Year 6: Solving problems with decimals
Part of the Year 6 decimals chapter. Chapter overview · whole chapter on one page
Solving problems with decimals
Most problems about money and measures need more than one step. Read the question twice. Decide which operations to use, in what order, and whether the answer needs rounding. Estimate first, so that you know roughly what the answer should be.
Money is usually written with two decimal places, so £3 is written as £3.00. Measures use the same idea: 2.5 metres is two and a half metres, and 0.5 of a litre is half a litre.
Key ideas
- Read the question twice. Write down what you need to find, then choose the operations.
- Estimate before you calculate, so you can see if your answer is about the right size.
- Keep two decimal places for money, and check the number of places in the final answer.
- Round only at the end, and to the degree of accuracy the question needs.
Key words. A decimal place is a digit after the decimal point. The answer to a multiplication has the same number of decimal places as the decimal number in the question.
Method. Multiply as if there were no decimal point. Then put the point back so the answer has 2 decimal places.
- Write the numbers in columns. Line up the numbers on the right. Ignore the decimal point for now.
0.89 × 3 ---- 2.67
- Multiply each digit, starting from the right. Write the last digit of each answer and carry the tens to the next column.9 × 3 = 27, write 7 and carry 2
8 × 3 = 24 + 2 carried = 26, write 6 and carry 2
write 2 - Count the decimal places and put the point back. The decimal number has 2 decimal places, so the answer has 2 decimal places.89 × 3 = 267, and 267 → 2.67
Check. About 1 × 3 = 3, and 2.67 is close to that. ✓
Watch out. Count the decimal places in the question. Do not count the zeros you carry.
Key words. A decimal place is a digit after the decimal point. The answer to a multiplication has the same number of decimal places as the decimal number in the question.
Method. Add the cost of the drink to the cost of the bags.
- Write the two amounts. Line up the decimal points.2.67 and 1.15
- Add them. Add the pence first, then the pounds.2.67 + 1.15 = 3.82
Check. Round to check: about £3 + £1 = £4, and 3.82 is close. ✓
Watch out. Add the bags first, then the drink. Each step must use the correct number of decimal places.
Key words. A decimal place is a digit after the decimal point. The answer to a multiplication has the same number of decimal places as the decimal number in the question.
Method. Multiply as if there were no decimal point. Then put the point back so the answer has 2 decimal places.
- Write the numbers in columns. Line up the numbers on the right. Ignore the decimal point for now.
0.35 × 8 ---- 2.80
- Multiply each digit, starting from the right. Write the last digit of each answer and carry the tens to the next column.5 × 8 = 40, write 0 and carry 4
3 × 8 = 24 + 4 carried = 28, write 8 and carry 2
write 2 - Count the decimal places and put the point back. The decimal number has 2 decimal places, so the answer has 2 decimal places.35 × 8 = 280, and 280 → 2.80
Check. About 0 × 8 = 0, and 2.80 is close to that. ✓
Watch out. Count the decimal places in the question. Do not count the zeros you carry.
Key words. In long division, the dividend is the number being divided, the divisor is the number we divide by, and the remainder is what is left over. Bring down a 0 after the decimal point to keep going.
Method. Share the number out one digit at a time. Put the decimal point in the answer when you reach it in the number.
- Set out the layout. The number being divided goes inside the bracket. The answer goes above it.
1.66 6 ) 10.00 0 - 10 6 -- 40 36 -- 40 36 -- 4 - Start with the first digit, 1. 6 does not go into 1, so this digit of the answer is 0.1 ÷ 6 = 0, remainder 1
- Bring down the next digit, so we have 10. How many 6s go into 10? 1, because 1 × 6 = 6.10 ÷ 6 = 1, remainder 4
- Bring down a 0, so we have 40. Write the decimal point in the answer now, above the point. How many 6s go into 40? 6, because 6 × 6 = 36.40 ÷ 6 = 6, remainder 4
- Bring down a 0, so we have 40. How many 6s go into 40? 6, because 6 × 6 = 36.40 ÷ 6 = 6, remainder 4
- Write the answer. Keep going until the remainder is 0, or until you have the places you need.1.66
Check. Multiply back: 1.66 × 6 is close to 10.
Watch out. Do not forget the decimal point in the answer. Bring down zeros only after you have used all the digits of the number.
