Year 6: Fractions as division and decimals
Part of the Year 6 fractions chapter. Chapter overview · whole chapter on one page
Fractions as division and decimals
A fraction is a division: 38 means 3 ÷ 8. Long division gives the answer 0.375. The number line below is cut into eighths. The orange dot is 38, which is exactly the same place as 0.375.
Decimals are fractions with bottoms of 10, 100, 1000 and so on. So 0.6 = 610 = 35 in its simplest form.
Key ideas
- A fraction is a division: the top number ÷ the bottom number.
- Use long division to change a fraction to a decimal. Add zeros after the point until the remainder is 0.
- A decimal stops only when the simplest bottom number has no factors except 2 and 5. So 13 never stops, but 38 does.
- Change a decimal to a fraction by writing it over 10, 100 or 1000, then simplify.
Key words. A decimal uses a point to show tenths, hundredths and thousandths. A terminating decimal stops after a few digits, such as 0.375.
Method. Use long division. Keep adding zeros until the remainder is 0.
- Add a decimal point and zeros to the top number. The top number is smaller than the bottom, so the answer starts with 0 and a decimal point. Adding zeros after the point does not change the value.3 becomes 3.000
- Divide one step at a time. Multiply the remainder by 10, then divide by the bottom number. Each answer is the next digit.30 ÷ 8 = 3, remainder 6
60 ÷ 8 = 7, remainder 4
40 ÷ 8 = 5, remainder 0 - Write the digits after the point. Stop when the remainder is 0. The decimal stops.38 = 0.375
Check. 0.375 × 8 = 3. ✓
Watch out. If the remainder never reaches 0, the decimal goes on forever. Check the bottom number has only factors 2 and 5.
Key words. A decimal uses a point to show tenths, hundredths and thousandths. A terminating decimal stops after a few digits, such as 0.375.
Method. Write the decimal as a fraction over a power of ten, then simplify.
- Write the decimal over a power of ten. Each digit after the point is a tenth, hundredth or thousandth. There are 1 digit after the point, so the bottom number is 10.0.6 = 610
- Find the HCF of the top and the bottom. Divide both by the biggest number that goes into both.HCF of 6 and 10 = 2
- Simplify. Divide the top and the bottom by the HCF.6 ÷ 2 = 3 and 10 ÷ 2 = 5, so 35
Check. 3 ÷ 5 = 0.6, and 0.6 is the same. ✓
Watch out. Simplify the fraction fully. A decimal like 0.6 is 6/10, which simplifies to 3/5.
Key words. A decimal uses a point to show tenths, hundredths and thousandths. A terminating decimal stops after a few digits, such as 0.375.
Method. Use long division. Keep adding zeros until the remainder is 0.
- Add a decimal point and zeros to the top number. The top number is smaller than the bottom, so the answer starts with 0 and a decimal point. Adding zeros after the point does not change the value.7 becomes 7.000
- Divide one step at a time. Multiply the remainder by 10, then divide by the bottom number. Each answer is the next digit.70 ÷ 20 = 3, remainder 10
100 ÷ 20 = 5, remainder 0 - Write the digits after the point. Stop when the remainder is 0. The decimal stops.720 = 0.35
Check. 0.35 × 20 = 7. ✓
Watch out. If the remainder never reaches 0, the decimal goes on forever. Check the bottom number has only factors 2 and 5.
Key words. A decimal uses a point to show tenths, hundredths and thousandths. A terminating decimal stops after a few digits, such as 0.375.
Method. Write the fraction as a decimal, then compare the decimals digit by digit.
- Write the fraction as a decimal. Use long division. The answer is a decimal that stops.38 = 0.375
- Write both decimals with the same number of places. Add zeros at the end so the decimal points line up.0.375 and 0.400
- Compare the digits from the left. The first digit that is different tells you which one is bigger.0.375 and 0.4: the first difference is at the same place value
- Say which one is bigger. Write the bigger decimal.0.4 is bigger
Check. 0.375 is 38 exactly, and 0.4 is the bigger value. ✓
Watch out. Compare decimals from the left, one place at a time. 0.4 is bigger than 0.375, because 0.400 is bigger than 0.375.
