Standard form and bounds
By the end of this chapter you will be able to
- Multiply, divide and add numbers written in standard form, and give the answer in standard form
- Read a calculator display in standard form and write the number in full
- Work out upper and lower bounds for a sum or product of two rounded measures
- Work out upper and lower bounds for a difference or quotient, choosing the correct end of each bound
- Work out bounds for a longer calculation, and decide whether an estimate is accurate enough
You should already know
- Standard form for large and small numbers, and the index laws with powers of 10 (Year 8, Chapter 1)
- Rounding to decimal places and to the nearest unit, and the idea of a bound for one rounded value (Year 8, Chapter 2)
- Working with decimals to 2 decimal places (Year 7 and Year 8)
Where this leads
- Recurring decimals and percentages (Year 9, Chapter 3)
- Compound measures and rates such as density and speed (Year 9, Chapter 10)
- Limits of accuracy and error intervals in GCSE measurement work
National Curriculum for England, Key Stage 3 mathematics, Year 9. Content: Calculate with numbers in standard form, including multiplying, dividing and adding, and read a calculator display, Find the upper and lower bounds of a sum or product of rounded values, Find the upper and lower bounds of a difference or quotient, where the bound is not always the same end, Find bounds for calculations with two or more measures, and use bounds to decide whether an estimate is accurate enough.
Exercise P Do you remember?
Non-calculatorMethod. Count the places from the decimal point to the end of the number.
- Write the front number. The front number must be between 1 and 10.4
- Count the places. The decimal point moves 6 places to the left, from 4 000 000.4 000 000 = 4 × 106
- Write the answer. 4 × 106
Check. 4 × 106 = 4 × 1 000 000 = 4 000 000.
Method. Move the decimal point until there is one digit before it.
- Find the front number. Move the decimal point 3 places to the right to get 3.5.0.0035 → 3.5
- Write the power of 10. Moving right 3 places makes the index negative.10−3
- Write the answer. 3.5 × 10−3
Check. 3.5 × 10−3 = 0.0035.
Method. Look at the third decimal place to decide whether to round up.
- Look at the third decimal place. The third decimal place is 5, so round the second place up.6.48|57 → 6.49
- Write the answer. 6.49
Check. 6.49 is closer to 6.4857 than 6.48 is.
Method. Look at the tens digit to decide.
- Find the hundreds digit. The hundreds digit is 8, and the tens digit is 4.2 8|49
- Decide whether to round up. The tens digit is 4, which is less than 5, so the hundreds stay the same.2 849 → 2 800
- Write the answer. 2,800
Check. 2,800 is the nearest hundred to 2,849.
Method. Look at the second decimal place. A 5 rounds up.
- Look at the second decimal place. The 5 in the second place means round the first place up.3.2|5 → 3.3
- Write the answer. 3.3
Check. 3.3 is the nearest 1 d.p. value to 3.25, because 3.25 is halfway and rounds up.
Calculating with standard form
A number in standard form is written as A × 10n, where 1 ≤ A < 10 and n is a whole number. To multiply numbers in standard form, multiply the front numbers and add the powers of 10. To divide, divide the front numbers and subtract the powers of 10. To add or subtract, first write both numbers with the same power of 10.
A calculator shows standard form using E. The display 4.2E-5 means 4.2 × 10−5.
Key ideas
- Multiply: multiply the front numbers, add the indices.
- Divide: divide the front numbers, subtract the indices.
- Add or subtract: make the powers of 10 the same first.
- After any calculation, check that the front number is between 1 and 10.
Method. Multiply the front numbers, add the powers of 10, then write the answer in standard form.
- Multiply the numbers at the front. Multiply the two numbers that are between 1 and 10.4 × 2.5 = 10
- Add the powers of 10. When powers of 10 are multiplied, their indices are added.103 × 102 = 105
- Put the front number between 1 and 10. Standard form needs one digit before the decimal point. Move the point and change the index to match.10 × 105 = 1 × 106
- Write the answer. 1 × 106
Check. Check: 1,000,000 is the same as 1,000,000.
