Number reasoning
By the end of this chapter you will be able to
- Say the value of a digit from its column in a large number
- Round a number to the nearest 10, 100, 1000 or 10,000 and explain the choice
- Count through zero and find the difference between negative numbers
- Use brackets and the order of operations to get the right answer
- Solve a multi-step word problem and check it with an estimate
You should already know
- Place value to 1,000,000 and rounding (Year 4 and Year 5 number)
- The four operations with whole numbers (Year 5 number)
Where this leads
- Fractions and decimals reasoning (Chapter 2 of this book)
KS2 SATs practice for reasoning (Year 6). Original questions in the style of the national tests. Content: Read, write, order and compare whole numbers up to 10,000,000 and find the value of each digit (Year 5-6), Use negative numbers in context and count through zero (Year 6), Round any whole number to a required degree of accuracy (Year 5-6), Use the order of operations, including brackets (Year 6), Solve multi-step problems and decide which operation to use (Year 5-6), Use factors, multiples and primes in problems (Year 5), Use estimation to check whether an answer is sensible, and explain a method (Year 5-6).
Exercise P Do you remember?
Non-calculatorMethod. Find the column of the 5, then multiply it by the column value.
- Find the column. The 5 is in the hundreds column.hundreds
- Work out the value. The value is 5 x 100.5 x 100 = 500
- Write the answer. 500
Check. 500 = 5 x 100. ✓
Method. Adding 1,000 changes only the thousands digit.
- Look at the thousands digit. It is 6.6 thousands
- Add one thousand. Six thousand becomes seven thousand.6,850 + 1,000 = 7,850
- Write the answer. 7,850
Check. 7,850 - 1,000 = 6,850. ✓
Method. Multiply the tens and the units separately, then add.
- Multiply the tens. 40 x 5 = 200.200
- Multiply the units. 8 x 5 = 40.40
- Add the two parts. Add 200 and 40.240
- Write the answer. 240
Check. 48 x 5 = 240, and 240 / 5 = 48. ✓
Method. The list goes up in twos.
- Find the step. Each number is 2 more than the one before.+2
- Add the step to 8. 8 + 2 = 10.10
- Write the answer. 10
Check. 10 - 8 = 2, the same step. ✓
Method. A multiple of 9 is a number in the 9 times table.
- Count in nines. 9, 18, 27 is in the 9 times table.9 x 3 = 27
- Write the answer. Because 9 x 3 = 27, it is a multiple of 9.yes
- Write the answer. yes
Check. 9 x 3 = 27. ✓
Place value, comparing and rounding
Rule
Each digit has a value that depends on its column. In 4,862,139 the 6 is in the ten thousands column, so it is worth 6 x 10,000 = 60,000.
To round, look at the digit next to the column you are rounding to. 5 or more: round up. 4 or less: round down.
For the full method of place value and rounding, read Year 5 and Year 6 maths, chapter 1.
Key ideas
- The value of a digit = the digit x the value of its column.
- Compare numbers by their number of digits first, then by the leading digit.
- To round to the nearest 10,000, look at the thousands digit.
Method. Find the column, then multiply the digit by its column value.
- Find the column of the digit. The digit 6 is in the ten thousands column.ten thousands
- Work out the value. The value is the digit times the column value 10,000.6 x 10,000 = 60,000
- Write the answer. 60,000
Check. 60,000 = 6 x 10,000. ✓
Watch out. The value of a digit depends on its column, not just on the digit.
Method. Look at the digit next to the rounding column.
- Find the rounding column and the digit next to it. Rounding to the nearest 1,000 looks at the hundreds digit.digit to check = 7
- Decide up or down. A 5 or more rounds up; 4 or less rounds down.7 is 5 or more, so round up
- Write the rounded number. Keep the digits to the left and make the rest zeros.57,000
- Write the answer. 57,000
Check. 57,000 is a multiple of 1,000 and is the closest one to 56,750. ✓
Watch out. Do not round a 5 down. A 5 always rounds up.
Method. Compare the digits, starting from the left.
- Count the digits in each number. A number with more digits is larger.45,000 has 5 digits, 4,500 has 4 digits, 405,000 has 6 digits, 54,000 has 5 digits
- Compare numbers with the same number of digits. Look at the leading digit, then the next one.compare from the left
- Write them in order. Put them smallest to largest.4,500, 45,000, 54,000, 405,000
- Write the answer. 4,500, 45,000, 54,000, 405,000
Check. Each number is smaller than the one after it. ✓
Watch out. 4,500 has four digits and 45,000 has five, so 45,000 is larger.
Watch out
Watch out. The 7 in 700,000 is worth 700,000, not 7. A digit is worth more the further left it is.
Exam tip
Practice guide. Check each rounding answer: it must be a multiple of the rounding unit and close to the original number.
Exercise 1A Practice: place value, comparing and rounding
Non-calculatorMethod. Find the column, then multiply the digit by its column value.
- Find the column of the digit. The digit 3 is in the hundreds column.hundreds
- Work out the value. The value is the digit times the column value 100.3 x 100 = 300
- Write the answer. 300
Check. 300 = 3 x 100. ✓
Watch out. The value of a digit depends on its column, not just on the digit.
Method. Find the column, then multiply the digit by its column value.
- Find the column of the digit. The digit 6 is in the hundred thousands column.hundred thousands
- Work out the value. The value is the digit times the column value 100,000.6 x 100,000 = 600,000
- Write the answer. 600,000
Check. 600,000 = 6 x 100,000. ✓
Watch out. The value of a digit depends on its column, not just on the digit.
Method. Write each part of the number in its column.
