Standard form, rounding and estimation
By the end of this chapter you will be able to
- Write large and small numbers in standard form, and change standard form back to an ordinary number
- Compare and order numbers in standard form, and multiply and divide them, writing the answer in standard form
- Round a number to a given number of decimal places or significant figures
- Estimate the answer to a calculation by rounding to one significant figure, and say whether an answer is reasonable
- Write the error interval for a measurement that has been rounded
- Use a calculator with brackets, round the answer as asked, and check it with an estimate
You should already know
- Place value of decimals, and powers of 10 (Chapter 1)
- Rounding to the nearest 10, 100 and 1000 (Year 6, Chapter 1)
- Fractions and decimals (Chapter 2)
Where this leads
- Standard form with the index laws in Year 8
- Bounds and error in calculations in Years 8 and 9
National Curriculum for England, Key Stage 3 mathematics, Year 7. Content: N8 interpret and compare numbers in standard form A × 10n, 1 ≤ A < 10, with n positive, negative or 0, N13 round to a given number of decimal places or significant figures, N14 estimate answers by rounding, and state the error interval for a measurement using inequality notation, N15 use a calculator and interpret the display; check with an estimate (calculator section only).
Exercise P Do you remember?
Non-calculatorMethod. Look at the column that holds the 7, then multiply the digit by the value of that column.
- Find the column of the 7. Counting from the right: ones, tens, hundreds, thousands.4,725: the 7 is in the hundreds column
- Multiply by the column value. The hundreds column is worth 100.7 × 100 = 700
Check. 700 + 25 = 725. ✓
Watch out. Look at the column first. A 7 in the hundreds column is worth 700, not 7.
Method. Look at the digit in the units column. It decides whether to round up or down.
- Find the units digit. The units digit is the last digit.3,452: the units digit is 2
- Decide. A 2 is less than 5, so we round down and keep the tens digit.3,450
- Write the answer. The number is the nearest multiple of 10.3,450
Check. 3,450 is 2 away from 3,452, and 3,460 would be 8 away. ✓
Watch out. The rounding digit is the one just to the right of the place you round to.
Method. Multiplying by 100 moves the decimal point two places to the right.
- Count the zeros in 100. There are two zeros, so move the point two places.two zeros
- Move the point. Move it two places to the right, adding a zero if needed.2.6 → 260
- Write the answer. The answer is bigger than 2.6, as it should be.260
Check. 260 ÷ 100 = 2.6. ✓
Watch out. Multiplying by 10 or 100 makes the number bigger. The point moves right, not left.
Method. Multiplying by 10 moves the decimal point one place to the right.
- Count the zeros in 10. There is one zero.one zero
- Move the point one place right. The digit 3 is now in the units column.0.3 → 3
- Write the answer. The answer is 3.3
Check. 3 ÷ 10 = 0.3. ✓
Watch out. 0.3 × 10 is three tenths of ten, which is 3.
Method. The small raised number tells us how many tens to multiply together.
- Write the multiplication. The power is 3, so there are three tens.10 × 10 × 10
- Multiply. Ten times ten is 100, and 100 times 10 is 1000.100 × 10 = 1000
Check. 1000 has three zeros, and the power is 3. ✓
Watch out. 103 is not 10 × 3, which is 30.
Standard form
A number is in standard form when it is written as A × 10n, where 1 ≤ A < 10 and n is a whole number. The power n tells us how many places to move the decimal point.
A big number has a positive power: 3,500,000 = 3.5 × 106, because the point moves 6 places left. A number between 0 and 1 has a negative power: 0.0042 = 4.2 × 10−3, because the point moves 3 places right.
Key ideas
- Standard form is A × 10n, where 1 ≤ A < 10.
- Big numbers (10 or more): n is positive. Count the places the point moves left.
- Numbers between 0 and 1: n is negative. Count the places the point moves right.
- To change back to an ordinary number, move the point n places: right if n is positive, left if n is negative.
Method. Move the decimal point so that there is one non-zero digit in front of it. Then count the places moved.
- Move the point to make a number from 1 to 10. The number must be at least 1 and less than 10.3500000 → 3.500000
- Count the places the point moved. A big number has a positive power of 10.6 places left, so the power is +6
- Write the answer in standard form. The number is the mantissa times 10 to the power.3.500000 × 106
Check. 3.500000 × 10 to the power 6 gives back 3500000. ✓
Watch out. Count the places carefully. A number between 0 and 1 gets a NEGATIVE power of 10.
Method. Move the decimal point so that there is one non-zero digit in front of it. Then count the places moved.
- Move the point to make a number from 1 to 10. The number must be at least 1 and less than 10.0.0042 → 4.2
- Count the places the point moved. A small number (less than 1) has a negative power of 10.3 places right, so the power is −3
- Write the answer in standard form. The number is the mantissa times 10 to the power.4.2 × 10−3
Check. 4.2 × 10 to the power −3 gives back 0.0042. ✓
Watch out. Count the places carefully. A number between 0 and 1 gets a NEGATIVE power of 10.
Method. The power of 10 tells us how far to move the decimal point. Add zeros if you need them.
- Look at the power of 10. The power is 5, so the point moves 5 places.6.1 × 105: move the point 5 places right
- Move the point and fill in zeros. Move the point 5 places to the right.6.1 with the point moved gives 610000
- Write the answer. Check the size: a big power gives a big number, a negative power gives a number less than 1.610000
Check. 610000 × 10−5 gives back 6.1. ✓
Watch out. Count the places. Writing too many or too few zeros changes the size by a factor of 10.
Watch out
Watch out: 4.2 × 10−3 is small, not big. A negative power makes the number smaller. Also, 30.6 × 103 is not in standard form, because 30.6 is not between 1 and 10.
Exam tip
Grown-up tip. Have the pupil say the number out loud: "three point five million" is 3.5 × 106. Count the zeros in 3,500,000 to check the power: there are six.
Exercise 3A Writing numbers in standard form
Non-calculatorMethod. Move the decimal point so that there is one non-zero digit in front of it. Then count the places moved.
- Move the point to make a number from 1 to 10. The number must be at least 1 and less than 10.3500000 → 3.500000
- Count the places the point moved. A big number has a positive power of 10.6 places left, so the power is +6
- Write the answer in standard form. The number is the mantissa times 10 to the power.3.500000 × 106
Check. 3.500000 × 10 to the power 6 gives back 3500000. ✓
Watch out. Count the places carefully. A number between 0 and 1 gets a NEGATIVE power of 10.
Method. Move the decimal point so that there is one non-zero digit in front of it. Then count the places moved.
- Move the point to make a number from 1 to 10. The number must be at least 1 and less than 10.47000 → 4.7000
- Count the places the point moved. A big number has a positive power of 10.4 places left, so the power is +4
- Write the answer in standard form. The number is the mantissa times 10 to the power.4.7000 × 104
Check. 4.7000 × 10 to the power 4 gives back 47000. ✓
Watch out. Count the places carefully. A number between 0 and 1 gets a NEGATIVE power of 10.
Method. Move the decimal point so that there is one non-zero digit in front of it. Then count the places moved.
- Move the point to make a number from 1 to 10. The number must be at least 1 and less than 10.8200 → 8.200
- Count the places the point moved. A big number has a positive power of 10.3 places left, so the power is +3
- Write the answer in standard form. The number is the mantissa times 10 to the power.8.200 × 103
Check. 8.200 × 10 to the power 3 gives back 8200. ✓
Watch out. Count the places carefully. A number between 0 and 1 gets a NEGATIVE power of 10.
Method. Move the decimal point so that there is one non-zero digit in front of it. Then count the places moved.
- Move the point to make a number from 1 to 10. The number must be at least 1 and less than 10.620 → 6.20
- Count the places the point moved. A big number has a positive power of 10.2 places left, so the power is +2
- Write the answer in standard form. The number is the mantissa times 10 to the power.6.20 × 102
Check. 6.20 × 10 to the power 2 gives back 620. ✓
Watch out. Count the places carefully. A number between 0 and 1 gets a NEGATIVE power of 10.