Watch out. If the front number is 10 or more, or less than 1, adjust it. Do not leave a front number of 12.
Method. Divide the front numbers, subtract the powers of 10, then tidy the front number.
- Divide the numbers at the front. Divide the first front number by the second.8 ÷ 4 = 2
- Subtract the powers of 10. When powers of 10 are divided, subtract the indices.109 ÷ 103 = 106
- Put the front number between 1 and 10. Move the decimal point, and change the index to keep the same value.2 × 106 = 2 × 106
- Write the answer. 2 × 106
Check. Check: 2,000,000 × 4 × 103 is 8,000,000,000, so the division is correct.
Watch out. Subtract the index of the bottom number from the index of the top number: k1 − k2.
Method. Write both numbers with the same power of 10, then add or subtract the front numbers.
- Write both numbers with the same power of 10. Use the larger power, 104, for both. Change the other front number to match.3 × 104 = 3 × 104
5 × 103 = 0.5 × 104 - Add or subtract the front numbers. The powers of 10 are now the same, so only the front numbers change.3 + 0.5 = 3.5
- Put the answer into standard form. The front number must be between 1 and 10.3.5 × 104 = 3.5 × 104
- Write the answer. 3.5 × 104
Check. Check: 30,000 + 5,000 = 35,000.
Watch out. You cannot add the front numbers before the powers of 10 are the same. Do not add the indices.
Watch out
Watch out: 3 × 104 + 5 × 103 is not 8 × 107. You cannot add the indices when adding. Write both numbers with the same power of 10 first.
Exam tip
Exam tip. Use the same method as the question. A calculator answer such as 4.2E-5 must be written as standard form or as a decimal.
Exercise 2A Standard form calculations
Non-calculatorMethod. Multiply the front numbers, add the powers of 10, then write the answer in standard form.
- Multiply the numbers at the front. Multiply the two numbers that are between 1 and 10.3 × 2 = 6
- Add the powers of 10. When powers of 10 are multiplied, their indices are added.104 × 103 = 107
- Put the front number between 1 and 10. Standard form needs one digit before the decimal point. Move the point and change the index to match.6 × 107 = 6 × 107
- Write the answer. 6 × 107
Check. Check: 60,000,000 is the same as 60,000,000.
Watch out. If the front number is 10 or more, or less than 1, adjust it. Do not leave a front number of 12.
Method. Multiply the front numbers, add the powers of 10, then write the answer in standard form.
- Multiply the numbers at the front. Multiply the two numbers that are between 1 and 10.4.5 × 2 = 9
- Add the powers of 10. When powers of 10 are multiplied, their indices are added.105 × 10−3 = 102
- Put the front number between 1 and 10. Standard form needs one digit before the decimal point. Move the point and change the index to match.9 × 102 = 9 × 102
- Write the answer. 9 × 102
Check. Check: 900 is the same as 900.
Watch out. If the front number is 10 or more, or less than 1, adjust it. Do not leave a front number of 12.
Method. Multiply the front numbers, add the powers of 10, then write the answer in standard form.
- Multiply the numbers at the front. Multiply the two numbers that are between 1 and 10.7 × 1.5 = 10.5
- Add the powers of 10. When powers of 10 are multiplied, their indices are added.106 × 10−2 = 104
- Put the front number between 1 and 10. Standard form needs one digit before the decimal point. Move the point and change the index to match.10.5 × 104 = 1.05 × 105
- Write the answer. 1.05 × 105
Check. Check: 105,000 is the same as 105,000.
Watch out. If the front number is 10 or more, or less than 1, adjust it. Do not leave a front number of 12.
Method. Multiply the front numbers, add the powers of 10, then write the answer in standard form.
- Multiply the numbers at the front. Multiply the two numbers that are between 1 and 10.2.5 × 4 = 10
- Add the powers of 10. When powers of 10 are multiplied, their indices are added.10−4 × 10−2 = 10−6
- Put the front number between 1 and 10. Standard form needs one digit before the decimal point. Move the point and change the index to match.10 × 10−6 = 1 × 10−5
- Write the answer. 1 × 10−5
Check. Check: 0.00001 is the same as 0.00001.