- Write the millions. 3 million is 3,000,000.3,000,000
- Write the thousands. 40 thousand is 40,000.40,000
- Write the units. Seven units is 7.7
- Add the parts together. Put the parts in their columns.3,040,007
- Write the answer. 3,040,007
Check. 3,000,000 + 40,000 + 7 = 3,040,007. ✓
Method. Look at the digit next to the rounding column.
- Find the rounding column and the digit next to it. Rounding to the nearest 10,000 looks at the thousands digit.digit to check = 2
- Decide up or down. A 5 or more rounds up; 4 or less rounds down.2 is 4 or less, so round down
- Write the rounded number. Keep the digits to the left and make the rest zeros.4,860,000
- Write the answer. 4,860,000
Check. 4,860,000 is a multiple of 10,000 and is the closest one to 4,862,139. ✓
Watch out. Do not round a 5 down. A 5 always rounds up.
Method. Look at the digit next to the rounding column.
- Find the rounding column and the digit next to it. Rounding to the nearest 1,000 looks at the hundreds digit.digit to check = 7
- Decide up or down. A 5 or more rounds up; 4 or less rounds down.7 is 5 or more, so round up
- Write the rounded number. Keep the digits to the left and make the rest zeros.57,000
- Write the answer. 57,000
Check. 57,000 is a multiple of 1,000 and is the closest one to 56,750. ✓
Watch out. Do not round a 5 down. A 5 always rounds up.
Method. Look at the digit next to the rounding column.
- Find the rounding column and the digit next to it. Rounding to the nearest 100 looks at the tens digit.digit to check = 9
- Decide up or down. A 5 or more rounds up; 4 or less rounds down.9 is 5 or more, so round up
- Write the rounded number. Keep the digits to the left and make the rest zeros.10,000
- Write the answer. 10,000
Check. 10,000 is a multiple of 100 and is the closest one to 9,996. ✓
Watch out. Do not round a 5 down. A 5 always rounds up.
Method. Compare the digits, starting from the left.
- Count the digits in each number. A number with more digits is larger.4,500 has 4 digits, 45,000 has 5 digits, 405,000 has 6 digits, 54,000 has 5 digits
- Compare numbers with the same number of digits. Look at the leading digit, then the next one.compare from the left
- Write them in order. Put them smallest to largest.4,500, 45,000, 54,000, 405,000
- Write the answer. 4,500, 45,000, 54,000, 405,000
Check. Each number is smaller than the one after it. ✓
Watch out. 4,500 has four digits and 45,000 has five, so 45,000 is larger.
Method. Compare the digits, starting from the left.
- Count the digits in each number. A number with more digits is larger.7,063 has 4 digits, 1,700 has 4 digits, 70,630 has 5 digits, 1,070 has 4 digits
- Compare numbers with the same number of digits. Look at the leading digit, then the next one.compare from the left
- Write them in order. Put them largest to smallest.70,630, 7,063, 1,700, 1,070
- Write the answer. 70,630, 7,063, 1,700, 1,070
Check. Each number is larger than the one after it. ✓
Watch out. 4,500 has four digits and 45,000 has five, so 45,000 is larger.
Method. Compare the digits from the left until they differ.
- Compare the digits from the left. The first two digits, 3 and 4, are the same in both numbers.3 and 4 match
- Find the first place where they differ. The third digit is 0 in the first number and 2 in the second, so the second number is bigger.3402871 is less than 3420871
- Write the sign. The smaller number is on the left, so use the less than sign.<
- Write the answer. <
Check. 3,402,871 is less than 3,420,871. ✓
Method. Add 1,000 to the thousands column.
- Add 1,000. Add one thousand to the number.6,495,300 + 1,000
- Write the answer. 6,496,300
Check. 6,496,300 - 1,000 = 6,495,300. ✓
Method. Subtract 10,000 from the ten thousands column.
- Subtract 10,000. Take one ten thousand away.6,495,300 - 10,000
- Write the answer. 6,485,300
Check. 6,485,300 + 10,000 = 6,495,300. ✓
Method. Test each option.
- Check each option. Test each one against the rule or the calculation.
- Tick the true ones. Each ticked number has more than seven digits and is above 2,000,000.2,004,500, 2,400,000
- Leave the others unticked. These do not match.1,990,000, 2,000,000, 1,999,999 are not ticked
- Write the answer. 2,004,500, 2,400,000
Check. Each ticked option is true. ✓
Watch out. 1,999,999 is less than 2 million, so it is not ticked.
Method. Test each choice against the question.
- Read what the question asks. Decide what the answer must do or show.
- Test the choice. The 7 is in the hundred thousands column: 7 x 100,000 = 700,000.No, the 7 is worth 700,000 fits
- Reject the other choices. Yes, Sam is right is wrong because 7 is not in the thousands column.Yes, Sam is right and No, the 7 is worth 70 do not fit
- Write the answer. No, the 7 is worth 700,000
Check. No, the 7 is worth 700,000 is the one that fits. ✓
Method. Put the largest digit first, then the next largest.
- Find the largest digit. Nine is the largest digit, so it goes first in the millions column.9 million
- Put the rest in order. Put the remaining digits in order from largest to smallest.8, 6, 4, 2, 1, 0
- Write the number. Each digit is used once.9,864,210
- Write the answer. 9,864,210.
Check. All seven digits are used once. ✓
Method. Numbers that round to 4,500 to the nearest 100 are from 4,450 up to 4,549.
- Find the range. Half of 100 is 50, so numbers from 4,450 to 4,549 round to 4,500.4,450 to 4,549
- Pick one. Any number in this range is correct.4,450
- Write the answer. 4,450.
Check. 4,450 rounds to 4,500 to the nearest 100. ✓
Method. Look at the digit next to the rounding column.
