use the fraction rules.3 = 31
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.31 ÷ 14 = 31 × 41
- Look for common factors to cancel. A top number and a bottom number from different fractions can be cancelled if they share a factor.There is nothing to cancel here.
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.3 × 4 = 12
1 × 1 = 1
121 - Write the answer. The bottom number is 1, so the answer is a whole number.121 = 12
Check. Multiply back: 12 × 14 should give 3. 12 × 0.25 = 3 and 3 = 3. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.5 ÷ 13
- Write the whole number as a fraction over 1. Any whole number can be written over 1 without changing its value. This lets us use the fraction rules.5 = 51
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.51 ÷ 13 = 51 × 31
- Look for common factors to cancel. A top number and a bottom number from different fractions can be cancelled if they share a factor.There is nothing to cancel here.
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.5 × 3 = 15
1 × 1 = 1
151 - Write the answer. The bottom number is 1, so the answer is a whole number.151 = 15
Check. Multiply back: 15 × 13 should give 5. 15 × 0.3333 = 5 and 5 = 5. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.4 ÷ 23
- Write the whole number as a fraction over 1. Any whole number can be written over 1 without changing its value. This lets us use the fraction rules.4 = 41
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.41 ÷ 23 = 41 × 32
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.41 × 32 = 21 × 31
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.2 × 3 = 6
1 × 1 = 1
61 - Write the answer. The bottom number is 1, so the answer is a whole number.61 = 6
Check. Multiply back: 6 × 23 should give 4. 6 × 0.6667 = 4 and 4 = 4. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.6 ÷ 34
- Write the whole number as a fraction over 1. Any whole number can be written over 1 without changing its value. This lets us use the fraction rules.6 = 61
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.61 ÷ 34 = 61 × 43
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.61 × 43 = 21 × 41
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.2 × 4 = 8
1 × 1 = 1
81 - Write the answer. The bottom number is 1, so the answer is a whole number.81 = 8
Check. Multiply back: 8 × 34 should give 6. 8 × 0.75 = 6 and 6 = 6. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.12 ÷ 14
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.12 ÷ 14 = 12 × 41
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.12 × 41 = 11 × 21
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.1 × 2 = 2
1 × 1 = 1
21 - Write the answer. The bottom number is 1, so the answer is a whole number.21 = 2
Check. Multiply back: 2 × 14 should give 12. 2 × 0.25 = 0.5 and 12 = 0.5. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.34 ÷ 12
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.34 ÷ 12 = 34 × 21
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.34 × 21 = 32 × 11
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.3 × 1 = 3
2 × 1 = 2
32 - Write the improper fraction as a mixed number. The top is bigger than the bottom, so the answer is more than 1. Divide to find the whole number and the remainder.3 ÷ 2 = 1 remainder 1
32 = 112
Check. Multiply back: 112 × 12 should give 34. 1.5 × 0.5 = 0.75 and 34 = 0.75. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.23 ÷ 49
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.23 ÷ 49 = 23 × 94
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.23 × 94 = 11 × 32
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.1 × 3 = 3
1 × 2 = 2
32 - Write the improper fraction as a mixed number. The top is bigger than the bottom, so the answer is more than 1. Divide to find the whole number and the remainder.3 ÷ 2 = 1 remainder 1
32 = 112
Check. Multiply back: 112 × 49 should give 23. 1.5 × 0.4444 = 0.6667 and 23 = 0.6667. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.25 ÷ 34
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.25 ÷ 34 = 25 × 43
- Look for common factors to cancel. A top number and a bottom number from different fractions can be cancelled if they share a factor.There is nothing to cancel here.
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.2 × 4 = 8
5 × 3 = 15
815 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.8 and 15 have no common factor except 1, so 815 is the answer.
Check. Multiply back: 815 × 34 should give 25. 0.5333 × 0.75 = 0.4 and 25 = 0.4. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.78 ÷ 14
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.78 ÷ 14 = 78 × 41
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.78 × 41 = 72 × 11
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.7 × 1 = 7
2 × 1 = 2
72 - Write the improper fraction as a mixed number. The top is bigger than the bottom, so the answer is more than 1. Divide to find the whole number and the remainder.7 ÷ 2 = 3 remainder 1
72 = 312
Check. Multiply back: 312 × 14 should give 78. 3.5 × 0.25 = 0.875 and 78 = 0.875. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.49 ÷ 23
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.49 ÷ 23 = 49 × 32
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.49 × 32 = 23 × 11
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.2 × 1 = 2
3 × 1 = 3
23 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.2 and 3 have no common factor except 1, so 23 is the answer.