Watch out. If the front number is 10 or more, or less than 1, adjust it. Do not leave a front number of 12.
Method. Divide the front numbers, subtract the powers of 10, then tidy the front number.
- Divide the numbers at the front. Divide the first front number by the second.8 ÷ 4 = 2
- Subtract the powers of 10. When powers of 10 are divided, subtract the indices.109 ÷ 103 = 106
- Put the front number between 1 and 10. Move the decimal point, and change the index to keep the same value.2 × 106 = 2 × 106
- Write the answer. 2 × 106
Check. Check: 2,000,000 × 4 × 103 is 8,000,000,000, so the division is correct.
Watch out. Subtract the index of the bottom number from the index of the top number: k1 − k2.
Method. Divide the front numbers, subtract the powers of 10, then tidy the front number.
- Divide the numbers at the front. Divide the first front number by the second.6.3 ÷ 9 = 0.7
- Subtract the powers of 10. When powers of 10 are divided, subtract the indices.108 ÷ 102 = 106
- Put the front number between 1 and 10. Move the decimal point, and change the index to keep the same value.0.7 × 106 = 7 × 105
- Write the answer. 7 × 105
Check. Check: 700,000 × 9 × 102 is 630,000,000, so the division is correct.
Watch out. Subtract the index of the bottom number from the index of the top number: k1 − k2.
Method. Divide the front numbers, subtract the powers of 10, then tidy the front number.
- Divide the numbers at the front. Divide the first front number by the second.1.2 ÷ 3 = 0.4
- Subtract the powers of 10. When powers of 10 are divided, subtract the indices.10−3 ÷ 104 = 10−7
- Put the front number between 1 and 10. Move the decimal point, and change the index to keep the same value.0.4 × 10−7 = 4 × 10−8
- Write the answer. 4 × 10−8
Check. Check: 0.00000004 × 3 × 104 is 0.0012, so the division is correct.
Watch out. Subtract the index of the bottom number from the index of the top number: k1 − k2.
Method. Divide the front numbers, subtract the powers of 10, then tidy the front number.
- Divide the numbers at the front. Divide the first front number by the second.9.6 ÷ 3 = 3.2
- Subtract the powers of 10. When powers of 10 are divided, subtract the indices.105 ÷ 10−2 = 107
- Put the front number between 1 and 10. Move the decimal point, and change the index to keep the same value.3.2 × 107 = 3.2 × 107
- Write the answer. 3.2 × 107
Check. Check: 32,000,000 × 3 × 10−2 is 960,000, so the division is correct.
Watch out. Subtract the index of the bottom number from the index of the top number: k1 − k2.
Method. Write both numbers with the same power of 10, then add or subtract the front numbers.
- Write both numbers with the same power of 10. Use the larger power, 104, for both. Change the other front number to match.3 × 104 = 3 × 104
5 × 103 = 0.5 × 104 - Add or subtract the front numbers. The powers of 10 are now the same, so only the front numbers change.3 + 0.5 = 3.5
- Put the answer into standard form. The front number must be between 1 and 10.3.5 × 104 = 3.5 × 104
- Write the answer. 3.5 × 104
Check. Check: 30,000 + 5,000 = 35,000.
Watch out. You cannot add the front numbers before the powers of 10 are the same. Do not add the indices.
Method. Write both numbers with the same power of 10, then add or subtract the front numbers.
- Write both numbers with the same power of 10. Use the larger power, 10−3, for both. Change the other front number to match.2.4 × 10−3 = 2.4 × 10−3
6 × 10−4 = 0.6 × 10−3 - Add or subtract the front numbers. The powers of 10 are now the same, so only the front numbers change.2.4 + 0.6 = 3
- Put the answer into standard form. The front number must be between 1 and 10.3 × 10−3 = 3 × 10−3
- Write the answer. 3 × 10−3
Check. Check: 0.0024 + 0.0006 = 0.003.