Watch out
Do not stop long division too early. 38 is 0.375, not 0.3 or 0.38. The decimal stops when the remainder is 0. Also, 0.38 is a rounded answer, so it is not exactly 38.Exercise 4G Fractions and decimals
Non-calculatorKey words. A decimal uses a point to show tenths, hundredths and thousandths. A terminating decimal stops after a few digits, such as 0.375.
Method. Use long division. Keep adding zeros until the remainder is 0.
- Add a decimal point and zeros to the top number. The top number is smaller than the bottom, so the answer starts with 0 and a decimal point. Adding zeros after the point does not change the value.1 becomes 1.000
- Divide one step at a time. Multiply the remainder by 10, then divide by the bottom number. Each answer is the next digit.10 ÷ 4 = 2, remainder 2
20 ÷ 4 = 5, remainder 0 - Write the digits after the point. Stop when the remainder is 0. The decimal stops.14 = 0.25
Check. 0.25 × 4 = 1. ✓
Watch out. If the remainder never reaches 0, the decimal goes on forever. Check the bottom number has only factors 2 and 5.
Key words. A decimal uses a point to show tenths, hundredths and thousandths. A terminating decimal stops after a few digits, such as 0.375.
Method. Use long division. Keep adding zeros until the remainder is 0.
- Add a decimal point and zeros to the top number. The top number is smaller than the bottom, so the answer starts with 0 and a decimal point. Adding zeros after the point does not change the value.3 becomes 3.000
- Divide one step at a time. Multiply the remainder by 10, then divide by the bottom number. Each answer is the next digit.30 ÷ 5 = 6, remainder 0
- Write the digits after the point. Stop when the remainder is 0. The decimal stops.35 = 0.6
Check. 0.6 × 5 = 3. ✓
Watch out. If the remainder never reaches 0, the decimal goes on forever. Check the bottom number has only factors 2 and 5.
Key words. A decimal uses a point to show tenths, hundredths and thousandths. A terminating decimal stops after a few digits, such as 0.375.
Method. Use long division. Keep adding zeros until the remainder is 0.
- Add a decimal point and zeros to the top number. The top number is smaller than the bottom, so the answer starts with 0 and a decimal point. Adding zeros after the point does not change the value.5 becomes 5.000
- Divide one step at a time. Multiply the remainder by 10, then divide by the bottom number. Each answer is the next digit.50 ÷ 8 = 6, remainder 2
20 ÷ 8 = 2, remainder 4
40 ÷ 8 = 5, remainder 0 - Write the digits after the point. Stop when the remainder is 0. The decimal stops.58 = 0.625
Check. 0.625 × 8 = 5. ✓
Watch out. If the remainder never reaches 0, the decimal goes on forever. Check the bottom number has only factors 2 and 5.
Key words. A decimal uses a point to show tenths, hundredths and thousandths. A terminating decimal stops after a few digits, such as 0.375.
Method. Use long division. Keep adding zeros until the remainder is 0.
- Add a decimal point and zeros to the top number. The top number is smaller than the bottom, so the answer starts with 0 and a decimal point. Adding zeros after the point does not change the value.7 becomes 7.000
- Divide one step at a time. Multiply the remainder by 10, then divide by the bottom number. Each answer is the next digit.70 ÷ 20 = 3, remainder 10
100 ÷ 20 = 5, remainder 0 - Write the digits after the point. Stop when the remainder is 0. The decimal stops.720 = 0.35
Check. 0.35 × 20 = 7. ✓
Watch out. If the remainder never reaches 0, the decimal goes on forever. Check the bottom number has only factors 2 and 5.
Key words. A decimal uses a point to show tenths, hundredths and thousandths. A terminating decimal stops after a few digits, such as 0.375.
Method. Use long division. Keep adding zeros until the remainder is 0.
- Add a decimal point and zeros to the top number. The top number is smaller than the bottom, so the answer starts with 0 and a decimal point. Adding zeros after the point does not change the value.9 becomes 9.000
- Divide one step at a time. Multiply the remainder by 10, then divide by the bottom number. Each answer is the next digit.90 ÷ 25 = 3, remainder 15
150 ÷ 25 = 6, remainder 0 - Write the digits after the point. Stop when the remainder is 0. The decimal stops.925 = 0.36
Check. 0.36 × 25 = 9. ✓
Watch out. If the remainder never reaches 0, the decimal goes on forever. Check the bottom number has only factors 2 and 5.