- Find the rounding column and the digit next to it. Rounding to the nearest 100 looks at the tens digit.digit to check = 5
- Decide up or down. A 5 or more rounds up; 4 or less rounds down.5 is 5 or more, so round up
- Write the rounded number. Keep the digits to the left and make the rest zeros.7,500
- Write the answer. 7,500
Check. 7,500 is a multiple of 100 and is the closest one to 7,458. ✓
Watch out. Do not round a 5 down. A 5 always rounds up.
Method. The smallest number that rounds up to 60,000 is the first one at or above 55,000.
- Find the halfway point. Halfway between 50,000 and 60,000 is 55,000.55,000
- Check the round. From 55,000 upwards the number rounds to 60,000.55,000 rounds to 60,000
- Write the answer. 55,000
Check. 55,000 rounds up to 60,000. ✓
Watch out. The number 54,999 would round down to 50,000.
Negative numbers and counting through zero
Rule
Negative numbers are less than zero. On a number line, numbers get smaller as you move left.
To find the change from a negative number, count the steps across zero. From -3 up 7 is -3 + 7 = 4.
Key ideas
- -3 is smaller than -1, because it is further left on the number line.
- Going up by 7 from -3 passes through zero and ends at 4.
- The difference between 9 and -4 is 13, because you count 4 steps to zero and then 9 more.
Method. Count up from -3 on a thermometer.
- Start at -3. The starting point is minus three.-3
- Count up 7 steps. Three steps go up to zero, and four more steps go on from zero.-3 -> 0 is 3 steps, then 4 more
- Write the new temperature. After 7 steps up from -3, we reach 4.4 degrees
- Write the answer. 4
Check. -3 + 7 = 4, and 4 - 7 = -3. ✓
Method. The colder temperature is the smaller number.
- Compare the numbers. Both are negative, so the one further from zero is smaller.-12 is 12 below zero, -3 is 3 below zero
- Use the number line. -12 is further to the left.-12 < -3
- Write the answer. The colder temperature is the smaller number.-12
- Write the answer. -12
Check. -12 < -3, so -12 is colder. ✓
Method. Subtracting a negative number is the same as adding the positive number.
- Count across zero. From -4 to 0 is 4 steps.4 steps
- Count on from zero. Then 9 more steps from 0 to 9.4 + 9
- Add the steps. The total distance is 13.9 - (-4) = 13
- Write the answer. 13
Check. 13 + (-4) = 9. ✓
Watch out
Watch out. -8 is smaller than -2, even though 8 is bigger than 2. Look at the number line.
Exam tip
Practice guide. Draw a quick number line with zero in the middle when a question crosses zero.
Exercise 1B Practice: negative numbers
Non-calculatorMethod. Count up 6 from -4.
- Count up to zero. From -4 to 0 is 4 steps.4 steps
- Count on from zero. 6 - 4 = 2 steps more after zero.6 - 4 = 2
- Write the answer. 2
Check. -4 + 6 = 2. ✓
Method. Count down 8 from 3.
- Count down to zero. From 3 to 0 is 3 steps.3 steps
- Count on below zero. 8 - 3 = 5 steps below zero.8 - 3 = 5
- Write the answer. -5
Check. 3 - 8 = -5. ✓
Method. Count left from -2.
- Start at -2. The start is minus two.-2
- Count 5 steps left. Each step left makes the number smaller.-2 -> -7
- Write the answer. -7
Check. -7 + 5 = -2. ✓
Method. Count the steps from -6 up to 5.
- Count to zero. From -6 to 0 is 6 steps.6 steps
- Count from zero to 5. From 0 to 5 is 5 steps.6 + 5 = 11
- Write the answer. 11
Check. -6 + 11 = 5. ✓
Method. Count up from -8 until you reach 3.
- Count to zero. From -8 to 0 is 8 steps.8 steps
- Count to 3. From 0 to 3 is 3 more steps.8 + 3 = 11
- Write the answer. 11
Check. -8 + 11 = 3. ✓
Method. The coldest temperature is the smallest number.
- Compare the negative numbers. -6 is further below zero than -1, so -6 is colder.-6 < -1
- Place zero and the positive number. 0 is above all the negatives and 2 is above 0.0 < 2
- Write them in order. From coldest to warmest.-6, -1, 0, 2
- Write the answer. -6, -1, 0, 2
Check. -6 < -1 < 0 < 2. ✓
Method. Count up 25 from -18.
- Count up to zero. From -18 to 0 is 18 steps.18 steps
- Count on from zero. 25 - 18 = 7 steps after zero.25 - 18 = 7
- Write the answer. 7
Check. -18 + 25 = 7. ✓
Method. Test each option.
- Check each option. Test each one against the rule or the calculation.
- Tick the true ones. -9 and -6 are further left than -5 on the number line.-9, -6
- Leave the others unticked. These do not match.-4, 0, -5 are not ticked
- Write the answer. -9, -6
Check. Each ticked option is true. ✓
Watch out. -5 is not less than -5, so it is not ticked.
Method. The greater number is further to the right on the number line.
- Place both numbers. -7 is 7 below zero and -15 is 15 below zero.-7 is nearer zero
- Choose the greater. The one nearer zero is greater.-7 > -15
- Write the answer. -7
Check. -7 > -15. ✓
Method. Work out each change in order.
- Rise 45 metres. Rising adds to the number: -120 + 45.-120 + 45 = -75
- Sink 70 metres. Sinking takes away from the number: -75 - 70.-75 - 70 = -145
- Write the answer. -145
Check. -120 + 45 - 70 = -145. ✓
Method. Test each choice against the question.
- Read what the question asks. Decide what the answer must do or show.