Key words. To round to a degree of accuracy, find the place you are rounding to and look at the digit to its right. 5 or more means round up. Less than 5 means round down.
Method. Find the place, look at the digit to its right, then round up or stay the same.
- Find the place you are rounding to. We want 1 decimal place, so look at the tenths column.1.66: the digit in the tenths column is 6
- Look at the digit to its right. This digit decides whether the number goes up or stays the same.The digit to the right is 6
- Decide whether to round up or down. If the digit is 5 or more, round up. If it is less than 5, keep the same digit.6 is 5 or more, so round up
- Write the answer. Keep the zeros after the point so the accuracy is clear.1.7
Check. 1.66 lies between 1.6 and 1.7, and 1.7 is the nearer one. ✓
Watch out. Look only at the digit to the right of the place you are rounding to. Do not round the digits one at a time.
Watch out
Do not give the answer without checking its size. 3.5 × 4 cannot be 1.4, because 3 × 4 is 12. Estimate first, then check.
Exam tip
Exam tip. Write the units in the answer, as the question asks. "Give your answer in pounds" means £3.45, not 345 pence.
Exercise 5F Solving problems with decimals
Non-calculatorKey words. The decimal point separates the ones from the part smaller than 1. The places after it are tenths (0.1), hundredths (0.01) and thousandths (0.001).
Method. Line up the decimal points, then add each column from the right.
- Write the numbers under each other. Line up the decimal points in one column. Fill the gaps with zeros.
1.25 + 4.80 + 0.95 ------ 7.00
- Add column by column, from the right. Carry any ten into the next column to the left.1.25 + 4.80 + 0.95 = 7.00
- Write the answer with its decimal point in the same column. The decimal points line up in the question and in the answer.7.00
Check. Add the numbers in a different order and you get 7.00 again. ✓
Watch out. Line up the decimal points, not the last digits. Fill the gaps with zeros.
Key words. A decimal place is a digit after the decimal point. The answer to a multiplication has the same number of decimal places as the decimal number in the question.
Method. Work out the change from £10.00.
- Write £10 with two decimal places. This lines up the subtraction.10.00 − 7.00
- Subtract. Borrow from the column on the left if you need to.10.00 − 7.00 = 3.00
- Write the answer. This is the change.3.00
Check. 7.00 + 3.00 = 10.00. ✓
Watch out. Check that the total plus the change makes the amount paid.
Key words. In long division, the dividend is the number being divided, the divisor is the number we divide by, and the remainder is what is left over. Bring down a 0 after the decimal point to keep going.
Method. Share the number out one digit at a time. Put the decimal point in the answer when you reach it in the number.
- Set out the layout. The number being divided goes inside the bracket. The answer goes above it.
0.75 6 ) 4.50 0 - 45 42 -- 30 30 -- 0 - Start with the first digit, 4. 6 does not go into 4, so this digit of the answer is 0.4 ÷ 6 = 0, remainder 4
- Bring down the next digit, so we have 45. Write the decimal point in the answer now, above the point. How many 6s go into 45? 7, because 7 × 6 = 42.45 ÷ 6 = 7, remainder 3
- Bring down a 0, so we have 30. How many 6s go into 30? 5, because 5 × 6 = 30.30 ÷ 6 = 5, remainder 0
- Write the answer. Keep going until the remainder is 0, or until you have the places you need.0.75
Check. 0.75 × 6 = 4.5. ✓
Watch out. Do not forget the decimal point in the answer. Bring down zeros only after you have used all the digits of the number.
Key words. A decimal place is a digit after the decimal point. The answer to a multiplication has the same number of decimal places as the decimal number in the question.
Method. Multiply as if there were no decimal point. Then put the point back so the answer has 2 decimal places.
- Write the numbers in columns. Line up the numbers on the right. Ignore the decimal point for now.
3.85 × 5 ----- 19.25
- Multiply each digit, starting from the right. Write the last digit of each answer and carry the tens to the next column.5 × 5 = 25, write 5 and carry 2
8 × 5 = 40 + 2 carried = 42, write 2 and carry 4
3 × 5 = 15 + 4 carried = 19, write 9 and carry 1
write 1 - Count the decimal places and put the point back. The decimal number has 2 decimal places, so the answer has 2 decimal places.385 × 5 = 1925, and 1925 → 19.25
Check. About 4 × 5 = 20, and 19.25 is close to that. ✓
Watch out. Count the decimal places in the question. Do not count the zeros you carry.
