Year 6: Simplifying fractions
Part of the Year 6 fractions chapter. Chapter overview · whole chapter on one page
Simplifying fractions
A fraction can look different and still have the same value. The bars below are the same length, so 68 and 34 are equal. The second bar is simpler to read: each part is bigger, and there are fewer parts.
A fraction is in its simplest form when the top and the bottom have no common factor except 1. To simplify a fraction, find the highest common factor (HCF) of the top and bottom. Then divide both numbers by it. The value stays the same, because the top and the bottom are divided by the same number.
Key ideas
- A fraction is in its simplest form when the HCF of the top and bottom is 1.
- To simplify, divide the top and the bottom by the same number, which must divide into both exactly.
- The HCF gives the quickest answer, because you divide once. Dividing in small steps also works.
- Check the answer: if the top and bottom still share a factor, you can simplify again.
Key words. A fraction is in its simplest form when the only number that divides exactly into its top and bottom is 1. The highest common factor (HCF) of the top and bottom gives you the number to divide by.
Method. Find the HCF of the top and the bottom, then divide both numbers by it.
- Find the highest common factor (HCF) of the top and the bottom. List the factors of both numbers. The HCF is the biggest factor that they share.Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
HCF = 6 - Divide the top and the bottom by the HCF. Doing the same to both numbers keeps the fraction the same size.12 ÷ 6 = 2 and 18 ÷ 6 = 3
- Check that nothing else divides into both. If the only factor they share is 1, the fraction is in its simplest form.Factors of 2: 1, 2. Factors of 3: 1, 3. Only 1 is shared, so the simplest form is 23.
Check. 2 × 6 = 12 and 3 × 6 = 18. ✓
Watch out. Divide the top AND the bottom. Dividing only one of them changes the amount.
Key words. A fraction is in its simplest form when the only number that divides exactly into its top and bottom is 1. The highest common factor (HCF) of the top and bottom gives you the number to divide by.
Method. Find the HCF of the top and the bottom, then divide both numbers by it.
- Find the highest common factor (HCF) of the top and the bottom. List the factors of both numbers. The HCF is the biggest factor that they share.Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
HCF = 12 - Divide the top and the bottom by the HCF. Doing the same to both numbers keeps the fraction the same size.36 ÷ 12 = 3 and 48 ÷ 12 = 4
- Check that nothing else divides into both. If the only factor they share is 1, the fraction is in its simplest form.Factors of 3: 1, 3. Factors of 4: 1, 2, 4. Only 1 is shared, so the simplest form is 34.
Check. 3 × 12 = 36 and 4 × 12 = 48. ✓
Watch out. Divide the top AND the bottom. Dividing only one of them changes the amount.
Key words. A fraction is in its simplest form when the only number that divides exactly into its top and bottom is 1. The highest common factor (HCF) of the top and bottom gives you the number to divide by.
Method. You can simplify in more than one step. Keep dividing by common factors until only 1 is shared.
- Divide the top and the bottom by 2. Both numbers divide by 2, so 2 is a common factor.2440 = 1220
- Divide the top and the bottom by 4. Both numbers still share a factor, so divide again.1220 = 35
- Check that the answer cannot be simplified. The top and the bottom have no shared factor except 1.1, 3 and 1, 5: only 1 is shared, so the simplest form is 35
Check. 3 × 8 = 24 and 5 × 8 = 40. ✓
Watch out. Two small steps are fine. The final answer must be the simplest form, not just a smaller fraction.
Watch out
Do not divide only the top or only the bottom. 1218 is not 618 in simplest form, and the value has changed. Also, a common factor is not always 2: check the HCF first.Exercise 4A Simplifying fractions
Non-calculatorKey words. A fraction is in its simplest form when the only number that divides exactly into its top and bottom is 1. The highest common factor (HCF) of the top and bottom gives you the number to divide by.
Method. Find the HCF of the top and the bottom, then divide both numbers by it.
- Find the highest common factor (HCF) of the top and the bottom. List the factors of both numbers. The HCF is the biggest factor that they share.Factors of 6: 1, 2, 3, 6
Factors of 8: 1, 2, 4, 8
HCF = 2 - Divide the top and the bottom by the HCF. Doing the same to both numbers keeps the fraction the same size.6 ÷ 2 = 3 and 8 ÷ 2 = 4
- Check that nothing else divides into both. If the only factor they share is 1, the fraction is in its simplest form.Factors of 3: 1, 3. Factors of 4: 1, 2, 4. Only 1 is shared, so the simplest form is 34.