- Test the choice. Subtracting -9 is the same as adding 9, so the answer is 5.positive fits
- Reject the other choices. 5 is positive, so negative and zero are wrong.negative and zero do not fit
- Write the answer. positive
Check. positive is the one that fits. ✓
Method. Count 10 steps on a number line from the negative number.
- Pick a negative number. Choose -4.-4
- Count 10 steps up. From -4 to 0 is 4 steps; 6 more steps reach 6.-4 + 10 = 6
- Write the answer. -4 and 6.
Check. 6 - (-4) = 10. ✓
Method. Place both numbers on a number line.
- Place -5 and 5. -5 is 5 steps left of zero and 5 is 5 steps right of zero.-5 < 0 < 5
- Write the reason. A negative number is always below zero.-5 < 5
- Write the answer. Yes, because -5 is below zero and 5 is above zero, so -5 is smaller.
Check. -5 is further left than 5. ✓
Order of operations and brackets
Rule
Work out brackets first. Then do multiplication and division (left to right). Then do addition and subtraction (left to right).
Brackets change the order. (2 + 3) x 4 = 20, but 2 + 3 x 4 = 14.
Key ideas
- Brackets first, then x and divide, then add and subtract.
- Work from left to right when the operations have the same priority.
- A bracket can change the answer completely.
Method. Multiplication comes before addition and subtraction.
- Multiply first. 8 x 3 is done before adding.8 x 3 = 24
- Add and subtract from left to right. 12 + 24 = 36, then 36 - 4.12 + 24 - 4 = 32
- Write the answer. 32
Check. 12 + 24 - 4 = 32. ✓
Method. Brackets first, then divide, then multiply and subtract.
- Work out the bracket. 12 + 8 = 20.(12 + 8) = 20
- Divide and multiply. Divide 4 by 2 and multiply 20 by 3.20 x 3 = 60 and 4 / 2 = 2
- Subtract. Take 2 from 60.60 - 2 = 58
- Write the answer. 58
Check. (20 x 3) - 2 = 58. ✓
Method. Work out each group, then add.
- Work out the adults. 4 x 6 = 24 pounds.24 pounds
- Work out the children. 3 x 4 = 12 pounds.12 pounds
- Add the two totals. Add the money from both groups.24 + 12 = 36 pounds
- Write the answer. 36
Check. 4 x 6 + 3 x 4 = 36. ✓
Watch out
Watch out. 2 + 3 x 4 is 14, not 20. Multiply first, then add.
Exam tip
Practice guide. Underline the multiplication and division first, then do the sum from left to right.
Exercise 1C Practice: order of operations
Non-calculatorMethod. Multiply first.
- Multiply. 6 x 3 = 18.18
- Add. 5 + 18 = 23.23
- Write the answer. 23
Check. 5 + 18 = 23. ✓
Method. Divide first.
- Divide. 12 / 4 = 3.3
- Subtract. 20 - 3 = 17.17
- Write the answer. 17
Check. 20 - 3 = 17. ✓
Method. Brackets first.
- Work out the bracket. 7 + 5 = 12.12
- Multiply. 12 x 4 = 48.48
- Write the answer. 48
Check. 12 x 4 = 48. ✓
Method. Brackets, then multiply, then add.
- Work out the bracket. 9 - 4 = 5.5
- Multiply. 6 x 5 = 30.30
- Add. 30 + 3 = 33.33
- Write the answer. 33
Check. 30 + 3 = 33. ✓
Method. Brackets, then divide and multiply, then subtract.
- Work out the bracket. 4 + 2 = 6.6
- Divide. 48 / 6 = 8.8
- Multiply. 8 x 3 = 24.24
- Subtract. 24 - 1 = 23.23
- Write the answer. 23
Check. 48 / 6 = 8, 8 x 3 = 24, 24 - 1 = 23. ✓
Method. Brackets, then multiply, then add and subtract.
- Work out the bracket. 2 + 8 = 10.10
- Multiply. 3 x 10 = 30.30
- Subtract and add from left to right. 100 - 30 = 70, then 70 + 5 = 75.75
- Write the answer. 75
Check. 100 - 30 + 5 = 75. ✓
Method. Put the brackets around the addition, so it is worked out first.
- Work out the bracket first. Brackets come first, so 2 + 3 is worked out before the multiplication.2 + 3 = 5
- Multiply. Then multiply 5 by 4.5 x 4 = 20
- Write the answer. (2 + 3) x 4
Check. (2 + 3) x 4 = 20. ✓
Method. Try a bracket around the subtraction and the addition, then check.
- Try brackets around 10 - 4. (10 - 4) / 2 + 1 = 3 + 1 = 4, which is not 2.4
- Try brackets around 2 + 1 as well. (10 - 4) / (2 + 1) = 6 / 3 = 2.2 is the answer
- Write the answer. (10 - 4) / (2 + 1) = 2.
Check. 6 / 3 = 2. ✓
Method. Multiply and divide first, then add and subtract.
- Work out the multiplications and division. 3 x 4 = 12, 2 x 5 = 10 and 6 / 3 = 2.12 + 10 - 2
- Add and subtract. 12 + 10 = 22, then 22 - 2 = 20.20
- Write the answer. 20
Check. 12 + 10 - 2 = 20. ✓
Method. Work out the cost first, then the change.
- Work out the cost. 3 bags at 3 pounds each is 3 x 3 = 9 pounds.9 pounds
- Work out the change. 10 - 9 = 1 pound.1 pound
- Write the answer. 1
Check. 9 + 1 = 10. ✓
Method. Brackets, then multiply and divide from left to right.