Watch out. You cannot add the front numbers before the powers of 10 are the same. Do not add the indices.
Method. Write both numbers with the same power of 10, then add or subtract the front numbers.
- Write both numbers with the same power of 10. Use the larger power, 106, for both. Change the other front number to match.9 × 106 = 9 × 106
2.5 × 105 = 0.25 × 106 - Add or subtract the front numbers. The powers of 10 are now the same, so only the front numbers change.9 − 0.25 = 8.75
- Put the answer into standard form. The front number must be between 1 and 10.8.75 × 106 = 8.75 × 106
- Write the answer. 8.75 × 106
Check. Check: 9,000,000 − 250,000 = 8,750,000.
Watch out. You cannot add the front numbers before the powers of 10 are the same. Do not add the indices.
Method. Write both numbers with the same power of 10, then add or subtract the front numbers.
- Write both numbers with the same power of 10. Use the larger power, 10−2, for both. Change the other front number to match.4 × 10−2 = 4 × 10−2
1.2 × 10−3 = 0.12 × 10−2 - Add or subtract the front numbers. The powers of 10 are now the same, so only the front numbers change.4 − 0.12 = 3.88
- Put the answer into standard form. The front number must be between 1 and 10.3.88 × 10−2 = 3.88 × 10−2
- Write the answer. 3.88 × 10−2
Check. Check: 0.04 − 0.0012 = 0.0388.
Watch out. You cannot add the front numbers before the powers of 10 are the same. Do not add the indices.
Method. Divide the front numbers, subtract the powers of 10, then tidy the front number.
- Divide the numbers at the front. Divide the first front number by the second.1.5 ÷ 3 = 0.5
- Subtract the powers of 10. When powers of 10 are divided, subtract the indices.108 ÷ 105 = 103
- Put the front number between 1 and 10. Move the decimal point, and change the index to keep the same value.0.5 × 103 = 5 × 102
- Write the answer. 5 × 102
Check. Check: 500 × 3 × 105 is 150,000,000, so the division is correct.
Watch out. Subtract the index of the bottom number from the index of the top number: k1 − k2.
Method. Use the power of 10 to move the decimal point.
- Read the calculator display. The E means times 10 to the power of the number after it.4.2E−5 means 4.2 × 10−5
- Move the decimal point. A positive index makes the number bigger, so move the point to the right. A negative index makes it smaller, so move it to the left.4.2 × 10−5 = 0.000042
- Write the answer. 0.000042
Check. Write 0.000042 in standard form to get back to 4.2 × 10−5.
Watch out. Count the places carefully. Each place moved is one step in the index.
Bounds for sums and products
Every measurement is rounded, so the true value lies somewhere in a range. For a value rounded to a unit, the lower bound is half a unit below the value shown, and the upper bound is half a unit above it. For 6.4 cm to 1 d.p., the bounds are 6.35 ≤ length < 6.45.
For a product or a sum of two positive measurements, the upper bound uses the two upper bounds, and the lower bound uses the two lower bounds.
Key ideas
- Lower bound = value − half the unit. Upper bound = value + half the unit.
- Upper bound of a sum or product: add or multiply the two upper bounds.
- Lower bound of a sum or product: add or multiply the two lower bounds.
- Work out the bounds before you combine the measurements.
