Year 6: Solving number and practical problems
Part of the Year 6 place value and negative numbers chapter. Chapter overview · whole chapter on one page
Solving number and practical problems
In a problem, first find the numbers you need and the operation to use. Read the question carefully. Does it ask for an exact answer, or an estimate? If it says “to the nearest”, you must round. If it says “how many more”, you need to subtract. If it says “where is it now” with negative numbers, count the steps.
Always write the units with your answer: people, metres, degrees or pounds. Then check that your answer is sensible. Is it about the size you expected?
Key ideas
- Read the question twice. Note the numbers and the words that tell you the operation.
- “To the nearest” means round. Choose the place value given in the question.
- “How many more” or “how much bigger” means subtract.
- For a change below zero, count the steps through zero.
- Write the units, and check whether an estimate or an exact answer is needed.
Key words. To round a number to a given place value, find the two multiples of that place value either side of it. Then choose the one the number is nearer to. If the number is exactly halfway, we round up.
Method. Find the two multiples of 100,000 either side of the number, then choose the nearer one.
- Find the two multiples. The number lies between two multiples of 100,000.2,300,000 < 2,376,540 < 2,400,000
- Check the digit after the rounding digit. Halfway is 50,000. Look at how far the number is past 2,300,000.2,376,540 − 2,300,000 = 76,540, and 76,540 is at least 50,000
- Round up or down. Five or more means round up. Less than five means round down.Round up to 2,400,000
Check. 2,400,000 is a multiple of 100,000 and is the closer one to 2,376,540. ✓
Watch out. Look at the digit just to the right of the place you are rounding to. The other digits do not matter for the rule.
Key words. A negative number is a number less than zero. It has a minus sign in front of it, such as −4. Zero is neither positive nor negative. On a number line, numbers get smaller as you move left.
Method. Start at the first number and move 90 steps up. Count through zero if you need to.
- Start at the first number. Find it on the number line or thermometer.Start at −150 m
- Move 90 steps up. Count one step at a time. Each step changes the number by 1.−150 + 90
- Write the new number. −60 m
Check. −150 + 90 = −60. Going back: −60 − 90 = −150. ✓
Watch out. Going down past zero makes the number negative. Count through zero, do not stop at zero.
Method. This year is last year plus the extra visitors. Add them together.
- Find the extra visitors. This year it had 480,000 more than last year.480,000 more
- Add them to last year. Add the extra visitors to last year's number.3,140,000 + 480,000 = 3,620,000
- Check. Subtract the extra visitors to get back to last year.3,620,000 − 480,000 = 3,140,000
Check. 3,620,000 − 480,000 = 3,140,000. ✓
Watch out. Read carefully: “more” means add, and “less” means subtract.
Watch out
When a question asks for the answer to the nearest 100,000, do not give the exact answer. Also, do not round the numbers before you add them if the question asks for an exact answer. Round only at the end when you are asked to.Exercise 1F Problems with place value, rounding and negative numbers
Non-calculatorKey words. To round a number to a given place value, find the two multiples of that place value either side of it. Then choose the one the number is nearer to. If the number is exactly halfway, we round up.
Method. Find the two multiples of 1,000 either side of the number, then choose the nearer one.
- Find the two multiples. The number lies between two multiples of 1,000.48,000 < 48,650 < 49,000
- Check the digit after the rounding digit. Halfway is 500. Look at how far the number is past 48,000.48,650 − 48,000 = 650, and 650 is at least 500
- Round up or down. Five or more means round up. Less than five means round down.Round up to 49,000
Check. 49,000 is a multiple of 1,000 and is the closer one to 48,650. ✓
Watch out. Look at the digit just to the right of the place you are rounding to. The other digits do not matter for the rule.
Method. Subtract the January total from the February total.
- Write the two totals. February is the bigger number.3,725,000 and 3,450,000
- Subtract. How many more means subtract the smaller from the bigger.3,725,000 − 3,450,000 = 275,000
- Check. Add the difference back on to January.3,450,000 + 275,000 = 3,725,000
Check. 3,450,000 + 275,000 = 3,725,000. ✓
Watch out. Find the difference, not the total: 3,450,000 + 3,725,000 is not asked for.
Key words. A negative number is a number less than zero. It has a minus sign in front of it, such as −4. Zero is neither positive nor negative. On a number line, numbers get smaller as you move left.
Method. Start at the first number and move 1800 steps up. Count through zero if you need to.
- Start at the first number. Find it on the number line or thermometer.Start at −2,450 m
- Move 1800 steps up. Count one step at a time. Each step changes the number by 1.−2,450 + 1800
- Write the new number. −650 m
Check. −2,450 + 1800 = −650. Going back: −650 − 1800 = −2,450. ✓
Watch out. Going down past zero makes the number negative. Count through zero, do not stop at zero.
Key words. To round a number to a given place value, find the two multiples of that place value either side of it. Then choose the one the number is nearer to. If the number is exactly halfway, we round up.
Method. Find the two multiples of 10,000 either side of the number, then choose the nearer one.