Check. 3 × 2 = 6 and 4 × 2 = 8. ✓
Watch out. Divide the top AND the bottom. Dividing only one of them changes the amount.
Key words. A fraction is in its simplest form when the only number that divides exactly into its top and bottom is 1. The highest common factor (HCF) of the top and bottom gives you the number to divide by.
Method. Find the HCF of the top and the bottom, then divide both numbers by it.
- Find the highest common factor (HCF) of the top and the bottom. List the factors of both numbers. The HCF is the biggest factor that they share.Factors of 10: 1, 2, 5, 10
Factors of 15: 1, 3, 5, 15
HCF = 5 - Divide the top and the bottom by the HCF. Doing the same to both numbers keeps the fraction the same size.10 ÷ 5 = 2 and 15 ÷ 5 = 3
- Check that nothing else divides into both. If the only factor they share is 1, the fraction is in its simplest form.Factors of 2: 1, 2. Factors of 3: 1, 3. Only 1 is shared, so the simplest form is 23.
Check. 2 × 5 = 10 and 3 × 5 = 15. ✓
Watch out. Divide the top AND the bottom. Dividing only one of them changes the amount.
Key words. A fraction is in its simplest form when the only number that divides exactly into its top and bottom is 1. The highest common factor (HCF) of the top and bottom gives you the number to divide by.
Method. Find the HCF of the top and the bottom, then divide both numbers by it.
- Find the highest common factor (HCF) of the top and the bottom. List the factors of both numbers. The HCF is the biggest factor that they share.Factors of 9: 1, 3, 9
Factors of 12: 1, 2, 3, 4, 6, 12
HCF = 3 - Divide the top and the bottom by the HCF. Doing the same to both numbers keeps the fraction the same size.9 ÷ 3 = 3 and 12 ÷ 3 = 4
- Check that nothing else divides into both. If the only factor they share is 1, the fraction is in its simplest form.Factors of 3: 1, 3. Factors of 4: 1, 2, 4. Only 1 is shared, so the simplest form is 34.
Check. 3 × 3 = 9 and 4 × 3 = 12. ✓
Watch out. Divide the top AND the bottom. Dividing only one of them changes the amount.
Key words. A fraction is in its simplest form when the only number that divides exactly into its top and bottom is 1. The highest common factor (HCF) of the top and bottom gives you the number to divide by.
Method. Find the HCF of the top and the bottom, then divide both numbers by it.
- Find the highest common factor (HCF) of the top and the bottom. List the factors of both numbers. The HCF is the biggest factor that they share.Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 16: 1, 2, 4, 8, 16
HCF = 4 - Divide the top and the bottom by the HCF. Doing the same to both numbers keeps the fraction the same size.12 ÷ 4 = 3 and 16 ÷ 4 = 4
- Check that nothing else divides into both. If the only factor they share is 1, the fraction is in its simplest form.Factors of 3: 1, 3. Factors of 4: 1, 2, 4. Only 1 is shared, so the simplest form is 34.
Check. 3 × 4 = 12 and 4 × 4 = 16. ✓
Watch out. Divide the top AND the bottom. Dividing only one of them changes the amount.
Key words. A fraction is in its simplest form when the only number that divides exactly into its top and bottom is 1. The highest common factor (HCF) of the top and bottom gives you the number to divide by.
Method. Find the HCF of the top and the bottom, then divide both numbers by it.
- Find the highest common factor (HCF) of the top and the bottom. List the factors of both numbers. The HCF is the biggest factor that they share.Factors of 15: 1, 3, 5, 15
Factors of 25: 1, 5, 25
HCF = 5 - Divide the top and the bottom by the HCF. Doing the same to both numbers keeps the fraction the same size.15 ÷ 5 = 3 and 25 ÷ 5 = 5
- Check that nothing else divides into both. If the only factor they share is 1, the fraction is in its simplest form.Factors of 3: 1, 3. Factors of 5: 1, 5. Only 1 is shared, so the simplest form is 35.
Check. 3 × 5 = 15 and 5 × 5 = 25. ✓
Watch out. Divide the top AND the bottom. Dividing only one of them changes the amount.
Key words. A fraction is in its simplest form when the only number that divides exactly into its top and bottom is 1. The highest common factor (HCF) of the top and bottom gives you the number to divide by.
Method. Find the HCF of the top and the bottom, then divide both numbers by it.