Key words. In long division, the dividend is the number being divided, the divisor is the number we divide by, and the remainder is what is left over. Bring down a 0 after the decimal point to keep going.
Method. Share the number out one digit at a time. Put the decimal point in the answer when you reach it in the number.
- Set out the layout. The number being divided goes inside the bracket. The answer goes above it.
0.65 4 ) 2.60 0 - 26 24 -- 20 20 -- 0 - Start with the first digit, 2. 4 does not go into 2, so this digit of the answer is 0.2 ÷ 4 = 0, remainder 2
- Bring down the next digit, so we have 26. Write the decimal point in the answer now, above the point. How many 4s go into 26? 6, because 6 × 4 = 24.26 ÷ 4 = 6, remainder 2
- Bring down a 0, so we have 20. How many 4s go into 20? 5, because 5 × 4 = 20.20 ÷ 4 = 5, remainder 0
- Write the answer. Keep going until the remainder is 0, or until you have the places you need.0.65
Check. 0.65 × 4 = 2.6. ✓
Watch out. Do not forget the decimal point in the answer. Bring down zeros only after you have used all the digits of the number.
Key words. A decimal place is a digit after the decimal point. The answer to a multiplication has the same number of decimal places as the decimal number in the question.
Method. Multiply as if there were no decimal point. Then put the point back so the answer has 2 decimal places.
- Write the numbers in columns. Line up the numbers on the right. Ignore the decimal point for now.
2.40 × 6 ----- 14.40
- Multiply each digit, starting from the right. Write the last digit of each answer and carry the tens to the next column.0 × 6 = 0, write 0
4 × 6 = 24, write 4 and carry 2
2 × 6 = 12 + 2 carried = 14, write 4 and carry 1
write 1 - Count the decimal places and put the point back. The decimal number has 2 decimal places, so the answer has 2 decimal places.240 × 6 = 1440, and 1440 → 14.40
Check. About 2 × 6 = 12, and 14.40 is close to that. ✓
Watch out. Count the decimal places in the question. Do not count the zeros you carry.
Key words. In long division, the dividend is the number being divided, the divisor is the number we divide by, and the remainder is what is left over. Bring down a 0 after the decimal point to keep going.
Method. Share the number out one digit at a time. Put the decimal point in the answer when you reach it in the number.
- Set out the layout. The number being divided goes inside the bracket. The answer goes above it.
9.75 2 ) 19.50 0 - 19 18 -- 15 14 -- 10 10 -- 0 - Start with the first digit, 1. 2 does not go into 1, so this digit of the answer is 0.1 ÷ 2 = 0, remainder 1
- Bring down the next digit, so we have 19. How many 2s go into 19? 9, because 9 × 2 = 18.19 ÷ 2 = 9, remainder 1
- Bring down the next digit, so we have 15. Write the decimal point in the answer now, above the point. How many 2s go into 15? 7, because 7 × 2 = 14.15 ÷ 2 = 7, remainder 1
- Bring down a 0, so we have 10. How many 2s go into 10? 5, because 5 × 2 = 10.10 ÷ 2 = 5, remainder 0
- Write the answer. Keep going until the remainder is 0, or until you have the places you need.9.75
Check. 9.75 × 2 = 19.5. ✓
Watch out. Do not forget the decimal point in the answer. Bring down zeros only after you have used all the digits of the number.
Key words. To round to a degree of accuracy, find the place you are rounding to and look at the digit to its right. 5 or more means round up. Less than 5 means round down.
Method. A bottle holds 2 litres. Decide whether to round up or down.
- Read the answer. 9.75 bottles would be 9 full bottles and part of one.9.75 bottles
- Decide. The school needs all the juice, so an extra bottle is needed. Round up.9.75 → 10 bottles
- Write the answer. Ten bottles are needed.10 bottles
Check. 10 bottles hold 10 × 2 = 20 litres, which is enough for 19.5 litres. ✓
Watch out. Rounding down would leave some juice out. Read the context before rounding.
Key words. A decimal place is a digit after the decimal point. The answer to a multiplication has the same number of decimal places as the decimal number in the question.