Key words. A decimal uses a point to show tenths, hundredths and thousandths. A terminating decimal stops after a few digits, such as 0.375.
Method. Use long division. Keep adding zeros until the remainder is 0.
- Add a decimal point and zeros to the top number. The top number is smaller than the bottom, so the answer starts with 0 and a decimal point. Adding zeros after the point does not change the value.1 becomes 1.000
- Divide one step at a time. Multiply the remainder by 10, then divide by the bottom number. Each answer is the next digit.10 ÷ 8 = 1, remainder 2
20 ÷ 8 = 2, remainder 4
40 ÷ 8 = 5, remainder 0 - Write the digits after the point. Stop when the remainder is 0. The decimal stops.18 = 0.125
Check. 0.125 × 8 = 1. ✓
Watch out. If the remainder never reaches 0, the decimal goes on forever. Check the bottom number has only factors 2 and 5.
Key words. A decimal uses a point to show tenths, hundredths and thousandths. A terminating decimal stops after a few digits, such as 0.375.
Method. Write the decimal as a fraction over a power of ten, then simplify.
- Write the decimal over a power of ten. Each digit after the point is a tenth, hundredth or thousandth. There are 1 digit after the point, so the bottom number is 10.0.5 = 510
- Find the HCF of the top and the bottom. Divide both by the biggest number that goes into both.HCF of 5 and 10 = 5
- Simplify. Divide the top and the bottom by the HCF.5 ÷ 5 = 1 and 10 ÷ 5 = 2, so 12
Check. 1 ÷ 2 = 0.5, and 0.5 is the same. ✓
Watch out. Simplify the fraction fully. A decimal like 0.6 is 6/10, which simplifies to 3/5.
Key words. A decimal uses a point to show tenths, hundredths and thousandths. A terminating decimal stops after a few digits, such as 0.375.
Method. Write the decimal as a fraction over a power of ten, then simplify.
- Write the decimal over a power of ten. Each digit after the point is a tenth, hundredth or thousandth. There are 2 digits after the point, so the bottom number is 100.0.75 = 75100
- Find the HCF of the top and the bottom. Divide both by the biggest number that goes into both.HCF of 75 and 100 = 25
- Simplify. Divide the top and the bottom by the HCF.75 ÷ 25 = 3 and 100 ÷ 25 = 4, so 34
Check. 3 ÷ 4 = 0.75, and 0.75 is the same. ✓
Watch out. Simplify the fraction fully. A decimal like 0.6 is 6/10, which simplifies to 3/5.
Key words. A decimal uses a point to show tenths, hundredths and thousandths. A terminating decimal stops after a few digits, such as 0.375.
Method. Write the decimal as a fraction over a power of ten, then simplify.
- Write the decimal over a power of ten. Each digit after the point is a tenth, hundredth or thousandth. There are 1 digit after the point, so the bottom number is 10.0.2 = 210
- Find the HCF of the top and the bottom. Divide both by the biggest number that goes into both.HCF of 2 and 10 = 2
- Simplify. Divide the top and the bottom by the HCF.2 ÷ 2 = 1 and 10 ÷ 2 = 5, so 15
Check. 1 ÷ 5 = 0.2, and 0.2 is the same. ✓
Watch out. Simplify the fraction fully. A decimal like 0.6 is 6/10, which simplifies to 3/5.
Key words. A decimal uses a point to show tenths, hundredths and thousandths. A terminating decimal stops after a few digits, such as 0.375.
Method. Write the decimal as a fraction over a power of ten, then simplify.
- Write the decimal over a power of ten. Each digit after the point is a tenth, hundredth or thousandth. There are 2 digits after the point, so the bottom number is 100.0.45 = 45100
- Find the HCF of the top and the bottom. Divide both by the biggest number that goes into both.HCF of 45 and 100 = 5
- Simplify. Divide the top and the bottom by the HCF.45 ÷ 5 = 9 and 100 ÷ 5 = 20, so 920
Check. 9 ÷ 20 = 0.45, and 0.45 is the same. ✓
Watch out. Simplify the fraction fully. A decimal like 0.6 is 6/10, which simplifies to 3/5.