- Find the highest common factor (HCF) of the top and the bottom. List the factors of both numbers. The HCF is the biggest factor that they share.Factors of 20: 1, 2, 4, 5, 10, 20
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
HCF = 10 - Divide the top and the bottom by the HCF. Doing the same to both numbers keeps the fraction the same size.20 ÷ 10 = 2 and 30 ÷ 10 = 3
- Check that nothing else divides into both. If the only factor they share is 1, the fraction is in its simplest form.Factors of 2: 1, 2. Factors of 3: 1, 3. Only 1 is shared, so the simplest form is 23.
Check. 2 × 10 = 20 and 3 × 10 = 30. ✓
Watch out. Divide the top AND the bottom. Dividing only one of them changes the amount.
Key words. A fraction is in its simplest form when the only number that divides exactly into its top and bottom is 1. The highest common factor (HCF) of the top and bottom gives you the number to divide by.
Method. Find the HCF of the top and the bottom, then divide both numbers by it.
- Find the highest common factor (HCF) of the top and the bottom. List the factors of both numbers. The HCF is the biggest factor that they share.Factors of 14: 1, 2, 7, 14
Factors of 21: 1, 3, 7, 21
HCF = 7 - Divide the top and the bottom by the HCF. Doing the same to both numbers keeps the fraction the same size.14 ÷ 7 = 2 and 21 ÷ 7 = 3
- Check that nothing else divides into both. If the only factor they share is 1, the fraction is in its simplest form.Factors of 2: 1, 2. Factors of 3: 1, 3. Only 1 is shared, so the simplest form is 23.
Check. 2 × 7 = 14 and 3 × 7 = 21. ✓
Watch out. Divide the top AND the bottom. Dividing only one of them changes the amount.
Key words. A fraction is in its simplest form when the only number that divides exactly into its top and bottom is 1. The highest common factor (HCF) of the top and bottom gives you the number to divide by.
Method. Find the HCF of the top and the bottom, then divide both numbers by it.
- Find the highest common factor (HCF) of the top and the bottom. List the factors of both numbers. The HCF is the biggest factor that they share.Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 27: 1, 3, 9, 27
HCF = 9 - Divide the top and the bottom by the HCF. Doing the same to both numbers keeps the fraction the same size.18 ÷ 9 = 2 and 27 ÷ 9 = 3
- Check that nothing else divides into both. If the only factor they share is 1, the fraction is in its simplest form.Factors of 2: 1, 2. Factors of 3: 1, 3. Only 1 is shared, so the simplest form is 23.
Check. 2 × 9 = 18 and 3 × 9 = 27. ✓
Watch out. Divide the top AND the bottom. Dividing only one of them changes the amount.
Key words. A fraction is in its simplest form when the only number that divides exactly into its top and bottom is 1. The highest common factor (HCF) of the top and bottom gives you the number to divide by.
Method. Find the HCF of the top and the bottom, then divide both numbers by it.
- Find the highest common factor (HCF) of the top and the bottom. List the factors of both numbers. The HCF is the biggest factor that they share.Factors of 16: 1, 2, 4, 8, 16
Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40
HCF = 8 - Divide the top and the bottom by the HCF. Doing the same to both numbers keeps the fraction the same size.16 ÷ 8 = 2 and 40 ÷ 8 = 5
- Check that nothing else divides into both. If the only factor they share is 1, the fraction is in its simplest form.Factors of 2: 1, 2. Factors of 5: 1, 5. Only 1 is shared, so the simplest form is 25.
Check. 2 × 8 = 16 and 5 × 8 = 40. ✓
Watch out. Divide the top AND the bottom. Dividing only one of them changes the amount.
Key words. A fraction is in its simplest form when the only number that divides exactly into its top and bottom is 1. The highest common factor (HCF) of the top and bottom gives you the number to divide by.
Method. Find the HCF of the top and the bottom, then divide both numbers by it.
- Find the highest common factor (HCF) of the top and the bottom. List the factors of both numbers. The HCF is the biggest factor that they share.Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
HCF = 12 - Divide the top and the bottom by the HCF. Doing the same to both numbers keeps the fraction the same size.24 ÷ 12 = 2 and 36 ÷ 12 = 3
- Check that nothing else divides into both. If the only factor they share is 1, the fraction is in its simplest form.Factors of 2: 1, 2. Factors of 3: 1, 3. Only 1 is shared, so the simplest form is 23.
Check. 2 × 12 = 24 and 3 × 12 = 36. ✓
Watch out. Divide the top AND the bottom. Dividing only one of them changes the amount.
Key words. A fraction is in its simplest form when the only number that divides exactly into its top and bottom is 1. The highest common factor (HCF) of the top and bottom gives you the number to divide by.