Method. Move the decimal point so that there is one non-zero digit in front of it. Then count the places moved.
- Move the point to make a number from 1 to 10. The number must be at least 1 and less than 10.0.0042 → 4.2
- Count the places the point moved. A small number (less than 1) has a negative power of 10.3 places right, so the power is −3
- Write the answer in standard form. The number is the mantissa times 10 to the power.4.2 × 10−3
Check. 4.2 × 10 to the power −3 gives back 0.0042. ✓
Watch out. Count the places carefully. A number between 0 and 1 gets a NEGATIVE power of 10.
Method. Move the decimal point so that there is one non-zero digit in front of it. Then count the places moved.
- Move the point to make a number from 1 to 10. The number must be at least 1 and less than 10.0.57 → 5.7
- Count the places the point moved. A small number (less than 1) has a negative power of 10.1 places right, so the power is −1
- Write the answer in standard form. The number is the mantissa times 10 to the power.5.7 × 10−1
Check. 5.7 × 10 to the power −1 gives back 0.57. ✓
Watch out. Count the places carefully. A number between 0 and 1 gets a NEGATIVE power of 10.
Method. Move the decimal point so that there is one non-zero digit in front of it. Then count the places moved.
- Move the point to make a number from 1 to 10. The number must be at least 1 and less than 10.0.00009 → 9
- Count the places the point moved. A small number (less than 1) has a negative power of 10.5 places right, so the power is −5
- Write the answer in standard form. The number is the mantissa times 10 to the power.9 × 10−5
Check. 9 × 10 to the power −5 gives back 0.00009. ✓
Watch out. Count the places carefully. A number between 0 and 1 gets a NEGATIVE power of 10.
Method. The power of 10 tells us how far to move the decimal point. Add zeros if you need them.
- Look at the power of 10. The power is 5, so the point moves 5 places.6.1 × 105: move the point 5 places right
- Move the point and fill in zeros. Move the point 5 places to the right.6.1 with the point moved gives 610000
- Write the answer. Check the size: a big power gives a big number, a negative power gives a number less than 1.610000
Check. 610000 × 10−5 gives back 6.1. ✓
Watch out. Count the places. Writing too many or too few zeros changes the size by a factor of 10.
Method. The power of 10 tells us how far to move the decimal point. Add zeros if you need them.
- Look at the power of 10. The power is 4, so the point moves 4 places.2.07 × 104: move the point 4 places right
- Move the point and fill in zeros. Move the point 4 places to the right.2.07 with the point moved gives 20700
- Write the answer. Check the size: a big power gives a big number, a negative power gives a number less than 1.20700
Check. 20700 × 10−4 gives back 2.07. ✓
Watch out. Count the places. Writing too many or too few zeros changes the size by a factor of 10.
Method. The power of 10 tells us how far to move the decimal point. Add zeros if you need them.
- Look at the power of 10. The power is −2, so the point moves 2 places.4.5 × 10−2: move the point 2 places left
- Move the point and fill in zeros. Move the point 2 places to the left.4.5 with the point moved gives 0.045
- Write the answer. Check the size: a big power gives a big number, a negative power gives a number less than 1.0.045
Check. 0.045 × 102 gives back 4.5. ✓
Watch out. Count the places. Writing too many or too few zeros changes the size by a factor of 10.
Method. Move the decimal point so that there is one non-zero digit in front of it. Then count the places moved.
- Move the point to make a number from 1 to 10. The number must be at least 1 and less than 10.1000000 → 1.000000
- Count the places the point moved. A big number has a positive power of 10.6 places left, so the power is +6
- Write the answer in standard form. The number is the mantissa times 10 to the power.1.000000 × 106
Check. 1.000000 × 10 to the power 6 gives back 1000000. ✓
Watch out. Count the places carefully. A number between 0 and 1 gets a NEGATIVE power of 10.
Method. Move the decimal point so that there is one non-zero digit in front of it. Then count the places moved.
- Move the point to make a number from 1 to 10. The number must be at least 1 and less than 10.0.001 → 1
- Count the places the point moved. A small number (less than 1) has a negative power of 10.3 places right, so the power is −3
- Write the answer in standard form. The number is the mantissa times 10 to the power.1 × 10−3
Check. 1 × 10 to the power −3 gives back 0.001. ✓
Watch out. Count the places carefully. A number between 0 and 1 gets a NEGATIVE power of 10.
Method. Move the decimal point so that there is one non-zero digit in front of it. Then count the places moved.
- Move the point to make a number from 1 to 10. The number must be at least 1 and less than 10.45000 → 4.5000
- Count the places the point moved. A big number has a positive power of 10.4 places left, so the power is +4
- Write the answer in standard form. The number is the mantissa times 10 to the power.4.5000 × 104
Check. 4.5000 × 10 to the power 4 gives back 45000. ✓
Watch out. Count the places carefully. A number between 0 and 1 gets a NEGATIVE power of 10.
Method. In standard form the number in front must be from 1 to 10. Check that first.
- Check the number in front. It must be at least 1 and less than 10.30.6 is not less than 10
- Move the point. Move it one place left, and add 1 to the power (since the power goes from −3 to −2).30.6 × 10−3 = 3.06 × 10−2
- Write the answer. Now the number in front is between 1 and 10.3.06 × 10−2
Check. 3.06 × 10−2 = 0.0306. ✓
Watch out. Numbers like 30.6 × 10−3 look right but are not in standard form. Move the point until the number in front is from 1 to 10.
Comparing and calculating with standard form
To compare two numbers in standard form, first look at the power of 10. A bigger power gives a bigger number. If the powers are the same, compare the numbers in front.
To multiply, multiply the numbers in front and add the powers of 10. To divide, divide the numbers in front and subtract the powers. Then check that the number in front is from 1 to 10, and adjust the power if it is not.
Key ideas
- Compare the powers first: a bigger power of 10 gives a bigger number (for positive numbers).
- Multiplying: multiply the numbers in front and add the powers.
- Dividing: divide the numbers in front and subtract the powers.
- If the number in front is 10 or more, divide it by 10 and add 1 to the power.
Method. Multiply (or divide) the numbers in front, then deal with the powers of 10. Check the size of the number in front last.
- Work with the numbers in front of the powers. Do the multiplication of 3 and 2.3 × 2 = 6
- Add the powers of 10. When you multiply powers of 10, add the powers.104 × 103 = 107
- Write the answer. This is in standard form.6 × 107
Check. Estimate: the answer is about 6 × 107. ✓
Watch out. The number in front must be from 1 to 10. If it is not, adjust the power of 10 by one.
Method. Multiply (or divide) the numbers in front, then deal with the powers of 10. Check the size of the number in front last.
- Work with the numbers in front of the powers. Do the division of 8 and 2.8 ÷ 2 = 4
- Subtract the powers of 10. When you divide powers of 10, subtract the powers.106 ÷ 102 = 104
- Write the answer. This is in standard form.4 × 104
Check. Estimate: the answer is about 4 × 104. ✓
Watch out. The number in front must be from 1 to 10. If it is not, adjust the power of 10 by one.
Method. Multiply (or divide) the numbers in front, then deal with the powers of 10. Check the size of the number in front last.
- Work with the numbers in front of the powers. Do the multiplication of 6 and 3.6 × 3 = 18
- Add the powers of 10. When you multiply powers of 10, add the powers.105 × 103 = 108
- Make the number in front a number from 1 to 10. 18 is too big, so move the point one place and add 1 to the power.18 × 108 = 1.8 × 109
- Write the answer. This is in standard form.1.8 × 109
Check. Estimate: the answer is about 2 × 109. ✓
Watch out. The number in front must be from 1 to 10. If it is not, adjust the power of 10 by one.
Watch out
Watch out: 18 × 108 is not in standard form. Write it as 1.8 × 109.
Exam tip
Grown-up tip. Ask the pupil which number is bigger, 4 × 105 or 3.9 × 106, before doing any working. Then check by writing both out.
Exercise 3B Comparing and calculating in standard form
Non-calculatorMethod. Write both numbers as ordinary numbers, then compare them.