Method. Multiply as if there were no decimal point. Then put the point back so the answer has 2 decimal places.
- Write the numbers in columns. Line up the numbers on the right. Ignore the decimal point for now.
2.65 × 3 ---- 7.95
- Multiply each digit, starting from the right. Write the last digit of each answer and carry the tens to the next column.5 × 3 = 15, write 5 and carry 1
6 × 3 = 18 + 1 carried = 19, write 9 and carry 1
2 × 3 = 6 + 1 carried = 7, write 7 - Count the decimal places and put the point back. The decimal number has 2 decimal places, so the answer has 2 decimal places.265 × 3 = 795, and 795 → 7.95
Check. About 3 × 3 = 9, and 7.95 is close to that. ✓
Watch out. Count the decimal places in the question. Do not count the zeros you carry.
Key words. A decimal place is a digit after the decimal point. The answer to a multiplication has the same number of decimal places as the decimal number in the question.
Method. Take the cost of the cakes from the £20.00 Mia has.
- Write £20 with two decimal places. This lines up the subtraction.20.00 − 7.95
- Subtract. Borrow when you need to.20.00 − 7.95 = 12.05
- Write the answer. This is the money left.12.05
Check. 7.95 + 12.05 = 20.00. ✓
Watch out. The answer must be less than £20 because she spent money.
Key words. In long division, the dividend is the number being divided, the divisor is the number we divide by, and the remainder is what is left over. Bring down a 0 after the decimal point to keep going.
Method. Share the number out one digit at a time. Put the decimal point in the answer when you reach it in the number.
- Set out the layout. The number being divided goes inside the bracket. The answer goes above it.
0.60 8 ) 4.80 0 - 48 48 -- 0 0 - 0 - Start with the first digit, 4. 8 does not go into 4, so this digit of the answer is 0.4 ÷ 8 = 0, remainder 4
- Bring down the next digit, so we have 48. Write the decimal point in the answer now, above the point. How many 8s go into 48? 6, because 6 × 8 = 48.48 ÷ 8 = 6, remainder 0
- Bring down the next digit, so we have 0. 8 does not go into 0, so this digit of the answer is 0.0 ÷ 8 = 0, remainder 0
- Write the answer. Keep going until the remainder is 0, or until you have the places you need.0.60
Check. 0.60 × 8 = 4.80. ✓
Watch out. Do not forget the decimal point in the answer. Bring down zeros only after you have used all the digits of the number.
Key words. A decimal place is a digit after the decimal point. The answer to a multiplication has the same number of decimal places as the decimal number in the question.
Method. Multiply as if there were no decimal point. Then put the point back so the answer has 1 decimal place.
- Write the numbers in columns. Line up the numbers on the right. Ignore the decimal point for now.
2.6 × 4 ---- 10.4
- Multiply each digit, starting from the right. Write the last digit of each answer and carry the tens to the next column.6 × 4 = 24, write 4 and carry 2
2 × 4 = 8 + 2 carried = 10, write 0 and carry 1
write 1 - Count the decimal places and put the point back. The decimal number has 1 decimal place, so the answer has 1 decimal place.26 × 4 = 104, and 104 → 10.4
Check. About 3 × 4 = 12, and 10.4 is close to that. ✓
Watch out. Count the decimal places in the question. Do not count the zeros you carry.
Key words. In long division, the dividend is the number being divided, the divisor is the number we divide by, and the remainder is what is left over. Bring down a 0 after the decimal point to keep going.
Method. Share the number out one digit at a time. Put the decimal point in the answer when you reach it in the number.
- Set out the layout. The number being divided goes inside the bracket. The answer goes above it.
0.9 6 ) 5.4 0 - 54 54 -- 0 - Start with the first digit, 5. 6 does not go into 5, so this digit of the answer is 0.5 ÷ 6 = 0, remainder 5
- Bring down the next digit, so we have 54. Write the decimal point in the answer now, above the point. How many 6s go into 54? 9, because 9 × 6 = 54.54 ÷ 6 = 9, remainder 0
- Write the answer. Keep going until the remainder is 0, or until you have the places you need.0.9
Check. 0.9 × 6 = 5.4. ✓
Watch out. Do not forget the decimal point in the answer. Bring down zeros only after you have used all the digits of the number.