- Work out the bracket. 25 - 7 = 18.18
- Multiply. 18 x 5 = 90.90
- Divide. 90 / 6 = 15.15
- Write the answer. 15
Check. 18 x 5 / 6 = 15. ✓
Method. Test each option.
- Check each option. Test each one against the rule or the calculation.
- Tick the true ones. Each ticked calculation gives 14 when the multiplication or division is done first.2 + 3 x 4, 5 x 2 + 4, 20 - 3 x 2
- Leave the others unticked. These do not match.(2 + 3) x 4, 2 + 12 / 6 are not ticked
- Write the answer. 2 + 3 x 4, 5 x 2 + 4, 20 - 3 x 2
Check. Each ticked option is true. ✓
Watch out. (2 + 3) x 4 is 20, and 2 + 12 / 6 is 4, so they are not ticked.
Factors, multiples and primes
Rule
A factor divides exactly into a number. 36 has the factors 1, 2, 3, 4, 6, 9, 12, 18, 36.
A multiple is in the times table of a number. 12 is a multiple of 3 and of 4.
A prime number has exactly two factors: 1 and itself. 2, 3, 5, 7, 11 and 13 are primes. 1 is not a prime.
Key ideas
- Factors come in pairs: 4 x 9 = 36, so 4 and 9 are factors of 36.
- The common factors of two numbers are the factors they share.
- A prime number has exactly two factors.
Method. Find pairs of numbers that multiply to 36.
- Start with 1. 1 x 36 = 36, so 1 and 36 are factors.1, 36
- Try 2, 3, 4, 5 and 6. 2 x 18, 3 x 12, 4 x 9 and 6 x 6 work. 5 does not divide 36.2, 18, 3, 12, 4, 9, 6
- Write them in order. Each number is listed once.1, 2, 3, 4, 6, 9, 12, 18, 36
- Write the answer. 1, 2, 3, 4, 6, 9, 12, 18, 36
Check. Each factor divides 36 exactly. ✓
Method. List the factors of each number, then find the largest one in both lists.
- List the factors of 12. 1, 2, 3, 4, 6, 12.1, 2, 3, 4, 6, 12
- List the factors of 18. 1, 2, 3, 6, 9, 18.1, 2, 3, 6, 9, 18
- Find the largest one in both lists. The biggest factor they share is 6.6
- Write the answer. 6
Check. 12 / 6 = 2 and 18 / 6 = 3. ✓
Method. A prime has only two factors, 1 and itself.
- Test 39. 39 = 3 x 13, so it is not prime.39 is not prime
- Test 45 and 51. 45 = 5 x 9 and 51 = 3 x 17, so neither is prime.45 and 51 are not prime
- Test 47. No number from 2 to 6 divides 47 exactly, so its only factors are 1 and 47.47 is prime
- Write the answer. 47
Check. 47 has no factors other than 1 and 47. ✓
Watch out
Watch out. 1 is not a prime number, and 2 is the only even prime.
Exam tip
Practice guide. Check each candidate by dividing by the small primes, 2, 3, 5 and 7, before deciding it is prime.
Exercise 1D Practice: factors, multiples and primes
Non-calculatorMethod. Find pairs that multiply to 24.
- Find pairs. 1 x 24, 2 x 12, 3 x 8, 4 x 6.1, 24, 2, 12, 3, 8, 4, 6
- Write them in order. List each factor once.1, 2, 3, 4, 6, 8, 12, 24
- Write the answer. 1, 2, 3, 4, 6, 8, 12, 24
Check. Each factor divides 24 exactly. ✓
Method. Count in sevens.
- Count in sevens. 7, 14, 21, 28, 35.7 x 1 to 7 x 5
- Write the answer. 7, 14, 21, 28, 35
Check. 7 x 5 = 35. ✓
Method. List the factors of both numbers and find the largest one in both lists.
- Factors of 24. 1, 2, 3, 4, 6, 8, 12, 24.1, 2, 3, 4, 6, 8, 12, 24
- Factors of 36. 1, 2, 3, 4, 6, 9, 12, 18, 36.1, 2, 3, 4, 6, 9, 12, 18, 36
- Find the largest shared factor. The largest number in both lists is 12.12
- Write the answer. 12
Check. 24 / 12 = 2 and 36 / 12 = 3. ✓
Method. List the multiples of each number until you find one in both lists.
- Multiples of 6. 6, 12, 18, 24, 30.6, 12, 18, 24
- Multiples of 8. 8, 16, 24, 32.8, 16, 24
- Find the first shared multiple. 24 is the first number in both lists.24
- Write the answer. 24
Check. 24 / 6 = 4 and 24 / 8 = 3. ✓
Method. Test each option.
- Check each option. Test each one against the rule or the calculation.
- Tick the true ones. A prime has exactly two factors: 1 and itself.11, 17, 29
- Leave the others unticked. These do not match.9, 15, 21 are not ticked
- Write the answer. 11, 17, 29
Check. Each ticked option is true. ✓
Watch out. 9 = 3 x 3, 15 = 3 x 5 and 21 = 3 x 7, so they are not prime.
Method. Divide 84 by each number and check for a remainder.
- Test 5. 84 / 5 = 16 remainder 4, so 5 is not a factor.no
- Test 6. 84 / 6 = 14, with no remainder.6 is a factor
- Test 8. 84 / 8 = 10 remainder 4, so 8 is not a factor.no
- Write the answer. 6
Check. 6 x 14 = 84. ✓
Method. Divide by the smallest prime that works, and repeat until you reach 1.
- Divide by 2. 60 / 2 = 30.2 x 30
- Divide 30 by 2. 30 / 2 = 15.2 x 2 x 15
- Divide 15 by 3. 15 / 3 = 5, and 5 is prime.2 x 2 x 3 x 5
- Write the answer. 2 x 2 x 3 x 5
Check. 2 x 2 x 3 x 5 = 60. ✓
Method. A prime has no factors except 1 and itself.