- Write 4 × 105 as an ordinary number. Move the point 5 places right.4 × 105 = 400,000
- Write 3.9 × 106 as an ordinary number. Move the point 6 places right.3.9 × 106 = 3,900,000
- Compare. Three million nine hundred thousand is bigger than four hundred thousand.3.9 × 106 is bigger
Check. The power 6 is bigger than 5, so 3.9 × 106 is bigger. ✓
Watch out. Compare the powers first. A bigger power beats a bigger number in front, when both numbers are in standard form.
Method. Negative powers get smaller as the power gets more negative, so a more negative power means a smaller number.
- Write each as an ordinary decimal. Move the point the number of places shown by the power, to the left.0.002, 0.04, 0.0009
- Compare the powers. The power −4 is the most negative, so it is the smallest.−4 < −3 < −2
- Write the order. List them from smallest to biggest.9 × 10−4, 2 × 10−3, 4 × 10−2
Check. 0.0009 < 0.002 < 0.04. ✓
Watch out. For numbers less than 1, the more negative the power, the smaller the number.
Method. Multiply (or divide) the numbers in front, then deal with the powers of 10. Check the size of the number in front last.
- Work with the numbers in front of the powers. Do the multiplication of 3 and 2.3 × 2 = 6
- Add the powers of 10. When you multiply powers of 10, add the powers.104 × 103 = 107
- Write the answer. This is in standard form.6 × 107
Check. Estimate: the answer is about 6 × 107. ✓
Watch out. The number in front must be from 1 to 10. If it is not, adjust the power of 10 by one.
Method. Multiply (or divide) the numbers in front, then deal with the powers of 10. Check the size of the number in front last.
- Work with the numbers in front of the powers. Do the division of 8 and 2.8 ÷ 2 = 4
- Subtract the powers of 10. When you divide powers of 10, subtract the powers.106 ÷ 102 = 104
- Write the answer. This is in standard form.4 × 104
Check. Estimate: the answer is about 4 × 104. ✓
Watch out. The number in front must be from 1 to 10. If it is not, adjust the power of 10 by one.
Method. Multiply (or divide) the numbers in front, then deal with the powers of 10. Check the size of the number in front last.
- Work with the numbers in front of the powers. Do the multiplication of 4.5 and 2.4.5 × 2 = 9.0
- Add the powers of 10. When you multiply powers of 10, add the powers.103 × 100 = 103
- Write the answer. This is in standard form.9.0 × 103
Check. Estimate: the answer is about 9 × 103. ✓
Watch out. The number in front must be from 1 to 10. If it is not, adjust the power of 10 by one.
Method. Multiply (or divide) the numbers in front, then deal with the powers of 10. Check the size of the number in front last.
- Work with the numbers in front of the powers. Do the multiplication of 6 and 3.6 × 3 = 18
- Add the powers of 10. When you multiply powers of 10, add the powers.105 × 103 = 108
- Make the number in front a number from 1 to 10. 18 is too big, so move the point one place and add 1 to the power.18 × 108 = 1.8 × 109
- Write the answer. This is in standard form.1.8 × 109
Check. Estimate: the answer is about 2 × 109. ✓
Watch out. The number in front must be from 1 to 10. If it is not, adjust the power of 10 by one.
Method. Multiply (or divide) the numbers in front, then deal with the powers of 10. Check the size of the number in front last.
- Work with the numbers in front of the powers. Do the multiplication of 1.2 and 5.1.2 × 5 = 6.0
- Add the powers of 10. When you multiply powers of 10, add the powers.10−2 × 104 = 102
- Write the answer. This is in standard form.6.0 × 102
Check. Estimate: the answer is about 6 × 102. ✓
Watch out. The number in front must be from 1 to 10. If it is not, adjust the power of 10 by one.
Method. Multiply (or divide) the numbers in front, then deal with the powers of 10. Check the size of the number in front last.
- Work with the numbers in front of the powers. Do the division of 9 and 3.9 ÷ 3 = 3
- Subtract the powers of 10. When you divide powers of 10, subtract the powers.107 ÷ 10−2 = 109
- Write the answer. This is in standard form.3 × 109
Check. Estimate: the answer is about 3 × 109. ✓
Watch out. The number in front must be from 1 to 10. If it is not, adjust the power of 10 by one.
Method. Write each part as an ordinary number first, then add.
- Write 3 × 104 as an ordinary number. Move the point 4 places right.3 × 104 = 30,000
- Write 2 × 103 as an ordinary number. Move the point 3 places right.2 × 103 = 2,000
- Add. Add the two ordinary numbers.30,000 + 2,000 = 32,000
Check. 32,000 = 3.2 × 104. ✓
Watch out. You cannot add the powers of 10. Adding is not the same as multiplying.
Method. Move the decimal point so that there is one non-zero digit in front of it. Then count the places moved.
- Move the point to make a number from 1 to 10. The number must be at least 1 and less than 10.0.0005 → 5
- Count the places the point moved. A small number (less than 1) has a negative power of 10.4 places right, so the power is −4
- Write the answer in standard form. The number is the mantissa times 10 to the power.5 × 10−4
Check. 5 × 10 to the power −4 gives back 0.0005. ✓
Watch out. Count the places carefully. A number between 0 and 1 gets a NEGATIVE power of 10.
Method. The power of 10 tells us how far to move the decimal point. Add zeros if you need them.
- Look at the power of 10. The power is −4, so the point moves 4 places.5 × 10−4: move the point 4 places left
- Move the point and fill in zeros. Move the point 4 places to the left.5 with the point moved gives 0.0005
- Write the answer. Check the size: a big power gives a big number, a negative power gives a number less than 1.0.0005
Check. 0.0005 × 104 gives back 5. ✓
Watch out. Count the places. Writing too many or too few zeros changes the size by a factor of 10.
Method. The power of 10 tells us how far to move the decimal point. Add zeros if you need them.
- Look at the power of 10. The power is 5, so the point moves 5 places.3.84 × 105: move the point 5 places right
- Move the point and fill in zeros. Move the point 5 places to the right.3.84 with the point moved gives 384000
- Write the answer. Check the size: a big power gives a big number, a negative power gives a number less than 1.384000
Check. 384000 × 10−5 gives back 3.84. ✓
Watch out. Count the places. Writing too many or too few zeros changes the size by a factor of 10.
Method. Multiply the numbers in front, then add the powers of 10.
- Multiply the numbers in front. Two times three is six.2 × 3 = 6
- Add the powers. The powers are 3 and 2, so add them.103 × 102 = 105
- Write the answer. The number in front is from 1 to 10, so no change is needed.6 × 105
Check. 6 × 105 = 600,000, and 2,000 × 300 = 600,000. ✓
Watch out. Sam added the numbers in front and the powers wrongly: he should add the powers, 3 + 2, not multiply them.
Rounding to decimal places and significant figures
To round to a number of decimal places, look at the digit just after the last place you keep. If it is 5 or more, round up. If it is less than 5, keep the digit the same.
To round to significant figures, count from the first digit that is not zero. The first significant figure is that first non-zero digit. Zeros at the start of a decimal are not significant. Round in the same way, using the digit after the last figure you keep.
Key ideas
- Look at the digit after the last place you keep: 5 or more rounds up.
- Decimal places count digits after the decimal point.
- Significant figures start counting at the first digit that is not zero.
- Keep the zeros that are needed to show the place value, such as 6.10 to 3 significant figures.
Method. Look at the digit after the last decimal place you keep. If it is 5 or more, round up. If it is less than 5, leave it.
- Find the digit after the 2 decimal places. This is the digit that decides whether to round up or down.4.7635: the next digit is 3
- Round up or down. 3 is less than 5, so leave the digit the same.keep the digit
- Write the answer with exactly the decimal places asked for. Keep the zeros at the end, if the place needs them.4.76
Check. 4.76 is within 0.005 of 4.7635. ✓
Watch out. Look only at the digit after the last one you keep. Do not round the digits one at a time.
Method. Find the first non-zero digit, count n significant figures from there, then look at the next digit.