- Find the two multiples. The number lies between two multiples of 10,000.1,280,000 < 1,286,400 < 1,290,000
- Check the digit after the rounding digit. Halfway is 5,000. Look at how far the number is past 1,280,000.1,286,400 − 1,280,000 = 6,400, and 6,400 is at least 5,000
- Round up or down. Five or more means round up. Less than five means round down.Round up to 1,290,000
Check. 1,290,000 is a multiple of 10,000 and is the closer one to 1,286,400. ✓
Watch out. Look at the digit just to the right of the place you are rounding to. The other digits do not matter for the rule.
Key words. To round a number to a given place value, find the two multiples of that place value either side of it. Then choose the one the number is nearer to. If the number is exactly halfway, we round up.
Method. Find the two multiples of 100,000 either side of the number, then choose the nearer one.
- Find the two multiples. The number lies between two multiples of 100,000.1,200,000 < 1,240,000 < 1,300,000
- Check the digit after the rounding digit. Halfway is 50,000. Look at how far the number is past 1,200,000.1,240,000 − 1,200,000 = 40,000, and 40,000 is less than 50,000
- Round up or down. Five or more means round up. Less than five means round down.Round down to 1,200,000
Check. 1,200,000 is a multiple of 100,000 and is the closer one to 1,240,000. ✓
Watch out. Look at the digit just to the right of the place you are rounding to. The other digits do not matter for the rule.
Method. Divide 2,500,000 by 100,000 to count the hundred thousands.
- Think about one hundred thousand. One hundred thousand is 100,000.100,000 × 25 = 2,500,000
- Count how many. Count the 100,000s until you reach 2,500,000.2,500,000 ÷ 100,000 = 25
- Check. Multiply back to check.25 × 100,000 = 2,500,000
Check. 25 × 100,000 = 2,500,000. ✓
Watch out. Count the zeros: 2,500,000 has five zeros, and 100,000 has five zeros.
Method. Compare the three populations. The biggest number is the largest population.
- Write the three populations. Compare the millions first, then the hundred thousands.Ashford 2,400,000, Brindley 1,850,000, Carlton 3,250,000
- Find the biggest number. The biggest population has the most people.3,250,000 is the biggest
Check. 3,250,000 is bigger than 2,400,000 and 1,850,000. ✓
Watch out. Compare the populations, not the names of the towns.
Method. Subtract the smallest population from the largest.
- Find the largest and smallest. The largest is the biggest number, and the smallest is the least number.3,250,000 and 1,850,000
- Subtract. How many more means subtract.3,250,000 − 1,850,000 = 1,400,000
Check. 1,850,000 + 1,400,000 = 3,250,000. ✓
Watch out. Subtract the smaller number from the bigger one.
Method. Compare the two statements. Write 2,500 thousand as a number.
- Write 2,500 thousand. One thousand is 1,000, so 2,500 thousands means 2,500 × 1,000.2,500 × 1,000 = 2,500,000
- Compare with 2,500,000. The two numbers are the same.2,500,000 = 2,500,000
- Give the answer. Mia is right.Yes, Mia is right
Check. 2,500 × 1,000 = 2,500,000. ✓
Watch out. A thousand thousands is a million.
Method. Subtract 1,000 from 1,000,000. Count the thousands.
- Write the start number. One million is 1,000,000.1,000,000
- Subtract 1,000. Take away one thousand from the million.1,000,000 − 1,000 = 999,000
- Check. Add 1,000 back to get one million.999,000 + 1,000 = 1,000,000
Check. 999,000 + 1,000 = 1,000,000. ✓
Watch out. Take one thousand off the thousands, not off the millions.
Key words. To round a number to a given place value, find the two multiples of that place value either side of it. Then choose the one the number is nearer to. If the number is exactly halfway, we round up.
Method. Find the two multiples of 1,000 either side of the number, then choose the nearer one.
- Find the two multiples. The number lies between two multiples of 1,000.3,000 < 3,600 < 4,000
- Check the digit after the rounding digit. Halfway is 500. Look at how far the number is past 3,000.3,600 − 3,000 = 600, and 600 is at least 500
- Round up or down. Five or more means round up. Less than five means round down.Round up to 4,000
Check. 4,000 is a multiple of 1,000 and is the closer one to 3,600. ✓
Watch out. Look at the digit just to the right of the place you are rounding to. The other digits do not matter for the rule.
Method. Subtract 100,000 from 3,000,000.
- Write the start number. Three million is 3,000,000.3,000,000
- Subtract 100,000. Take one hundred thousand away.3,000,000 − 100,000 = 2,900,000
- Check. Add 100,000 back on.2,900,000 + 100,000 = 3,000,000
Check. 2,900,000 + 100,000 = 3,000,000. ✓
Watch out. Count the hundred thousands: three million is 30 hundred thousands.
Method. Find the numbers that round to 8,000,000 to the nearest million, then find the ones that round to 7,990,000 to the nearest 10,000.
- Nearest 1,000,000. Numbers from 7,500,000 up to 8,499,999 round to 8,000,000.7,500,000 to 8,499,999
- Nearest 10,000. Numbers from 7,985,000 up to 7,994,999 round to 7,990,000.7,985,000 to 7,994,999
- Find the overlap. The smallest number in both ranges is the answer.The overlap starts at 7,985,000, so the answer is 7,985,000
Check. 7,985,000 rounds to 8,000,000 to the nearest 1,000,000, and to 7,990,000 to the nearest 10,000. ✓
Watch out. Check both roundings. The number must follow both rules.