- Test 23. 23 is not in the 2, 3 or 5 times tables.23 is prime
- Write the answer. Either 23 or 29 is correct.23
- Write the answer. 23.
Check. 23 is prime. ✓
Method. List the pairs that multiply to 42, then pick the pair with both numbers in the range.
- List the pairs. 1 x 42, 2 x 21, 3 x 14, 6 x 7.6 x 7 = 42
- Check the range. 6 and 7 are both between 5 and 10.6 and 7
- Write the answer. 6 and 7
Check. 6 x 7 = 42. ✓
Method. Find the highest common factor of 48 and 60.
- Factors of 48. 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.48 / 12 = 4
- Factors of 60 that are also factors of 48. 12 divides 60 exactly, 5 times.60 / 12 = 5
- Find the largest. The largest shared factor is 12.12
- Write the answer. 12
Check. 5 groups of 12 make 60, and 12 divides both 48 and 60. ✓
Method. Test each choice against the question.
- Read what the question asks. Decide what the answer must do or show.
- Test the choice. 7 x 13 = 91, so 91 has factors other than 1 and itself.no, because 7 x 13 = 91 fits
- Reject the other choices. The number 91 is odd, so 2 is not a factor, and 2 x 45 is not 91.yes and no, because 2 x 45 = 91 do not fit
- Write the answer. no, because 7 x 13 = 91
Check. no, because 7 x 13 = 91 is the one that fits. ✓
Method. Think about how 6 is made.
- Look at 6. 6 = 2 x 3.6 = 2 x 3
- Check a multiple. 18 = 6 x 3 is also 3 times 6.18 = 3 x 6
- Write the answer. Yes, because 6 is 2 x 3, so any multiple of 6 is 3 times something.
Check. Each multiple of 6 is 6 x a whole number, which is 3 x 2 x that number. ✓
Method. Try two small primes.
- Pick two primes. 2 and 3 are both prime.2, 3
- Add them. 2 + 3 = 5, which is odd.5
- Write the answer. No. 2 + 3 = 5, which is odd.
Check. 5 is odd, so the answer is no. ✓
Choosing the operation and checking the answer
Rule
Read the question twice. Decide which operations are needed and in which order.
Check the answer with an estimate: round each number first, work out a rough answer, and see if your answer is close.
Key ideas
- Two steps: work out the total first, then share or take away.
- Round to 1 significant figure to estimate, for example 368 x 5 is about 400 x 5 = 2000.
- Always ask: is the answer about the right size?
Method. Find the total first, then take away the sales.
- Find the total rolls made. 368 x 5 = 1,840 rolls.1,840 rolls
- Take away the rolls sold. Subtract the number sold from the total.1,840 - 1,600 = 240
- Check the answer. Rounding 368 up to 400 gives 400 x 5 = 2,000, and 2,000 - 1,600 = 400. The exact answer is smaller, because 368 was rounded up.about 400, and 240 is the exact answer
- Write the answer. 240
Check. 368 x 5 = 1,840 and 1,840 - 1,600 = 240. ✓
Method. Round the first number, then multiply.
- Round 4,980 to the nearest 1,000. 4,980 is close to 5,000.5,000
- Estimate. 5,000 x 6 = 30,000.about 30,000
- Compare with the answer. 29,880 is very close to 30,000, so it is sensible.yes, 29,880 is sensible
- Write the answer. yes
Check. 5,000 x 6 = 30,000 and 30,000 - 29,880 = 120. ✓
Method. Divide, then decide whether to round up or down.
- Divide the pupils by the seats. 47 / 12 = 3 remainder 11.3 buses and 11 pupils
- Decide what the leftover means. The 11 pupils still need a seat, so one more bus is needed.3 + 1 = 4
- Write the answer. Four buses carry all 47 pupils.4
- Write the answer. 4
Check. 4 x 12 = 48, which is enough for 47. ✓
Watch out
Watch out. When the answer is a remainder, read the question to decide whether to round up, down or ignore it.
Exam tip
Practice guide. Write the operation in words first: "total, then take away", or "share equally".
Exercise 1E Practice: choosing the operation and checking
Non-calculatorMethod. Multiply the number of packs by the number in each pack.
- Choose the operation. The total is the packs times the pencils in each pack.12 x 36
- Work it out. 12 x 36 = 432.432
- Write the answer. 432
Check. 432 / 12 = 36. ✓
Method. Take away the empty seats from the total.
- Choose the operation. People sitting = total seats minus empty seats.485 - 136
- Work it out. 485 - 136 = 349.349
- Write the answer. 349
Check. 349 + 136 = 485. ✓
Method. Round each number, then multiply.
- Round the numbers. 61 rounds to 60 and 39 rounds to 40.60 and 40
- Multiply. 60 x 40 = 2,400.2,400
- Write the answer. 2,400
Check. 61 x 39 is close to 2,400. ✓
Method. Multiply the cost of one bag by 4.
- Work in pence. 3 pounds 20 pence is 320 pence.320 pence
- Multiply. 320 x 4 = 1,280 pence.1,280 pence
- Change to pounds. 1,280 pence is 1,280 / 100 = 12.80 pounds.12.80 pounds
- Write the answer. 12.80
Check. 4 x 3.20 = 12.80. ✓
Method. Find the total cost, then the change.
- Find the total. 5 x 7.50 = 37.50 pounds.37.50 pounds
- Find the change. 40 - 37.50 = 2.50 pounds.2.50 pounds
- Write the answer. 2.50
Check. 37.50 + 2.50 = 40. ✓
Method. Work out the gap between the amount in the tank and the full amount.