- Find the first significant figure. The first significant figure is the first digit that is not zero.84265: first significant figure is 8
- Look at the digit after the last significant figure you keep. That digit is 6, so round up.next digit 6
- Write the answer to n significant figures, filling with zeros for places. Zeros are needed to keep the place value of the number.84300
Check. The answer has 3 significant figures. ✓
Watch out. Zeros after the first digit count as significant figures when they come before a point, but zeros at the start do not.
Method. Find the first non-zero digit, count n significant figures from there, then look at the next digit.
- Find the first significant figure. The first significant figure is the first digit that is not zero.0.004672: first significant figure is 4
- Look at the digit after the last significant figure you keep. That digit is 7, so round up.next digit 7
- Write the answer to n significant figures, filling with zeros for places. Zeros are needed to keep the place value of the number.0.0047
Check. The answer has 2 significant figures. ✓
Watch out. Zeros after the first digit count as significant figures when they come before a point, but zeros at the start do not.
Watch out
Watch out: rounding 4.7635 to 2 decimal places is 4.76, not 4.8. Only the digit just after the last place counts. Also, 0.004672 to 2 significant figures is 0.0047, because the first two significant figures are 4 and 6, not 0 and 0.
Exam tip
Grown-up tip. Ask the pupil to circle the place they are rounding to, then the digit to its right, before doing anything else.
Exercise 3C Rounding
Non-calculatorMethod. Look at the digit after the last decimal place you keep. If it is 5 or more, round up. If it is less than 5, leave it.
- Find the digit after the 2 decimal places. This is the digit that decides whether to round up or down.4.7635: the next digit is 3
- Round up or down. 3 is less than 5, so leave the digit the same.keep the digit
- Write the answer with exactly the decimal places asked for. Keep the zeros at the end, if the place needs them.4.76
Check. 4.76 is within 0.005 of 4.7635. ✓
Watch out. Look only at the digit after the last one you keep. Do not round the digits one at a time.
Method. Look at the digit after the last decimal place you keep. If it is 5 or more, round up. If it is less than 5, leave it.
- Find the digit after the 3 decimal places. This is the digit that decides whether to round up or down.0.4568: the next digit is 8
- Round up or down. 8 is 5 or more, so round up.round up
- Write the answer with exactly the decimal places asked for. Keep the zeros at the end, if the place needs them.0.457
Check. 0.457 is within 0.0005 of 0.4568. ✓
Watch out. Look only at the digit after the last one you keep. Do not round the digits one at a time.
Method. Look at the digit after the last decimal place you keep. If it is 5 or more, round up. If it is less than 5, leave it.
- Find the digit after the 2 decimal places. This is the digit that decides whether to round up or down.9.996: the next digit is 6
- Round up or down. 6 is 5 or more, so round up.round up
- Write the answer with exactly the decimal places asked for. Keep the zeros at the end, if the place needs them.10.00
Check. 10.00 is within 0.005 of 9.996. ✓
Watch out. Look only at the digit after the last one you keep. Do not round the digits one at a time.
Method. Find the first non-zero digit, count n significant figures from there, then look at the next digit.
- Find the first significant figure. The first significant figure is the first digit that is not zero.84265: first significant figure is 8
- Look at the digit after the last significant figure you keep. That digit is 6, so round up.next digit 6
- Write the answer to n significant figures, filling with zeros for places. Zeros are needed to keep the place value of the number.84300
Check. The answer has 3 significant figures. ✓
Watch out. Zeros after the first digit count as significant figures when they come before a point, but zeros at the start do not.
Method. Find the first non-zero digit, count n significant figures from there, then look at the next digit.
- Find the first significant figure. The first significant figure is the first digit that is not zero.0.004672: first significant figure is 4
- Look at the digit after the last significant figure you keep. That digit is 7, so round up.next digit 7
- Write the answer to n significant figures, filling with zeros for places. Zeros are needed to keep the place value of the number.0.0047
Check. The answer has 2 significant figures. ✓
Watch out. Zeros after the first digit count as significant figures when they come before a point, but zeros at the start do not.
Method. Find the first non-zero digit, count n significant figures from there, then look at the next digit.
- Find the first significant figure. The first significant figure is the first digit that is not zero.6.0951: first significant figure is 6
- Look at the digit after the last significant figure you keep. That digit is 5, so round up.next digit 5
- Write the answer to n significant figures, filling with zeros for places. Zeros are needed to keep the place value of the number.6.10
Check. The answer has 3 significant figures. ✓
Watch out. Zeros after the first digit count as significant figures when they come before a point, but zeros at the start do not.
Method. Find the first non-zero digit, count n significant figures from there, then look at the next digit.
- Find the first significant figure. The first significant figure is the first digit that is not zero.2799: first significant figure is 2
- Look at the digit after the last significant figure you keep. That digit is 7, so round up.next digit 7
- Write the answer to n significant figures, filling with zeros for places. Zeros are needed to keep the place value of the number.3000
Check. The answer has 1 significant figures. ✓
Watch out. Zeros after the first digit count as significant figures when they come before a point, but zeros at the start do not.
Method. Find the first non-zero digit, count n significant figures from there, then look at the next digit.
- Find the first significant figure. The first significant figure is the first digit that is not zero.0.0796: first significant figure is 7
- Look at the digit after the last significant figure you keep. That digit is 9, so round up.next digit 9
- Write the answer to n significant figures, filling with zeros for places. Zeros are needed to keep the place value of the number.0.08
Check. The answer has 1 significant figures. ✓
Watch out. Zeros after the first digit count as significant figures when they come before a point, but zeros at the start do not.
Method. Look at the digit after the last decimal place you keep. If it is 5 or more, round up. If it is less than 5, leave it.
- Find the digit after the 2 decimal places. This is the digit that decides whether to round up or down.3.14159: the next digit is 1
- Round up or down. 1 is less than 5, so leave the digit the same.keep the digit
- Write the answer with exactly the decimal places asked for. Keep the zeros at the end, if the place needs them.3.14
Check. 3.14 is within 0.005 of 3.14159. ✓
Watch out. Look only at the digit after the last one you keep. Do not round the digits one at a time.
Method. Find the first non-zero digit, count n significant figures from there, then look at the next digit.
- Find the first significant figure. The first significant figure is the first digit that is not zero.56874: first significant figure is 5
- Look at the digit after the last significant figure you keep. That digit is 8, so round up.next digit 8
- Write the answer to n significant figures, filling with zeros for places. Zeros are needed to keep the place value of the number.57000
Check. The answer has 2 significant figures. ✓
Watch out. Zeros after the first digit count as significant figures when they come before a point, but zeros at the start do not.
Method. Find the first non-zero digit, count n significant figures from there, then look at the next digit.
- Find the first significant figure. The first significant figure is the first digit that is not zero.0.3049: first significant figure is 3
- Look at the digit after the last significant figure you keep. That digit is 4, so keep the digit.next digit 4
- Write the answer to n significant figures, filling with zeros for places. Zeros are needed to keep the place value of the number.0.30
Check. The answer has 2 significant figures. ✓
Watch out. Zeros after the first digit count as significant figures when they come before a point, but zeros at the start do not.
Method. Look at the units digit. It tells us whether to round the tens up or down.
- Find the digit that decides. The digit in the units place is 6.1,256: units digit is 6
- Round up. A 6 is 5 or more, so the tens digit goes up by 1. The 5 becomes 6.125 tens → 126 tens
- Write the answer. The units become zero.1,260
Check. 1,260 is 4 away from 1,256, and 1,250 would be 6 away. ✓
Watch out. To round to the nearest 10, look at the units. To round to the nearest 100, look at the tens.
Method. Look at the digit after the first decimal place.
- Find the digit after the first decimal place. The first decimal place is the 4 in 3.4951.The next digit is 9
- Round. 9 is 5 or more, so round up.3.4 → 3.5
- Write the answer. Sam forgot to look at the digit that decides.3.5
Check. 3.5 is 0.0049 away from 3.4951, and 3.4 is 0.0951 away. ✓
Watch out. Sam only looked at the digits 49, not the 9 that comes next.