Key words. A decimal uses a point to show tenths, hundredths and thousandths. A terminating decimal stops after a few digits, such as 0.375.
Method. Write the decimal as a fraction over a power of ten, then simplify.
- Write the decimal over a power of ten. Each digit after the point is a tenth, hundredth or thousandth. There are 2 digits after the point, so the bottom number is 100.0.64 = 64100
- Find the HCF of the top and the bottom. Divide both by the biggest number that goes into both.HCF of 64 and 100 = 4
- Simplify. Divide the top and the bottom by the HCF.64 ÷ 4 = 16 and 100 ÷ 4 = 25, so 1625
Check. 16 ÷ 25 = 0.64, and 0.64 is the same. ✓
Watch out. Simplify the fraction fully. A decimal like 0.6 is 6/10, which simplifies to 3/5.
Key words. A decimal uses a point to show tenths, hundredths and thousandths. A terminating decimal stops after a few digits, such as 0.375.
Method. Write the decimal as a fraction over a power of ten, then simplify.
- Write the decimal over a power of ten. Each digit after the point is a tenth, hundredth or thousandth. There are 3 digits after the point, so the bottom number is 1000.0.125 = 1251000
- Find the HCF of the top and the bottom. Divide both by the biggest number that goes into both.HCF of 125 and 1000 = 125
- Simplify. Divide the top and the bottom by the HCF.125 ÷ 125 = 1 and 1000 ÷ 125 = 8, so 18
Check. 1 ÷ 8 = 0.125, and 0.125 is the same. ✓
Watch out. Simplify the fraction fully. A decimal like 0.6 is 6/10, which simplifies to 3/5.
Key words. A decimal uses a point to show tenths, hundredths and thousandths. A terminating decimal stops after a few digits, such as 0.375.
Method. Change the fractions to decimals so that you can compare them easily.
- Write every number as a decimal. Long division turns each fraction into a decimal.14 = 0.25
0.3 is already a decimal
38 = 0.375 - Compare the decimals. Compare the tenths first, then the hundredths and so on.0.25, 0.3, 0.375
- Write them in order, using the numbers from the question. Smallest first means the smallest decimal comes first.14, 0.3, 38
Check. 0.25 is the smallest, and 0.375 is the biggest. ✓
Watch out. Write the decimals in order to check your answer, but give the answer in the form the question uses.
Key words. A decimal uses a point to show tenths, hundredths and thousandths. A terminating decimal stops after a few digits, such as 0.375.
Method. Use long division. Keep adding zeros until the remainder is 0.
- Add a decimal point and zeros to the top number. The top number is smaller than the bottom, so the answer starts with 0 and a decimal point. Adding zeros after the point does not change the value.11 becomes 11.000
- Divide one step at a time. Multiply the remainder by 10, then divide by the bottom number. Each answer is the next digit.110 ÷ 20 = 5, remainder 10
100 ÷ 20 = 5, remainder 0 - Write the digits after the point. Stop when the remainder is 0. The decimal stops.1120 = 0.55
Check. 0.55 × 20 = 11. ✓
Watch out. If the remainder never reaches 0, the decimal goes on forever. Check the bottom number has only factors 2 and 5.
Key words. A decimal uses a point to show tenths, hundredths and thousandths. A terminating decimal stops after a few digits, such as 0.375.
Method. Use long division. Keep adding zeros until the remainder is 0.
- Add a decimal point and zeros to the top number. The top number is smaller than the bottom, so the answer starts with 0 and a decimal point. Adding zeros after the point does not change the value.3 becomes 3.000
- Divide one step at a time. Multiply the remainder by 10, then divide by the bottom number. Each answer is the next digit.30 ÷ 8 = 3, remainder 6
60 ÷ 8 = 7, remainder 4
40 ÷ 8 = 5, remainder 0 - Write the digits after the point. Stop when the remainder is 0. The decimal stops.38 = 0.375
Check. 0.375 × 8 = 3. ✓
Watch out. If the remainder never reaches 0, the decimal goes on forever. Check the bottom number has only factors 2 and 5.