- Choose the operation. The space left = full amount minus the amount already there.2,000 - 1,375
- Work it out. 2,000 - 1,375 = 625.625
- Write the answer. 625
Check. 1,375 + 625 = 2,000. ✓
Method. Divide the total by the number of shelves.
- Choose the operation. Equal sharing is division.1,840 / 8
- Work it out. 1,840 / 8 = 230.230
- Write the answer. 230
Check. 230 x 8 = 1,840. ✓
Method. Compare the two divisors.
- Compare the divisors. 60 is 10 times 6.60 = 10 x 6
- Compare the answers. Dividing by 10 times as much gives an answer 10 times smaller: 500 / 10 = 50.50
- Write the answer. Yes, because 60 is 10 times 6, so the answer is 10 times smaller.
Check. 3,000 / 60 = 50. ✓
Method. Test each choice against the question.
- Read what the question asks. Decide what the answer must do or show.
- Test the choice. 4,980 is close to 5,000, so the estimate 35,000 is close to the exact answer 34,860.Yes, 35,000 is close to 34,860 fits
- Reject the other choices. The answer 3,486 is ten times too small, and 35,000 is not much bigger than 34,860.No, 4,980 x 7 must be 3,486 and No, 35,000 is much bigger than 34,860 do not fit
- Write the answer. Yes, 35,000 is close to 34,860
Check. Yes, 35,000 is close to 34,860 is the one that fits. ✓
Method. Start from the answer 60 and work out a calculation that makes it.
- Find a way to make 60. 8 x 5 = 40, then 40 + 20 = 60.8 x 5 + 20 = 60
- Write the problem. Eight bags with 5 sweets each, plus 20 more sweets.60 sweets
- Write the answer. Mia has 8 bags with 5 sweets in each, and her gran gives her 20 more sweets. How many sweets does she have now? 8 x 5 + 20 = 60.
Check. 8 x 5 + 20 = 60. ✓
Method. Find the total first, then take away the marbles given away.
- Total marbles. 3 x 24 = 72, then 72 + 14 = 86.86
- Take away the marbles given away. 86 - 25 = 61.61
- Write the answer. 61
Check. 86 - 25 = 61. ✓
Method. Work out the flour for one person first, or multiply by 3.
- Find the number of times. 24 is 3 times 8.24 / 8 = 3
- Multiply the flour. 250 x 3 = 750 g.750 g
- Write the answer. 750
Check. 750 / 3 = 250. ✓
Method. Multiply the rows by the chairs in each row.
- Multiply. 75 x 8 = 600.600
- Compare. The school said 600, which matches.yes, 600 chairs
- Write the answer. Yes, because 75 x 8 = 600, so the number is right.
Check. 75 x 8 = 600. ✓
Chapter summary
| Skill | Method |
|---|---|
| Place value | Digit x column value. 6 in 4,862,139 is worth 60,000. |
| Rounding | Look at the digit next to the rounding column: 5 or more rounds up. |
| Negative numbers | Count across zero on a number line; -8 is smaller than -2. |
| Order of operations | Brackets, then x and divide, then add and subtract. |
| Factors and multiples | Factors divide exactly; multiples are in the times table. |
| Primes | Exactly two factors: 1 and itself. 1 is not prime. |
| Choosing the operation | Read twice; decide total, take away, share or repeat. |
| Checking | Round and estimate; does the answer have the right size? |
Can you…?
- I can say the value of a digit in a number up to 10,000,000
- I can round to the nearest 10, 100, 1,000 or 10,000 and explain my choice
- I can count through zero and find differences between negative numbers
- I can use brackets and the order of operations
- I can choose the operation for a multi-step problem and check it with an estimate
Key terms
- factor
- a number that divides exactly into another number
- multiple
- a number in the times table of another number
- prime number
- a number with exactly two factors, 1 and itself
- negative number
- a number less than zero, such as -4
- order of operations
- the rule for which calculation to do first: brackets, then multiply and divide, then add and subtract
- estimate
- a rough answer found by rounding the numbers first
Exercise 1Z Check-up
Non-calculatorMethod. Find the column, then multiply the digit by its column value.
- Find the column of the digit. The digit 5 is in the thousands column.thousands
- Work out the value. The value is the digit times the column value 1,000.5 x 1,000 = 5,000
- Write the answer. 5,000
Check. 5,000 = 5 x 1,000. ✓
Watch out. The value of a digit depends on its column, not just on the digit.
Method. Look at the digit next to the rounding column.
- Find the rounding column and the digit next to it. Rounding to the nearest 100 looks at the tens digit.digit to check = 5
- Decide up or down. A 5 or more rounds up; 4 or less rounds down.5 is 5 or more, so round up
- Write the rounded number. Keep the digits to the left and make the rest zeros.8,400
- Write the answer. 8,400
Check. 8,400 is a multiple of 100 and is the closest one to 8,359. ✓
Watch out. Do not round a 5 down. A 5 always rounds up.
Method. Brackets, then multiply, then subtract.
- Brackets. 6 + 4 = 10.10
- Multiply. 9 x 10 = 90.90
- Subtract. 90 - 15 = 75.75
- Write the answer. 75
Check. 90 - 15 = 75. ✓
Method. Count up 12 from -7.
- Count to zero. From -7 to 0 is 7 steps.7 steps
- Count on. 12 - 7 = 5.5
- Write the answer. 5
Check. -7 + 12 = 5. ✓
Method. Find the factors of 30, then keep the ones that are multiples of 5.