Estimation and error intervals
An estimate is a sensible answer found by rounding each number to 1 significant figure and working out the easy calculation. Estimates help us check that a calculator answer makes sense.
A measurement that has been rounded has an error interval. If a length is 40 cm to the nearest 10 cm, the true length is at least 35 cm and less than 45 cm. Write this as 35 ≤ length < 45. The lower bound is included, and the upper bound is not.
Key ideas
- To estimate, round every number to 1 significant figure, then calculate.
- Half the rounding unit is the error. For a value to the nearest 10, the error is 5 in either direction.
- Error interval: lower bound ≤ true value < upper bound.
- When adding measurements, add the lower bounds for the least possible total.
Method. Round each number to 1 significant figure, then work out the easy calculation.
- Round each number to 1 significant figure. Keep the first digit and make the others zeros.48 ≈ 50 and 19 ≈ 20
- Work out the easy calculation. Use the rounded numbers, which are easy to work with.50 × 20 = 1000
- Write the estimate. An estimate is only close to the real answer.about 1000
Check. The real answer is 48 × 19 and it is close to 1000. ✓
Watch out. Estimate first to check your calculator answer. An answer far from the estimate is probably wrong.
Method. Round each number to 1 significant figure, then work out the easy calculation.
- Round each number to 1 significant figure. Keep the first digit and make the others zeros.412 ≈ 400 and 7.9 ≈ 8
- Work out the easy calculation. Use the rounded numbers, which are easy to work with.400 ÷ 8 = 50
- Write the estimate. An estimate is only close to the real answer.about 50
Check. The real answer is 412 ÷ 7.9 and it is close to 50. ✓
Watch out. Estimate first to check your calculator answer. An answer far from the estimate is probably wrong.
Method. Half the rounding unit is the most the true value can be above or below the rounded value.
- Find half the rounding unit. The value was rounded to the nearest 10 cm, so the error is at most half of that.10 ÷ 2 = 5
- Find the smallest possible value. Subtract half the unit from the rounded value. This value is included.40 − 5 = 35
- Find the largest possible value. Add half the unit. This value is NOT included, because it would round up.40 + 5 = 45
- Write the interval. Use ≤ for the lower bound and < for the upper bound.35 ≤ length < 45
Check. 35 rounds to 40 and so does every number up to, but not including, 45. ✓
Watch out. The lower bound is included (≤). The upper bound is not (<), because a value of exactly the upper bound would round up.
Watch out
Watch out: the upper bound is NOT included. A length of exactly 45 cm to the nearest 10 cm would be rounded up to 50 cm, so 45 is not in the interval.
Exam tip
Grown-up tip. Ask the pupil to estimate first, every time, before using a calculator. The estimate tells them if the calculator answer is roughly right.
Exercise 3D Estimating and error intervals
Non-calculatorMethod. Round each number to 1 significant figure, then work out the easy calculation.
- Round each number to 1 significant figure. Keep the first digit and make the others zeros.48 ≈ 50 and 19 ≈ 20
- Work out the easy calculation. Use the rounded numbers, which are easy to work with.50 × 20 = 1000
- Write the estimate. An estimate is only close to the real answer.about 1000
Check. The real answer is 48 × 19 and it is close to 1000. ✓
Watch out. Estimate first to check your calculator answer. An answer far from the estimate is probably wrong.
Method. Round each number to 1 significant figure, then work out the easy calculation.
- Round each number to 1 significant figure. Keep the first digit and make the others zeros.412 ≈ 400 and 7.9 ≈ 8
- Work out the easy calculation. Use the rounded numbers, which are easy to work with.400 ÷ 8 = 50
- Write the estimate. An estimate is only close to the real answer.about 50
Check. The real answer is 412 ÷ 7.9 and it is close to 50. ✓
Watch out. Estimate first to check your calculator answer. An answer far from the estimate is probably wrong.
Method. Round each number to 1 significant figure, then work out the easy calculation.
- Round each number to 1 significant figure. Keep the first digit and make the others zeros.6.3 ≈ 6 and 0.48 ≈ 0.5
- Work out the easy calculation. Use the rounded numbers, which are easy to work with.6 × 0.5 = 3.0
- Write the estimate. An estimate is only close to the real answer.about 3.0
Check. The real answer is 6.3 × 0.48 and it is close to 3.0. ✓
Watch out. Estimate first to check your calculator answer. An answer far from the estimate is probably wrong.
Method. Round each number to 1 significant figure, then work out the easy calculation.
- Round each number to 1 significant figure. Keep the first digit and make the others zeros.2873 ≈ 3000 and 1488 ≈ 1000
- Work out the easy calculation. Use the rounded numbers, which are easy to work with.3000 + 1000 = 4000
- Write the estimate. An estimate is only close to the real answer.about 4000
Check. The real answer is 2873 + 1488 and it is close to 4000. ✓
Watch out. Estimate first to check your calculator answer. An answer far from the estimate is probably wrong.
Method. Round each number to 1 significant figure, then work out the easy calculation.
- Round each number to 1 significant figure. Keep the first digit and make the others zeros.38 ≈ 40 and 21 ≈ 20
- Work out the easy calculation. Use the rounded numbers, which are easy to work with.40 × 20 = 800
- Write the estimate. An estimate is only close to the real answer.about 800
Check. The real answer is 38 × 21 and it is close to 800. ✓
Watch out. Estimate first to check your calculator answer. An answer far from the estimate is probably wrong.
Method. Round each number to 1 significant figure, then work out the easy calculation.
- Round each number to 1 significant figure. Keep the first digit and make the others zeros.0.37 ≈ 0.4 and 0.61 ≈ 0.6
- Work out the easy calculation. Use the rounded numbers, which are easy to work with.0.4 × 0.6 = 0.24
- Write the estimate. An estimate is only close to the real answer.about 0.24
Check. The real answer is 0.37 × 0.61 and it is close to 0.24. ✓
Watch out. Estimate first to check your calculator answer. An answer far from the estimate is probably wrong.
Method. Half the rounding unit is the most the true value can be above or below the rounded value.
- Find half the rounding unit. The value was rounded to the nearest 10 cm, so the error is at most half of that.10 ÷ 2 = 5
- Find the smallest possible value. Subtract half the unit from the rounded value. This value is included.40 − 5 = 35
- Find the largest possible value. Add half the unit. This value is NOT included, because it would round up.40 + 5 = 45
- Write the interval. Use ≤ for the lower bound and < for the upper bound.35 ≤ length < 45
Check. 35 rounds to 40 and so does every number up to, but not including, 45. ✓
Watch out. The lower bound is included (≤). The upper bound is not (<), because a value of exactly the upper bound would round up.
Method. Half the rounding unit is the most the true value can be above or below the rounded value.
- Find half the rounding unit. The value was rounded to the nearest 0.1 kg, so the error is at most half of that.0.1 ÷ 2 = 0.05
- Find the smallest possible value. Subtract half the unit from the rounded value. This value is included.2.5 − 0.05 = 2.45
- Find the largest possible value. Add half the unit. This value is NOT included, because it would round up.2.5 + 0.05 = 2.55
- Write the interval. Use ≤ for the lower bound and < for the upper bound.2.45 ≤ m < 2.55
Check. 2.45 rounds to 2.5 and so does every number up to, but not including, 2.55. ✓
Watch out. The lower bound is included (≤). The upper bound is not (<), because a value of exactly the upper bound would round up.
Method. Half the rounding unit is the most the true value can be above or below the rounded value.
- Find half the rounding unit. The value was rounded to the nearest 10 g, so the error is at most half of that.10 ÷ 2 = 5
- Find the smallest possible value. Subtract half the unit from the rounded value. This value is included.150 − 5 = 145
- Find the largest possible value. Add half the unit. This value is NOT included, because it would round up.150 + 5 = 155
- Write the interval. Use ≤ for the lower bound and < for the upper bound.145 ≤ m < 155
Check. 145 rounds to 150 and so does every number up to, but not including, 155. ✓
Watch out. The lower bound is included (≤). The upper bound is not (<), because a value of exactly the upper bound would round up.