Method. Find the HCF of the top and the bottom, then divide both numbers by it.
- Find the highest common factor (HCF) of the top and the bottom. List the factors of both numbers. The HCF is the biggest factor that they share.Factors of 35: 1, 5, 7, 35
Factors of 50: 1, 2, 5, 10, 25, 50
HCF = 5 - Divide the top and the bottom by the HCF. Doing the same to both numbers keeps the fraction the same size.35 ÷ 5 = 7 and 50 ÷ 5 = 10
- Check that nothing else divides into both. If the only factor they share is 1, the fraction is in its simplest form.Factors of 7: 1, 7. Factors of 10: 1, 2, 5, 10. Only 1 is shared, so the simplest form is 710.
Check. 7 × 5 = 35 and 10 × 5 = 50. ✓
Watch out. Divide the top AND the bottom. Dividing only one of them changes the amount.
Key words. A fraction is in its simplest form when the only number that divides exactly into its top and bottom is 1. The highest common factor (HCF) of the top and bottom gives you the number to divide by.
Method. Find the HCF of the top and the bottom, then divide both numbers by it.
- Find the highest common factor (HCF) of the top and the bottom. List the factors of both numbers. The HCF is the biggest factor that they share.Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42
Factors of 56: 1, 2, 4, 7, 8, 14, 28, 56
HCF = 14 - Divide the top and the bottom by the HCF. Doing the same to both numbers keeps the fraction the same size.42 ÷ 14 = 3 and 56 ÷ 14 = 4
- Check that nothing else divides into both. If the only factor they share is 1, the fraction is in its simplest form.Factors of 3: 1, 3. Factors of 4: 1, 2, 4. Only 1 is shared, so the simplest form is 34.
Check. 3 × 14 = 42 and 4 × 14 = 56. ✓
Watch out. Divide the top AND the bottom. Dividing only one of them changes the amount.
68 34 912
Key words. A fraction is in its simplest form when the only number that divides exactly into its top and bottom is 1. The highest common factor (HCF) of the top and bottom gives you the number to divide by.
Method. A fraction is in its simplest form when its HCF is 1.
- Find the HCF of the top and the bottom of each fraction. Look for a number that divides into both.68: HCF is 2, so it can be simplified to 34.
34: HCF is 1, so it is already in its simplest form.
912: HCF is 3, so it can be simplified to 34. - Choose the one whose HCF is 1. Only that fraction cannot be simplified any more.34
Check. Divide the others by their HCF and you get fractions that are the same size. ✓
Watch out. A fraction can be simplified if both numbers are even, or if both divide by 3, 5 and so on.
Key words. A fraction is in its simplest form when the only number that divides exactly into its top and bottom is 1. The highest common factor (HCF) of the top and bottom gives you the number to divide by.
Method. Find the HCF of the top and the bottom, then divide both numbers by it.
- Find the highest common factor (HCF) of the top and the bottom. List the factors of both numbers. The HCF is the biggest factor that they share.Factors of 16: 1, 2, 4, 8, 16
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
HCF = 8 - Divide the top and the bottom by the HCF. Doing the same to both numbers keeps the fraction the same size.16 ÷ 8 = 2 and 24 ÷ 8 = 3
- Check that nothing else divides into both. If the only factor they share is 1, the fraction is in its simplest form.Factors of 2: 1, 2. Factors of 3: 1, 3. Only 1 is shared, so the simplest form is 23.
Check. 2 × 8 = 16 and 3 × 8 = 24. ✓
Watch out. Divide the top AND the bottom. Dividing only one of them changes the amount.
Key words. A fraction is in its simplest form when the only number that divides exactly into its top and bottom is 1. The highest common factor (HCF) of the top and bottom gives you the number to divide by.
Method. Find the HCF of the top and the bottom, then divide both numbers by it.
- Find the highest common factor (HCF) of the top and the bottom. List the factors of both numbers. The HCF is the biggest factor that they share.Factors of 14: 1, 2, 7, 14
Factors of 35: 1, 5, 7, 35
HCF = 7 - Divide the top and the bottom by the HCF. Doing the same to both numbers keeps the fraction the same size.14 ÷ 7 = 2 and 35 ÷ 7 = 5
- Check that nothing else divides into both. If the only factor they share is 1, the fraction is in its simplest form.Factors of 2: 1, 2. Factors of 5: 1, 5. Only 1 is shared, so the simplest form is 25.