- Factors of 30. 1, 2, 3, 5, 6, 10, 15, 30.1, 2, 3, 5, 6, 10, 15, 30
- Keep the multiples of 5. Numbers in the 5 times table are 5, 10, 15 and 30.5, 10, 15, 30
- Write the answer. 5, 10, 15, 30
Check. Each of 5, 10, 15 and 30 divides 30. ✓
Method. Compare the digits, starting from the left.
- Count the digits in each number. A number with more digits is larger.3,050 has 4 digits, 30,500 has 5 digits, 3,005 has 4 digits, 35,000 has 5 digits
- Compare numbers with the same number of digits. Look at the leading digit, then the next one.compare from the left
- Write them in order. Put them smallest to largest.3,005, 3,050, 30,500, 35,000
- Write the answer. 3,005, 3,050, 30,500, 35,000
Check. Each number is smaller than the one after it. ✓
Watch out. 4,500 has four digits and 45,000 has five, so 45,000 is larger.
Method. Divide and multiply first, then add.
- Divide. 144 / 12 = 12.12
- Multiply. 3 x 4 = 12.12
- Add. 12 + 12 = 24.24
- Write the answer. 24
Check. 12 + 12 = 24. ✓
Method. Take away the Year 6 pupils.
- Choose the operation. Other years = total minus Year 6.642 - 185
- Work it out. 642 - 185 = 457.457
- Write the answer. 457
Check. 457 + 185 = 642. ✓
Method. Check the numbers 51 to 59 for factors other than 1 and themselves.
- Test 51. 51 = 3 x 17, so it is not prime.no
- Test 53. 53 is not in the 2, 3, 5 or 7 times tables.53 is prime
- Write the answer. 53
Check. 53 has only the factors 1 and 53. ✓
Method. Compare a number and its negative partner.
- Pick a number. Use 4.4
- Compare. 4 is above zero, -4 is below zero.4 > -4
- Write the answer. Yes, because 4 is above zero and -4 is below zero, so 4 is bigger than -4.
Check. 4 > -4. ✓
Method. Add the two totals.
- Add. 2,450 + 3,890.2,450 + 3,890
- Write the total. Add the thousands, hundreds, tens and units.6,340
- Write the answer. 6340
Check. 6,340 - 3,890 = 2,450. ✓
Total 17 marks. Allow about 17 minutes.
Exercise 1X Practice test questions
Non-calculatorMethod. Find the column, then multiply the digit by its column value.
- Find the column of the digit. The digit 5 is in the hundred thousands column.hundred thousands
- Work out the value. The value is the digit times the column value 100,000.5 x 100,000 = 500,000
- Write the answer. 500,000
Check. 500,000 = 5 x 100,000. ✓
Watch out. The value of a digit depends on its column, not just on the digit.
Method. Look at the digit next to the rounding column.
- Find the rounding column and the digit next to it. Rounding to the nearest 10,000 looks at the thousands digit.digit to check = 4
- Decide up or down. A 5 or more rounds up; 4 or less rounds down.4 is 4 or less, so round down
- Write the rounded number. Keep the digits to the left and make the rest zeros.260,000
- Write the answer. 260,000
Check. 260,000 is a multiple of 10,000 and is the closest one to 264,850. ✓
Watch out. Do not round a 5 down. A 5 always rounds up.
Method. Add two million.
- Add. 4,375,210 + 2,000,000.6,375,210
- Write the answer. 6375210
Check. 6,375,210 - 2,000,000 = 4,375,210. ✓
Method. Count down 5 from -2.
- Count down. -2 - 5 = -7.-7
- Write the answer. -7
Check. -7 + 5 = -2. ✓
Method. Divide and multiply first, then add and subtract.
- Divide. 20 / 4 = 5.5
- Multiply. 5 x 3 = 15.15
- Add and subtract. 5 + 15 - 1 = 19.19
- Write the answer. 19
Check. 5 + 15 - 1 = 19. ✓
Method. List multiples of 4 and 10 until one is in both lists.
- Multiples of 4. 4, 8, 12, 16, 20.4, 8, 12, 16, 20
- Multiples of 10. 10, 20.10, 20
- Find the first shared one. 20 is in both lists.20
- Write the answer. 20
Check. 20 / 4 = 5 and 20 / 10 = 2. ✓
Method. Find the total seats first, then take away the seats sold.
- Total seats. 24 x 36 = 864.864
- Empty seats. 864 - 700 = 164.164
- Write the answer. 164
Check. 700 + 164 = 864. ✓
Method. Recall the rule for a prime.
- Recall the rule. A prime has exactly two factors: 1 and itself.two factors
- Test 1. 1 has only the factor 1, so it has one factor.one factor only
- Write the answer. 1 has only one factor, so it does not have exactly two factors.
Check. 1 has one factor, not two. ✓
Method. Test each option.
- Check each option. Test each one against the rule or the calculation.
- Tick the true ones. A number in both the 3 and the 4 times tables is a multiple of 12.12, 24
- Leave the others unticked. These do not match.6, 18, 30 are not ticked
- Write the answer. 12, 24
Check. Each ticked option is true. ✓
Watch out. 6, 18 and 30 are not in the 4 times table, so they are not ticked.
Method. Look at the tens digit.
- Look at the tens digit. The tens digit is 4, so round down.4 is 4 or less
- Write the answer. 6,200 is the rounded number.6,200
- Write the answer. 6,200
Check. 6,200 is the nearest hundred to 6,249. ✓
Method. Smallest means the smallest digit first, but not zero first.
- Choose the first digit. Zero cannot be first in a four-digit number, so use 3.3
- Put the rest in order. Put 0, 5 and 7 in order.0, 5, 7
- Write the number. The smallest is 3,057.3,057
- Write the answer. 3,057.
Check. 3,057 uses each digit once. ✓