Method. Half the rounding unit is the most the true value can be above or below the rounded value.
- Find half the rounding unit. The value was rounded to the nearest 0.1 s, so the error is at most half of that.0.1 ÷ 2 = 0.05
- Find the smallest possible value. Subtract half the unit from the rounded value. This value is included.3.6 − 0.05 = 3.55
- Find the largest possible value. Add half the unit. This value is NOT included, because it would round up.3.6 + 0.05 = 3.65
- Write the interval. Use ≤ for the lower bound and < for the upper bound.3.55 ≤ t < 3.65
Check. 3.55 rounds to 3.6 and so does every number up to, but not including, 3.65. ✓
Watch out. The lower bound is included (≤). The upper bound is not (<), because a value of exactly the upper bound would round up.
Method. The least possible value of each stick is half a centimetre below its measured length. Add the lower bounds.
- Find the lower bound of the first stick. Half of 1 cm is 0.5 cm, so the least length is 12 − 0.5.12 − 0.5 = 11.5 cm
- Find the lower bound of the second stick. The same rule applies to the 8 cm stick.8 − 0.5 = 7.5 cm
- Add the lower bounds. The least possible total is the sum of the least possible lengths.11.5 + 7.5 = 19 cm
Check. The measured total is 20 cm, and the least possible total is 19 cm. ✓
Watch out. For the least possible total, use the lower bound of each measurement, not the measured value.
Method. Round each number to 1 significant figure, then check how close the estimate is to the real answer.
- Round the numbers. 0.48 rounds to 0.5 and 19.6 rounds to 20.0.48 ≈ 0.5 and 19.6 ≈ 20
- Work out the estimate. Use the easy calculation.0.5 × 20 = 10
- Compare with the real answer. The real answer is 0.48 × 19.6 = 9.408, which is close to 10.9.408 ≈ 10, so the estimate is reasonable
Check. 9.408 is close to 10, so the estimate is reasonable. ✓
Watch out. An estimate can be a little different from the real answer. It only needs to be close enough to check a calculation.
Using a calculator and checking the answer
Use brackets on a calculator when you want a whole part worked out first. For example, (3.6 + 2.4)2 is worked out as 62 = 36, but 3.6 + 2.42 would square 2.4 first.
Some calculators show standard form on the display. "4.5 E 6" means 4.5 × 106. Always check the answer with an estimate.
Key ideas
- Use brackets to control the order of the calculation.
- Round only at the end, using the rounding the question asks for.
- Check the answer with an estimate: if they are not close, look for a mistake.
- Some calculators use E for standard form: 4.5 E 6 means 4.5 × 106.
Method. Type the calculation into the calculator, using brackets where they are needed. Then round as the question asks.
- Type the calculation. Use brackets so the calculator does the right part first.1 ÷ 7
- Read the display. Write down all the digits the calculator shows, or the rounded value it gives.display: 0.142857142857
- Round to 3 decimal places Round only at the end, so that rounding does not spoil the answer.0.143
Check. Estimate first: 1 ÷ 7 is about the answer you got. ✓
Watch out. Round only at the end. Rounding the middle of a calculation can change the answer.
Method. Type the calculation into the calculator, using brackets where they are needed. Then round as the question asks.
- Type the calculation. Use brackets so the calculator does the right part first.(3.6 + 2.4)2 ÷ 4
- Read the display. Write down all the digits the calculator shows, or the rounded value it gives.display: 9
- Write the answer Round only at the end, so that rounding does not spoil the answer.9
Check. Estimate first: (3.6 + 2.4)^2 / 4 is about the answer you got. ✓
Watch out. Round only at the end. Rounding the middle of a calculation can change the answer.
Method. Type the calculation into the calculator, using brackets where they are needed. Then round as the question asks.
- Type the calculation. Use brackets so the calculator does the right part first.39.7 × 4.2
- Read the display. Write down all the digits the calculator shows, or the rounded value it gives.display: 166.74
- Write the answer Round only at the end, so that rounding does not spoil the answer.166.74
Check. Estimate first: 39.7 × 4.2 is about the answer you got. ✓
Watch out. Round only at the end. Rounding the middle of a calculation can change the answer.
Watch out
Watch out: 3 + 4 × 2 on a calculator is 3 + 8 = 11, not 14. Use brackets, (3 + 4) × 2, to make the calculator add first.
Exam tip
Grown-up tip. Ask the pupil to estimate before pressing any key. The calculator answer should be close to the estimate.
Exercise 3E Calculator questions
Calculator allowedMethod. Type the calculation into the calculator, using brackets where they are needed. Then round as the question asks.
- Type the calculation. Use brackets so the calculator does the right part first.47.6 × 3.8
- Read the display. Write down all the digits the calculator shows, or the rounded value it gives.display: 180.88
- Write the answer Round only at the end, so that rounding does not spoil the answer.180.88
Check. Estimate first: 47.6 × 3.8 is about the answer you got. ✓
Watch out. Round only at the end. Rounding the middle of a calculation can change the answer.
Method. Type the calculation into the calculator, using brackets where they are needed. Then round as the question asks.
- Type the calculation. Use brackets so the calculator does the right part first.(7.3 − 2.85)2
- Read the display. Write down all the digits the calculator shows, or the rounded value it gives.display: 19.8025
- Write the answer Round only at the end, so that rounding does not spoil the answer.19.8025
Check. Estimate first: (7.3 - 2.85)^2 is about the answer you got. ✓
Watch out. Round only at the end. Rounding the middle of a calculation can change the answer.
Method. Type the calculation into the calculator, using brackets where they are needed. Then round as the question asks.
- Type the calculation. Use brackets so the calculator does the right part first.5 ÷ 9
- Read the display. Write down all the digits the calculator shows, or the rounded value it gives.display: 0.5555555
- Round to 2 decimal places Round only at the end, so that rounding does not spoil the answer.0.56
Check. Estimate first: 5 / 9 is about the answer you got. ✓
Watch out. Round only at the end. Rounding the middle of a calculation can change the answer.
Method. Type the calculation into the calculator, using brackets where they are needed. Then round as the question asks.
- Type the calculation. Use brackets so the calculator does the right part first.√(64.8 + 3.2)
- Read the display. Write down all the digits the calculator shows, or the rounded value it gives.display: 8.246211251235321099642819711948154050294
- Round to 2 decimal places Round only at the end, so that rounding does not spoil the answer.8.25
Check. Estimate first: sqrt(68) is about the answer you got. ✓
Watch out. Round only at the end. Rounding the middle of a calculation can change the answer.
Method. Type the calculation into the calculator, using brackets where they are needed. Then round as the question asks.
- Type the calculation. Use brackets so the calculator does the right part first.1.54
- Read the display. Write down all the digits the calculator shows, or the rounded value it gives.display: 5.0625
- Write the answer Round only at the end, so that rounding does not spoil the answer.5.0625
Check. Estimate first: 1.5^4 is about the answer you got. ✓
Watch out. Round only at the end. Rounding the middle of a calculation can change the answer.
Method. Type the calculation into the calculator, using brackets where they are needed. Then round as the question asks.
- Type the calculation. Use brackets so the calculator does the right part first.17.9 ÷ 0.46
- Read the display. Write down all the digits the calculator shows, or the rounded value it gives.display: 38.91304348
- Round to 3 significant figures Round only at the end, so that rounding does not spoil the answer.38.9
Check. Estimate first: 17.9 / 0.46 is about the answer you got. ✓
Watch out. Round only at the end. Rounding the middle of a calculation can change the answer.
Method. Type each part with the E key, or use the power of 10 keys, then write the display in standard form.
- Type the calculation. Use brackets around each part.(2.8 E 4) ÷ (4 E −2)
- Read the display. The calculator shows 700000.display: 700000
- Write in standard form. Move the point 5 places left so the number in front is from 1 to 10.700,000 = 7 × 105
Check. Estimate: 3 ÷ 4 is less than 1, so 3 × 104 ÷ 4 × 10−2 is about 7 × 105. ✓
Watch out. Use the power key for each power of 10. Adding the powers, and dividing the numbers, is exactly the same as the calculator does.