Check. 2 × 7 = 14 and 5 × 7 = 35. ✓
Watch out. Divide the top AND the bottom. Dividing only one of them changes the amount.
Key words. A fraction is in its simplest form when the only number that divides exactly into its top and bottom is 1. The highest common factor (HCF) of the top and bottom gives you the number to divide by.
Method. Find the HCF of the top and the bottom, then divide both numbers by it.
- Find the highest common factor (HCF) of the top and the bottom. List the factors of both numbers. The HCF is the biggest factor that they share.Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
HCF = 6 - Divide the top and the bottom by the HCF. Doing the same to both numbers keeps the fraction the same size.18 ÷ 6 = 3 and 48 ÷ 6 = 8
- Check that nothing else divides into both. If the only factor they share is 1, the fraction is in its simplest form.Factors of 3: 1, 3. Factors of 8: 1, 2, 4, 8. Only 1 is shared, so the simplest form is 38.
Check. 3 × 6 = 18 and 8 × 6 = 48. ✓
Watch out. Divide the top AND the bottom. Dividing only one of them changes the amount.
Key words. A fraction is in its simplest form when the only number that divides exactly into its top and bottom is 1. The highest common factor (HCF) of the top and bottom gives you the number to divide by.
Method. Find the HCF of the top and the bottom, then divide both numbers by it.
- Find the highest common factor (HCF) of the top and the bottom. List the factors of both numbers. The HCF is the biggest factor that they share.Factors of 50: 1, 2, 5, 10, 25, 50
Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
HCF = 10 - Divide the top and the bottom by the HCF. Doing the same to both numbers keeps the fraction the same size.50 ÷ 10 = 5 and 60 ÷ 10 = 6
- Check that nothing else divides into both. If the only factor they share is 1, the fraction is in its simplest form.Factors of 5: 1, 5. Factors of 6: 1, 2, 3, 6. Only 1 is shared, so the simplest form is 56.
Check. 5 × 10 = 50 and 6 × 10 = 60. ✓
Watch out. Divide the top AND the bottom. Dividing only one of them changes the amount.
Key words. A fraction is in its simplest form when the only number that divides exactly into its top and bottom is 1. The highest common factor (HCF) of the top and bottom gives you the number to divide by.
Method. Find the HCF of the top and the bottom, then divide both numbers by it.
- Find the highest common factor (HCF) of the top and the bottom. List the factors of both numbers. The HCF is the biggest factor that they share.Factors of 7: 1, 7
Factors of 21: 1, 3, 7, 21
HCF = 7 - Divide the top and the bottom by the HCF. Doing the same to both numbers keeps the fraction the same size.7 ÷ 7 = 1 and 21 ÷ 7 = 3
- Check that nothing else divides into both. If the only factor they share is 1, the fraction is in its simplest form.Factors of 1: 1. Factors of 3: 1, 3. Only 1 is shared, so the simplest form is 13.
Check. 1 × 7 = 7 and 3 × 7 = 21. ✓
Watch out. Divide the top AND the bottom. Dividing only one of them changes the amount.
Key words. A fraction is in its simplest form when the only number that divides exactly into its top and bottom is 1. The highest common factor (HCF) of the top and bottom gives you the number to divide by.
Method. Test some examples, then explain why the rule always works.
- Try some fractions where both numbers are even. Test the rule with examples you know.46 → 23
1014 → 57
812 → 46 - Look at why it works. When the top and the bottom are both even, 2 divides into both, so 2 is a common factor and the fraction can be simplified.Dividing both by 2 always works
- Decide. The rule works for every example, and the reason always applies.Always
Check. 2 divides into any even number, so it divides into both the top and the bottom. ✓
Watch out. One example is not a proof. Give a reason that works for every fraction of this kind.
Key words. A fraction is in its simplest form when the only number that divides exactly into its top and bottom is 1. The highest common factor (HCF) of the top and bottom gives you the number to divide by.
Method. Multiply the top and the bottom by k, and count the values of k that keep the bottom number below the limit.
- Write the fraction as a multiple: 25 = 2 × k over 5 × k. Each time we multiply the top and bottom by k, the fraction stays the same. The bottom must be below 50.5 × k must be less than 50, so k can be 1 to 9
- List the fractions for each k. Check that each one has the same value as the target.25
410
615
820
1025
1230
1435
1640
1845 - Count them. Each fraction in the list is one answer.9 fractions
Check. The next one would have a bottom number of 50, which is not below 50. ✓
Watch out. Multiply the top and the bottom by the same k. Adding the same number to both changes the value.