Method. Use the upper bound of each measure, so that the answer is as large as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.l: 6.35 ≤ l < 6.45
w: 2.45 ≤ w < 2.55 - Choose the upper bound of each measure. To make the answer as large as possible, use the bound that pushes it that way.l = 6.45 (upper bound), w = 2.55 (upper bound)
- Work out the answer with these values. Use the same order of working as the calculation.6.45 × 2.55 = 16.4475
- Write the answer. 16.4475 cm2
Check. The other bound is 15.5575, so the true value lies between 15.5575 and 16.4475.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the lower bound of each measure, so that the answer is as small as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.p: 4.65 ≤ p < 4.75
q: 3.15 ≤ q < 3.25 - Choose the lower bound of each measure. To make the answer as small as possible, use the bound that pushes it that way.p = 4.65 (lower bound), q = 3.15 (lower bound)
- Work out the answer with these values. Use the same order of working as the calculation.4.65 + 3.15 = 7.8
- Write the answer. 7.8 m
Check. The other bound is 8, so the true value lies between 7.8 and 8.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the lower bound of each measure, so that the answer is as small as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.s: 2.5 ≤ s < 3.5
s: 2.5 ≤ s < 3.5 - Choose the lower bound of each measure. To make the answer as small as possible, use the bound that pushes it that way.s = 2.5 (lower bound), s = 2.5 (lower bound)
- Work out the answer with these values. Use the same order of working as the calculation.2.5 × 2.5 = 6.25
- Write the answer. 6.25 cm2
Check. The other bound is 12.25, so the true value lies between 6.25 and 12.25.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Watch out
Watch out: the bounds of 6.4 are 6.35 and 6.45, not 6.3 and 6.5. Using the wrong unit doubles or halves the error.
Exam tip
Exam tip. Write both bounds of each measurement on the page before you multiply or add. This keeps the working tidy.
Exercise 2B Bounds for sums and products
Non-calculatorMethod. Use the upper bound of each measure, so that the answer is as large as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.l: 6.35 ≤ l < 6.45
w: 2.45 ≤ w < 2.55 - Choose the upper bound of each measure. To make the answer as large as possible, use the bound that pushes it that way.l = 6.45 (upper bound), w = 2.55 (upper bound)
- Work out the answer with these values. Use the same order of working as the calculation.6.45 × 2.55 = 16.4475
- Write the answer. 16.4475 cm2
Check. The other bound is 15.5575, so the true value lies between 15.5575 and 16.4475.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the lower bound of each measure, so that the answer is as small as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.l: 6.35 ≤ l < 6.45
w: 2.45 ≤ w < 2.55 - Choose the lower bound of each measure. To make the answer as small as possible, use the bound that pushes it that way.l = 6.35 (lower bound), w = 2.45 (lower bound)
- Work out the answer with these values. Use the same order of working as the calculation.6.35 × 2.45 = 15.5575
- Write the answer. 15.5575 cm2
Check. The other bound is 16.4475, so the true value lies between 15.5575 and 16.4475.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the upper bound of each measure, so that the answer is as large as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.a: 11.5 ≤ a < 12.5
b: 7.5 ≤ b < 8.5 - Choose the upper bound of each measure. To make the answer as large as possible, use the bound that pushes it that way.a = 12.5 (upper bound), b = 8.5 (upper bound)
- Work out the answer with these values. Use the same order of working as the calculation.12.5 × 8.5 = 106.25
- Write the answer. 106.25 cm2
Check. The other bound is 86.25, so the true value lies between 86.25 and 106.25.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the lower bound of each measure, so that the answer is as small as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.a: 11.5 ≤ a < 12.5
b: 7.5 ≤ b < 8.5 - Choose the lower bound of each measure. To make the answer as small as possible, use the bound that pushes it that way.a = 11.5 (lower bound), b = 7.5 (lower bound)
- Work out the answer with these values. Use the same order of working as the calculation.11.5 × 7.5 = 86.25
- Write the answer. 86.25 cm2
Check. The other bound is 106.25, so the true value lies between 86.25 and 106.25.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the upper bound of each measure, so that the answer is as large as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.p: 4.65 ≤ p < 4.75
q: 3.15 ≤ q < 3.25 - Choose the upper bound of each measure. To make the answer as large as possible, use the bound that pushes it that way.p = 4.75 (upper bound), q = 3.25 (upper bound)