Method. The E 6 means × 106. Move the decimal point 6 places to the right.
- Read the display. 4.5 E 6 means 4.5 × 106.4.5 × 106
- Move the point 6 places right. Add zeros to fill the gaps.4.5 → 4500000 → 4,500,000
- Write the answer. Check it has 7 digits, since 106 has six zeros.4,500,000
Check. 4,500,000 = 4.5 × 106. ✓
Watch out. The E number is a power of 10, not a number of zeros to type in.
Method. Type the calculation into the calculator, using brackets where they are needed. Then round as the question asks.
- Type the calculation. Use brackets so the calculator does the right part first.15% of 368 = 0.15 × 368
- Read the display. Write down all the digits the calculator shows, or the rounded value it gives.display: 55.2
- Write the answer Round only at the end, so that rounding does not spoil the answer.55.2
Check. Estimate first: 0.15 * 368 is about the answer you got. ✓
Watch out. Round only at the end. Rounding the middle of a calculation can change the answer.
Method. Type the calculation into the calculator, using brackets where they are needed. Then round as the question asks.
- Type the calculation. Use brackets so the calculator does the right part first.2 ÷ 9
- Read the display. Write down all the digits the calculator shows, or the rounded value it gives.display: 0.2222222222
- Round to 4 decimal places Round only at the end, so that rounding does not spoil the answer.0.2222
Check. Estimate first: 2 / 9 is about the answer you got. ✓
Watch out. Round only at the end. Rounding the middle of a calculation can change the answer.
Method. Type the calculation into the calculator, using brackets where they are needed. Then round as the question asks.
- Type the calculation. Use brackets so the calculator does the right part first.11 ÷ 13
- Read the display. Write down all the digits the calculator shows, or the rounded value it gives.display: 0.8461538462
- Round to 3 decimal places Round only at the end, so that rounding does not spoil the answer.0.846
Check. Estimate first: 11 / 13 is about the answer you got. ✓
Watch out. Round only at the end. Rounding the middle of a calculation can change the answer.
Method. Type the calculation into the calculator, using brackets where they are needed. Then round as the question asks.
- Type the calculation. Use brackets so the calculator does the right part first.(3.4 + 5.6 × 2) ÷ 0.5
- Read the display. Write down all the digits the calculator shows, or the rounded value it gives.display: 29.2
- Write the answer Round only at the end, so that rounding does not spoil the answer.29.2
Check. Estimate first: (3.4 + 5.6 * 2) / 0.5 is about the answer you got. ✓
Watch out. Round only at the end. Rounding the middle of a calculation can change the answer.
Method. Work out each calculation separately, using the order of operations for the second one.
- Work out (3.2 + 4.8) ÷ 2. The brackets are done first: 3.2 + 4.8 = 8, then divide by 2.8 ÷ 2 = 4
- Work out 3.2 + 4.8 ÷ 2. Division comes before addition, so divide 4.8 by 2 first.4.8 ÷ 2 = 2.4, then 3.2 + 2.4 = 5.6
- Explain. The brackets change which part is worked out first.4 and 5.6: the brackets change the answer
Check. Estimate: 3 + 5 ÷ 2 is about 5.5, which is close to 5.6, not 4. ✓
Watch out. Use brackets on the calculator when you want to add first, so that the calculator does what you mean.
Chapter summary
| Skill | Method |
|---|---|
| Standard form | A × 10n, where 1 ≤ A < 10; positive n for big numbers, negative n for numbers below 1 |
| Multiplying and dividing | Multiply or divide the numbers in front; add or subtract the powers; adjust so that 1 ≤ A < 10 |
| Rounding decimal places | Look at the digit after the last place you keep: 5 or more rounds up |
| Significant figures | Count from the first non-zero digit; zeros that show place value are kept |
| Estimating | Round each number to 1 significant figure, then calculate the easy answer |
| Error interval | Half the rounding unit either side of the value: lower ≤ value < upper |
| Calculator use | Use brackets; round only at the end; check with an estimate |
Can you…?
- I can write large and small numbers in standard form, and change back to ordinary numbers
- I can compare, multiply and divide numbers in standard form
- I can round to decimal places and to significant figures
- I can estimate an answer by rounding to 1 significant figure
- I can write the error interval for a rounded measurement
- I can use a calculator with brackets, and check the answer with an estimate
Key terms
- standard form
- a number written as A × 10n where 1 ≤ A < 10
- power of 10
- the number of tens multiplied together, such as 103 = 1000
- decimal place
- a place after the decimal point, such as the tenths or hundredths place
- significant figure
- a digit that counts for the size of a number, starting from the first digit that is not zero
- estimate
- an approximate answer found by rounding the numbers first
- error interval
- the range of values a rounded measurement could have, written with ≤ and <
- bracket
- a symbol ( ) that makes the calculator, or you, work out the part inside first
Exercise 3Z Check-up
Non-calculatorMethod. Move the decimal point so that there is one non-zero digit in front of it. Then count the places moved.
- Move the point to make a number from 1 to 10. The number must be at least 1 and less than 10.73000 → 7.3000
- Count the places the point moved. A big number has a positive power of 10.4 places left, so the power is +4
- Write the answer in standard form. The number is the mantissa times 10 to the power.7.3000 × 104
Check. 7.3000 × 10 to the power 4 gives back 73000. ✓
Watch out. Count the places carefully. A number between 0 and 1 gets a NEGATIVE power of 10.
Method. The power of 10 tells us how far to move the decimal point. Add zeros if you need them.
- Look at the power of 10. The power is −3, so the point moves 3 places.9.2 × 10−3: move the point 3 places left
- Move the point and fill in zeros. Move the point 3 places to the left.9.2 with the point moved gives 0.0092
- Write the answer. Check the size: a big power gives a big number, a negative power gives a number less than 1.0.0092
Check. 0.0092 × 103 gives back 9.2. ✓
Watch out. Count the places. Writing too many or too few zeros changes the size by a factor of 10.
Method. Look at the digit after the last decimal place you keep. If it is 5 or more, round up. If it is less than 5, leave it.
- Find the digit after the 3 decimal places. This is the digit that decides whether to round up or down.6.8285: the next digit is 5
- Round up or down. 5 is 5 or more, so round up.round up
- Write the answer with exactly the decimal places asked for. Keep the zeros at the end, if the place needs them.6.829
Check. 6.829 is within 0.0005 of 6.8285. ✓
Watch out. Look only at the digit after the last one you keep. Do not round the digits one at a time.
Method. Find the first non-zero digit, count n significant figures from there, then look at the next digit.
- Find the first significant figure. The first significant figure is the first digit that is not zero.31482: first significant figure is 3
- Look at the digit after the last significant figure you keep. That digit is 4, so keep the digit.next digit 4
- Write the answer to n significant figures, filling with zeros for places. Zeros are needed to keep the place value of the number.31000
Check. The answer has 2 significant figures. ✓
Watch out. Zeros after the first digit count as significant figures when they come before a point, but zeros at the start do not.
Method. Multiply (or divide) the numbers in front, then deal with the powers of 10. Check the size of the number in front last.
- Work with the numbers in front of the powers. Do the multiplication of 4 and 5.4 × 5 = 20
- Add the powers of 10. When you multiply powers of 10, add the powers.103 × 10−1 = 102
- Make the number in front a number from 1 to 10. 20 is too big, so move the point one place and add 1 to the power.20 × 102 = 2 × 103
- Write the answer. This is in standard form.2 × 103
Check. Estimate: the answer is about 2 × 103. ✓
Watch out. The number in front must be from 1 to 10. If it is not, adjust the power of 10 by one.
Method. Round each number to 1 significant figure, then work out the easy calculation.
- Round each number to 1 significant figure. Keep the first digit and make the others zeros.312 ≈ 300 and 0.49 ≈ 0.5
- Work out the easy calculation. Use the rounded numbers, which are easy to work with.300 × 0.5 = 150
- Write the estimate. An estimate is only close to the real answer.about 150
Check. The real answer is 312 × 0.49 and it is close to 150. ✓
Watch out. Estimate first to check your calculator answer. An answer far from the estimate is probably wrong.