Key words. A decimal uses a point to show tenths, hundredths and thousandths. A terminating decimal stops after a few digits, such as 0.375.
Method. Test some examples. A decimal with a bottom of 10 has one digit after the point, because one tenth is one place.
- Try some fractions with a bottom number of 10. Find the decimal for each one.310 = 0.3
2710 = 2.7
910 = 0.9 - Look at the place value. Every fraction with a bottom of 10 has its value in tenths, so the decimal has one place.One digit after the point, every time
- Decide. The rule works for every top number, so it is always true.Always
Check. 2710 = 2.7, which has one digit after the point. ✓
Watch out. A fraction bigger than 1, such as 27/10, still has one decimal place: 2.7.
Key words. A decimal uses a point to show tenths, hundredths and thousandths. A terminating decimal stops after a few digits, such as 0.375.
Method. Simplify each fraction, then check its bottom number. Only bottoms made from 2s and 5s give a decimal that stops.
- Simplify each fraction and look at its bottom number. Simplify first, so that you see the real bottom number.13: simplest bottom number is 3, so the decimal goes on forever
38: simplest bottom number is 8, so the decimal stops
27: simplest bottom number is 7, so the decimal goes on forever
520: simplest bottom number is 4, so the decimal stops - Check the bottom number. A decimal stops only if the bottom number has no factors except 2 and 5.So 8 and 4 stop, but 3 and 7 do not
- Write the ones that stop. These fractions have a decimal that stops.38 and 520
Check. 0.375 stops, but 1/3 = 0.333... goes on forever. ✓
Watch out. 1/3 and 2/7 never stop. The decimal repeats, so it cannot be written exactly with a few digits.
Key words. A decimal uses a point to show tenths, hundredths and thousandths. A terminating decimal stops after a few digits, such as 0.375.
Method. Write the fraction as a decimal, then compare the decimals digit by digit.
- Write the fraction as a decimal. Use long division. The answer is a decimal that stops.38 = 0.375
- Write both decimals with the same number of places. Add zeros at the end so the decimal points line up.0.375 and 0.400
- Compare the digits from the left. The first digit that is different tells you which one is bigger.0.375 and 0.4: the first difference is at the same place value
- Say which one is bigger. Write the bigger decimal.0.4 is bigger
Check. 0.375 is 38 exactly, and 0.4 is the bigger value. ✓
Watch out. Compare decimals from the left, one place at a time. 0.4 is bigger than 0.375, because 0.400 is bigger than 0.375.
Key words. A decimal uses a point to show tenths, hundredths and thousandths. A terminating decimal stops after a few digits, such as 0.375.
Method. Write the decimal as a fraction over a power of ten, then simplify.
- Write the decimal over a power of ten. Each digit after the point is a tenth, hundredth or thousandth. There are 1 digit after the point, so the bottom number is 10.0.8 = 810
- Find the HCF of the top and the bottom. Divide both by the biggest number that goes into both.HCF of 8 and 10 = 2
- Simplify. Divide the top and the bottom by the HCF.8 ÷ 2 = 4 and 10 ÷ 2 = 5, so 45
Check. 4 ÷ 5 = 0.8, and 0.8 is the same. ✓
Watch out. Simplify the fraction fully. A decimal like 0.6 is 6/10, which simplifies to 3/5.
Key words. A decimal uses a point to show tenths, hundredths and thousandths. A terminating decimal stops after a few digits, such as 0.375.
Method. Write 0.15 as a fraction, then multiply it by 8.
- Write 0.15 as a fraction in its simplest form. There are two digits after the point, so the bottom is 100. Then simplify.0.15 = 15100 = 320
- Multiply by 8. Multiply the top number by 8, and keep the bottom number.320 × 8 = 2420
- Simplify, then write as a decimal. Divide the top and bottom by their HCF, 4. Then 1 1/5 is one whole and one fifth, and 1/5 = 0.2.2420 = 65 = 1 15 = 1.2
Check. 1.2 × 5 = 6, so 1.2 = 6/5. ✓
Watch out. Remember that 1 1/5 is 1.2, because 1/5 = 0.2.