- Work out the answer with these values. Use the same order of working as the calculation.4.75 + 3.25 = 8
- Write the answer. 8 m
Check. The other bound is 7.8, so the true value lies between 7.8 and 8.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the lower bound of each measure, so that the answer is as small as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.p: 4.65 ≤ p < 4.75
q: 3.15 ≤ q < 3.25 - Choose the lower bound of each measure. To make the answer as small as possible, use the bound that pushes it that way.p = 4.65 (lower bound), q = 3.15 (lower bound)
- Work out the answer with these values. Use the same order of working as the calculation.4.65 + 3.15 = 7.8
- Write the answer. 7.8 m
Check. The other bound is 8, so the true value lies between 7.8 and 8.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the upper bound of each measure, so that the answer is as large as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.s: 2.5 ≤ s < 3.5
s: 2.5 ≤ s < 3.5 - Choose the upper bound of each measure. To make the answer as large as possible, use the bound that pushes it that way.s = 3.5 (upper bound), s = 3.5 (upper bound)
- Work out the answer with these values. Use the same order of working as the calculation.3.5 × 3.5 = 12.25
- Write the answer. 12.25 cm2
Check. The other bound is 6.25, so the true value lies between 6.25 and 12.25.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the lower bound of each measure, so that the answer is as small as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.s: 2.5 ≤ s < 3.5
s: 2.5 ≤ s < 3.5 - Choose the lower bound of each measure. To make the answer as small as possible, use the bound that pushes it that way.s = 2.5 (lower bound), s = 2.5 (lower bound)
- Work out the answer with these values. Use the same order of working as the calculation.2.5 × 2.5 = 6.25
- Write the answer. 6.25 cm2
Check. The other bound is 12.25, so the true value lies between 6.25 and 12.25.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the upper bound of each measure, so that the answer is as large as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.m: 3.55 ≤ m < 3.65
n: 2.845 ≤ n < 2.855 - Choose the upper bound of each measure. To make the answer as large as possible, use the bound that pushes it that way.m = 3.65 (upper bound), n = 2.855 (upper bound)
- Work out the answer with these values. Use the same order of working as the calculation.3.65 + 2.855 = 6.505
- Write the answer. 6.505 kg
Check. The other bound is 6.395, so the true value lies between 6.395 and 6.505.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the lower bound of each measure, so that the answer is as small as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.m: 3.55 ≤ m < 3.65
n: 2.845 ≤ n < 2.855 - Choose the lower bound of each measure. To make the answer as small as possible, use the bound that pushes it that way.m = 3.55 (lower bound), n = 2.845 (lower bound)
- Work out the answer with these values. Use the same order of working as the calculation.3.55 + 2.845 = 6.395
- Write the answer. 6.395 kg
Check. The other bound is 6.505, so the true value lies between 6.395 and 6.505.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the lower bound of each measure, so that the answer is as small as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.l: 7.45 ≤ l < 7.55
w: 4.15 ≤ w < 4.25 - Choose the lower bound of each measure. To make the answer as small as possible, use the bound that pushes it that way.l = 7.45 (lower bound), w = 4.15 (lower bound)
- Work out the answer with these values. Use the same order of working as the calculation.7.45 × 4.15 = 30.9175
- Write the answer. 30.9175 cm2
Check. The other bound is 32.0875, so the true value lies between 30.9175 and 32.0875.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the upper bound of each measure, so that the answer is as large as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.l: 1.245 ≤ l < 1.255
w: 0.35 ≤ w < 0.45 - Choose the upper bound of each measure. To make the answer as large as possible, use the bound that pushes it that way.l = 1.255 (upper bound), w = 0.45 (upper bound)
- Work out the answer with these values. Use the same order of working as the calculation.1.255 × 0.45 = 0.56475
- Write the answer. 0.56475 m2
Check. The other bound is 0.43575, so the true value lies between 0.43575 and 0.56475.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Find the true bounds of each number first.
- Find the bounds of 6.4. The value 6.4 is to 1 d.p., so it is between 6.35 and 6.45.6.35 ≤ x < 6.45
- Find the bounds of 2.5. The value 2.5 is also to 1 d.p., so it is between 2.45 and 2.55.2.45 ≤ y < 2.55
- Work out the upper bound. Multiply the two upper bounds.6.45 × 2.55 = 16.4475
- Write the answer. Ali used 6.5 instead of 6.45, which is too large.16.4475
Check. The true area is between 15.5575 and 16.4475, so 16.4475 is the upper bound.