Check. Multiply back: 23 × 23 should give 49. 0.6667 × 0.6667 = 0.4444 and 49 = 0.4444. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.34 ÷ 3
- Write the whole number as a fraction over 1. Any whole number can be written over 1 without changing its value. This lets us use the fraction rules.3 = 31
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.34 ÷ 31 = 34 × 13
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.34 × 13 = 14 × 11
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.1 × 1 = 1
4 × 1 = 4
14 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.1 and 4 have no common factor except 1, so 14 is the answer.
Check. Multiply back: 14 × 3 should give 34. 0.25 × 3 = 0.75 and 34 = 0.75. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.45 ÷ 2
- Write the whole number as a fraction over 1. Any whole number can be written over 1 without changing its value. This lets us use the fraction rules.2 = 21
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.45 ÷ 21 = 45 × 12
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.45 × 12 = 25 × 11
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.2 × 1 = 2
5 × 1 = 5
25 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.2 and 5 have no common factor except 1, so 25 is the answer.
Check. Multiply back: 25 × 2 should give 45. 0.4 × 2 = 0.8 and 45 = 0.8. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.67 ÷ 4
- Write the whole number as a fraction over 1. Any whole number can be written over 1 without changing its value. This lets us use the fraction rules.4 = 41
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.67 ÷ 41 = 67 × 14
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.67 × 14 = 37 × 12
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.3 × 1 = 3
7 × 2 = 14
314 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.3 and 14 have no common factor except 1, so 314 is the answer.
Check. Multiply back: 314 × 4 should give 67. 0.2143 × 4 = 0.8571 and 67 = 0.8571. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.56 ÷ 10
- Write the whole number as a fraction over 1. Any whole number can be written over 1 without changing its value. This lets us use the fraction rules.10 = 101
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.56 ÷ 101 = 56 × 110
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.56 × 110 = 16 × 12
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.1 × 1 = 1
6 × 2 = 12
112 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.1 and 12 have no common factor except 1, so 112 is the answer.
Check. Multiply back: 112 × 10 should give 56. 0.0833 × 10 = 0.8333 and 56 = 0.8333. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.212 ÷ 114
- Change the mixed numbers to improper fractions. A mixed number is a whole number plus a fraction. Multiply the whole number by the denominator, add the numerator, and keep the denominator. This lets us calculate with a single fraction.212 = (2 × 2 + 1) over 2 = 52
114 = (1 × 4 + 1) over 4 = 54 - Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.52 ÷ 54 = 52 × 45
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.52 × 45 = 11 × 21
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.1 × 2 = 2
1 × 1 = 1
21 - Write the answer. The bottom number is 1, so the answer is a whole number.21 = 2
Check. Multiply back: 2 × 114 should give 212. 2 × 1.25 = 2.5 and 212 = 2.5. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.313 ÷ 23
- Change the mixed number to improper fraction. A mixed number is a whole number plus a fraction. Multiply the whole number by the denominator, add the numerator, and keep the denominator. This lets us calculate with a single fraction.313 = (3 × 3 + 1) over 3 = 103
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.103 ÷ 23 = 103 × 32
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.103 × 32 = 51 × 11
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.5 × 1 = 5
1 × 1 = 1
51 - Write the answer. The bottom number is 1, so the answer is a whole number.51 = 5
Check. Multiply back: 5 × 23 should give 313. 5 × 0.6667 = 3.3333 and 313 = 3.3333. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.134 ÷ 213
- Change the mixed numbers to improper fractions. A mixed number is a whole number plus a fraction. Multiply the whole number by the denominator, add the numerator, and keep the denominator. This lets us calculate with a single fraction.134 = (1 × 4 + 3) over 4 = 74
213 = (2 × 3 + 1) over 3 = 73 - Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.74 ÷ 73 = 74 × 37
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.74 × 37 = 14 × 31
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.1 × 3 = 3
4 × 1 = 4
34 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.3 and 4 have no common factor except 1, so 34 is the answer.