Key words. In long division, the dividend is the number being divided, the divisor is the number we divide by, and the remainder is what is left over. Bring down a 0 after the decimal point to keep going.
Method. Multiply the answer by 6 to check it.
- Multiply the answer by the divisor. If the division is correct, the answer times 6 is 5.4.0.9 × 6 = 5.4
- Compare. The result matches the number being divided.5.4 = 5.4
- Decide. Tom is right.Yes, 0.9 × 6 = 5.4.
Check. 0.9 × 6 = 5.4. ✓
Watch out. Multiplying is the way to check a division.
Key words. A decimal place is a digit after the decimal point. The answer to a multiplication has the same number of decimal places as the decimal number in the question.
Method. Test some examples, then decide whether the rule always works.
- Test an example. Pick a number bigger than 0 and multiply it by 2.0.4 × 2 = 0.8, and 0.8 is bigger than 0.4
- Test another example. Try a number with two decimal places.2.75 × 2 = 5.5, and 5.5 is bigger than 2.75
- Decide. Multiplying a number bigger than 0 by 2 always gives a bigger number.Always
Check. Doubling a number bigger than 0 always makes it bigger. ✓
Watch out. A number bigger than 0 gets bigger when you double it. The question only asks about numbers bigger than 0.
Key words. In long division, the dividend is the number being divided, the divisor is the number we divide by, and the remainder is what is left over. Bring down a 0 after the decimal point to keep going.
Method. Divide the length by the piece length, and look at what is left over.
- Find how many 0.5 m pieces fit. Each piece is 0.5 m, so dividing by 0.5 tells us how many pieces fit.3.75 ÷ 0.5 = 7.5
- Count the full pieces. Only whole pieces count, so take the whole number part.7 full pieces
- Find what is left. The leftover is 3.75 minus 7 pieces of 0.5 m.7 × 0.5 = 3.5, and 3.75 − 3.5 = 0.25 m left
- Write the answer. Ben cuts 7 full pieces.7
Check. 7 × 0.5 = 3.5, and 3.5 + 0.25 = 3.75. ✓
Watch out. Count only the full pieces, not the 0.25 m that is left over.
Key words. A decimal place is a digit after the decimal point. The answer to a multiplication has the same number of decimal places as the decimal number in the question.
Method. Multiply as if there were no decimal point. Then put the point back so the answer has 2 decimal places.
- Write the numbers in columns. Line up the numbers on the right. Ignore the decimal point for now.
2.35 × 3 ---- 7.05
- Multiply each digit, starting from the right. Write the last digit of each answer and carry the tens to the next column.5 × 3 = 15, write 5 and carry 1
3 × 3 = 9 + 1 carried = 10, write 0 and carry 1
2 × 3 = 6 + 1 carried = 7, write 7 - Count the decimal places and put the point back. The decimal number has 2 decimal places, so the answer has 2 decimal places.235 × 3 = 705, and 705 → 7.05
Check. About 2 × 3 = 6, and 7.05 is close to that. ✓
Watch out. Count the decimal places in the question. Do not count the zeros you carry.
Key words. The decimal point separates the ones from the part smaller than 1. The places after it are tenths (0.1), hundredths (0.01) and thousandths (0.001).
Method. Line up the decimal points, then add each column from the right.
- Write the numbers under each other. Line up the decimal points in one column. Fill the gaps with zeros.
7.05 + 0.40 ------ 7.45
- Add column by column, from the right. Carry any ten into the next column to the left.7.05 + 0.40 = 7.45
- Write the answer with its decimal point in the same column. The decimal points line up in the question and in the answer.7.45
Check. Add the numbers in a different order and you get 7.45 again. ✓
Watch out. Line up the decimal points, not the last digits. Fill the gaps with zeros.
Key words. A decimal place is a digit after the decimal point. The answer to a multiplication has the same number of decimal places as the decimal number in the question.
Method. Multiply first, then add. The order matters.
- Work out the multiplication first. Multiplication comes before addition.2.35 × 3 = 7.05
- Add. Then add 0.4.7.05 + 0.4 = 7.45
- Write the answer. 7.45
Check. Adding first would give 2.75 × 3, which is wrong. ✓
Watch out. Do the multiplication before the addition, as in the order of operations.