Method. Half the rounding unit is the most the true value can be above or below the rounded value.
- Find half the rounding unit. The value was rounded to the nearest 10 kg, so the error is at most half of that.10 ÷ 2 = 5
- Find the smallest possible value. Subtract half the unit from the rounded value. This value is included.70 − 5 = 65
- Find the largest possible value. Add half the unit. This value is NOT included, because it would round up.70 + 5 = 75
- Write the interval. Use ≤ for the lower bound and < for the upper bound.65 ≤ m < 75
Check. 65 rounds to 70 and so does every number up to, but not including, 75. ✓
Watch out. The lower bound is included (≤). The upper bound is not (<), because a value of exactly the upper bound would round up.
Method. Move the decimal point so that there is one non-zero digit in front of it. Then count the places moved.
- Move the point to make a number from 1 to 10. The number must be at least 1 and less than 10.0.00043 → 4.3
- Count the places the point moved. A small number (less than 1) has a negative power of 10.4 places right, so the power is −4
- Write the answer in standard form. The number is the mantissa times 10 to the power.4.3 × 10−4
Check. 4.3 × 10 to the power −4 gives back 0.00043. ✓
Watch out. Count the places carefully. A number between 0 and 1 gets a NEGATIVE power of 10.
Method. Look at the digit after the last decimal place you keep. If it is 5 or more, round up. If it is less than 5, leave it.
- Find the digit after the 2 decimal places. This is the digit that decides whether to round up or down.0.7149: the next digit is 4
- Round up or down. 4 is less than 5, so leave the digit the same.keep the digit
- Write the answer with exactly the decimal places asked for. Keep the zeros at the end, if the place needs them.0.71
Check. 0.71 is within 0.005 of 0.7149. ✓
Watch out. Look only at the digit after the last one you keep. Do not round the digits one at a time.
Method. Multiply (or divide) the numbers in front, then deal with the powers of 10. Check the size of the number in front last.
- Work with the numbers in front of the powers. Do the division of 9 and 3.9 ÷ 3 = 3
- Subtract the powers of 10. When you divide powers of 10, subtract the powers.108 ÷ 104 = 104
- Write the answer. This is in standard form.3 × 104
Check. Estimate: the answer is about 3 × 104. ✓
Watch out. The number in front must be from 1 to 10. If it is not, adjust the power of 10 by one.
Total 21 marks. Allow about 21 minutes.
Exercise 3X Practice test questions
Non-calculatorMethod. Move the decimal point so that there is one non-zero digit in front of it. Then count the places moved.
- Move the point to make a number from 1 to 10. The number must be at least 1 and less than 10.2400000 → 2.400000
- Count the places the point moved. A big number has a positive power of 10.6 places left, so the power is +6
- Write the answer in standard form. The number is the mantissa times 10 to the power.2.400000 × 106
Check. 2.400000 × 10 to the power 6 gives back 2400000. ✓
Watch out. Count the places carefully. A number between 0 and 1 gets a NEGATIVE power of 10.
Method. Move the decimal point so that there is one non-zero digit in front of it. Then count the places moved.
- Move the point to make a number from 1 to 10. The number must be at least 1 and less than 10.0.00035 → 3.5
- Count the places the point moved. A small number (less than 1) has a negative power of 10.4 places right, so the power is −4
- Write the answer in standard form. The number is the mantissa times 10 to the power.3.5 × 10−4
Check. 3.5 × 10 to the power −4 gives back 0.00035. ✓
Watch out. Count the places carefully. A number between 0 and 1 gets a NEGATIVE power of 10.
Method. Multiply (or divide) the numbers in front, then deal with the powers of 10. Check the size of the number in front last.
- Work with the numbers in front of the powers. Do the multiplication of 2 and 4.2 × 4 = 8
- Add the powers of 10. When you multiply powers of 10, add the powers.105 × 103 = 108
- Write the answer. This is in standard form.8 × 108
Check. Estimate: the answer is about 8 × 108. ✓
Watch out. The number in front must be from 1 to 10. If it is not, adjust the power of 10 by one.
Method. Find the first non-zero digit, count n significant figures from there, then look at the next digit.
- Find the first significant figure. The first significant figure is the first digit that is not zero.0.07483: first significant figure is 7
- Look at the digit after the last significant figure you keep. That digit is 8, so round up.next digit 8
- Write the answer to n significant figures, filling with zeros for places. Zeros are needed to keep the place value of the number.0.075
Check. The answer has 2 significant figures. ✓
Watch out. Zeros after the first digit count as significant figures when they come before a point, but zeros at the start do not.
Method. Look at the tens digit. It tells us whether to round the hundreds up or down.
- Find the digit that decides. The tens digit is 7, which is 5 or more.5,672: tens digit is 7
- Round up. The hundreds digit goes up by 1: 56 hundreds becomes 57 hundreds.5,600 → 5,700
- Write the answer. The tens and units become zero.5,700
Check. 5,700 is 28 away from 5,672, and 5,600 is 72 away. ✓
Watch out. To round to the nearest 100, look at the tens digit, not the hundreds.
Method. Round each number to 1 significant figure, then work out the easy calculation.
- Round each number to 1 significant figure. Keep the first digit and make the others zeros.589 ≈ 600 and 0.41 ≈ 0.4
- Work out the easy calculation. Use the rounded numbers, which are easy to work with.600 × 0.4 = 240
- Write the estimate. An estimate is only close to the real answer.about 240
Check. The real answer is 589 × 0.41 and it is close to 240. ✓
Watch out. Estimate first to check your calculator answer. An answer far from the estimate is probably wrong.
Method. Half the rounding unit is the most the true value can be above or below the rounded value.
- Find half the rounding unit. The value was rounded to the nearest 0.1 cm, so the error is at most half of that.0.1 ÷ 2 = 0.05
- Find the smallest possible value. Subtract half the unit from the rounded value. This value is included.6.8 − 0.05 = 6.75
- Find the largest possible value. Add half the unit. This value is NOT included, because it would round up.6.8 + 0.05 = 6.85
- Write the interval. Use ≤ for the lower bound and < for the upper bound.6.75 ≤ l < 6.85
Check. 6.75 rounds to 6.8 and so does every number up to, but not including, 6.85. ✓
Watch out. The lower bound is included (≤). The upper bound is not (<), because a value of exactly the upper bound would round up.
Method. Multiply (or divide) the numbers in front, then deal with the powers of 10. Check the size of the number in front last.
- Work with the numbers in front of the powers. Do the division of 6 and 3.6 ÷ 3 = 2
- Subtract the powers of 10. When you divide powers of 10, subtract the powers.107 ÷ 102 = 105
- Write the answer. This is in standard form.2 × 105
Check. Estimate: the answer is about 2 × 105. ✓
Watch out. The number in front must be from 1 to 10. If it is not, adjust the power of 10 by one.
Method. Multiply (or divide) the numbers in front, then deal with the powers of 10. Check the size of the number in front last.
- Work with the numbers in front of the powers. Do the multiplication of 3 and 2.3 × 2 = 6
- Add the powers of 10. When you multiply powers of 10, add the powers.10−2 × 10−4 = 10−6
- Write the answer. This is in standard form.6 × 10−6
Check. Estimate: the answer is about 6 × 10−6. ✓
Watch out. The number in front must be from 1 to 10. If it is not, adjust the power of 10 by one.
Method. Round to 3 significant figures: the first significant figure is 4, and the digits 4, 5 and 7 are already three significant figures.
- Count the significant figures in 0.0457. The first significant figure is 4, so the figures are 4, 5 and 7.4, 5, 7: three figures
- Check whether rounding changes it. It has exactly three significant figures, so there is nothing to round.0.0457 stays as it is
- Write the answer. Sam rounded to the wrong number of figures.0.0457
Check. 0.046 has only two significant figures: 4 and 6. ✓
Watch out. Count significant figures from the first digit that is not zero. Zeros at the start do not count.