Check. Multiply back: 34 × 213 should give 134. 0.75 × 2.3333 = 1.75 and 134 = 1.75. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Read the question and decide the plan. Dividing by a fraction is done by multiplying by its reciprocal. First look at the type of numbers we have.412 ÷ 118
- Change the mixed numbers to improper fractions. A mixed number is a whole number plus a fraction. Multiply the whole number by the denominator, add the numerator, and keep the denominator. This lets us calculate with a single fraction.412 = (4 × 2 + 1) over 2 = 92
118 = (1 × 8 + 1) over 8 = 98 - Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.92 ÷ 98 = 92 × 89
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.92 × 89 = 11 × 41
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.1 × 4 = 4
1 × 1 = 1
41 - Write the answer. The bottom number is 1, so the answer is a whole number.41 = 4
Check. Multiply back: 4 × 118 should give 412. 4 × 1.125 = 4.5 and 412 = 4.5. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Decide the operation. “How many pieces can be cut” asks how many times the piece length goes into the ribbon length, so divide the total length by the length of one piece.412 m ÷ 34 m
- Change the mixed number to improper fraction. A mixed number is a whole number plus a fraction. Multiply the whole number by the denominator, add the numerator, and keep the denominator. This lets us calculate with a single fraction.412 = (4 × 2 + 1) over 2 = 92
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.92 ÷ 34 = 92 × 43
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.92 × 43 = 31 × 21
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.3 × 2 = 6
1 × 1 = 1
61 - Write the answer. The bottom number is 1, so the answer is a whole number.61 = 6
- Answer in words. The answer is a whole number of pieces and there is nothing left over.6 pieces can be cut.
Check. Multiply back: 6 × 34 should give 412. 6 × 0.75 = 4.5 and 412 = 4.5. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Decide the operation. “Shared equally between 3 people” means divide by 3.35 ÷ 3
- Write the whole number as a fraction over 1. Any whole number can be written over 1 without changing its value. This lets us use the fraction rules.3 = 31
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.35 ÷ 31 = 35 × 13
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.35 × 13 = 15 × 11
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.1 × 1 = 1
5 × 1 = 5
15 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.1 and 5 have no common factor except 1, so 15 is the answer.
- Answer in words. Each person gets this fraction of the whole pizza.Each person gets 15 of the pizza.
Check. Multiply back: 15 × 3 should give 35. 0.2 × 3 = 0.6 and 35 = 0.6. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.
Method. Work out both divisions and compare the answers. If division were commutative they would be the same.
- Work out the first division. Keep, change, flip: turn the second fraction upside down and multiply.12 ÷ 14 = 12 × 41 = 42 = 2
- Work out the second division. Same method, but the numbers are the other way round.14 ÷ 12 = 14 × 21 = 24 = 12
- Compare the answers. The first answer is 2 and the second is a half.2 ≠ 12
- Write the conclusion. Different answers mean the two divisions are not the same.No, they are not the same.
Check. Think of it in words. How many quarters fit into a half? Two. How many halves fit into a quarter? Only half of one. Two and a half are not the same. ✓
Watch out. Division is not commutative: the order matters (unlike multiplication).
Method. Work from left to right, because multiplying and dividing have the same priority in BIDMAS.
- Say the order of working. There are no brackets or powers. Multiplication and division have equal priority, so we work from left to right.23 × 34 ÷ 12: multiply first, then divide
- Multiply first. Cancel the 3s, then multiply the tops and the bottoms.23 × 34 = 24 = 12
- Then divide. Keep, change, flip.12 ÷ 12 = 12 × 21
- Multiply and simplify. Multiply the tops and the bottoms.22 = 1
Check. Any number divided by itself is 1, and the first step gave 1/2 divided by 1/2. ✓
Watch out. Do not do the division first. Work left to right.
Method. To divide by a fraction, turn the second fraction upside down and multiply: keep, change, flip.
- Decide the operation. Area of a rectangle = length × width, so width = area ÷ length.Width = 34 ÷ 56
- Keep, change, flip. Dividing by a fraction is the same as multiplying by its reciprocal (the fraction turned upside down). For example, dividing by 1/2 asks "how many halves fit?", which is the same as multiplying by 2.34 ÷ 56 = 34 × 65
- Cancel common factors. Divide a top and a bottom by a number they share, to keep the numbers small.34 × 65 = 32 × 35
- Multiply the numerators, then the denominators. Now it is an ordinary multiplication: top × top and bottom × bottom.3 × 3 = 9
2 × 5 = 10
910 - Check the answer is in its simplest form. See whether any number except 1 divides into both the top and the bottom. If not, we are finished.9 and 10 have no common factor except 1, so 910 is the answer.
- Write the answer with its units. The width is a length, so it is in metres.Width = 910 m
Check. Multiply back: 910 × 56 should give 34. 0.9 × 0.8333 = 0.75 and 34 = 0.75. ✓
Watch out. Only flip the second fraction (the one you are dividing by), never the first.