Watch out. Do not use the rounded value 6.5. The upper bound of 6.4 is 6.45, not 6.5.
Bounds for differences and quotients
For a difference the bounds do not come from the same ends. To make a difference as large as possible, use the upper bound of the number you take away from, and the lower bound of the number subtracted. For example, the upper bound of 4.7 − 3.2 is 4.75 − 3.15 = 1.6.
For a quotient (divide), the same idea applies. To make the answer as large as possible, use the upper bound of the top, and the lower bound of the bottom.
Key ideas
- Upper bound of a − b = upper bound of a − lower bound of b.
- Lower bound of a − b = lower bound of a − upper bound of b.
- Upper bound of a ÷ b = upper bound of a ÷ lower bound of b.
- Lower bound of a ÷ b = lower bound of a ÷ upper bound of b.
Method. Use the upper bound of each measure, so that the answer is as large as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.p: 4.65 ≤ p < 4.75
q: 3.15 ≤ q < 3.25 - Choose the upper bound of each measure. To make the answer as large as possible, use the bound that pushes it that way.p = 4.75 (upper bound), q = 3.15 (lower bound)
- Work out the answer with these values. Use the same order of working as the calculation.4.75 − 3.15 = 1.6
- Write the answer. 1.6
Check. The other bound is 1.4, so the true value lies between 1.4 and 1.6.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the lower bound of each measure, so that the answer is as small as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.m: 235 ≤ m < 245
v: 47.5 ≤ v < 52.5 - Choose the lower bound of each measure. To make the answer as small as possible, use the bound that pushes it that way.m = 235 (lower bound), v = 52.5 (upper bound)
- Work out the answer with these values. Use the same order of working as the calculation.235 ÷ 52.5 = 4.4762 (to 4 d.p.)
- Round to 2 decimal places. The question asks for this many decimal places.4.4762 (to 4 d.p.) = 4.48 g/cm3
Check. The other bound is 5.16, so the true value lies between 4.48 and 5.16.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the lower bound of each measure, so that the answer is as small as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.x: 9.55 ≤ x < 9.65
y: 3.5 ≤ y < 4.5 - Choose the lower bound of each measure. To make the answer as small as possible, use the bound that pushes it that way.x = 9.55 (lower bound), y = 4.5 (upper bound)
- Work out the answer with these values. Use the same order of working as the calculation.9.55 ÷ 4.5 = 2.1222 (to 4 d.p.)
- Round to 2 decimal places. The question asks for this many decimal places.2.1222 (to 4 d.p.) = 2.12
Check. The other bound is 2.76, so the true value lies between 2.12 and 2.76.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Watch out
Watch out: the upper bound of a difference is not the upper bound of each number subtracted. Use the upper bound of the first number and the lower bound of the second.
Exam tip
Exam tip. Ask yourself which end makes the answer as big (or small) as possible. Then use that bound for each measurement.
Exercise 2C Bounds for differences and quotients
Non-calculatorMethod. Use the upper bound of each measure, so that the answer is as large as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.p: 4.65 ≤ p < 4.75
q: 3.15 ≤ q < 3.25 - Choose the upper bound of each measure. To make the answer as large as possible, use the bound that pushes it that way.p = 4.75 (upper bound), q = 3.15 (lower bound)
- Work out the answer with these values. Use the same order of working as the calculation.4.75 − 3.15 = 1.6
- Write the answer. 1.6
Check. The other bound is 1.4, so the true value lies between 1.4 and 1.6.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the lower bound of each measure, so that the answer is as small as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.p: 4.65 ≤ p < 4.75
q: 3.15 ≤ q < 3.25 - Choose the lower bound of each measure. To make the answer as small as possible, use the bound that pushes it that way.p = 4.65 (lower bound), q = 3.25 (upper bound)
- Work out the answer with these values. Use the same order of working as the calculation.4.65 − 3.25 = 1.4
- Write the answer. 1.4
Check. The other bound is 1.6, so the true value lies between 1.4 and 1.6.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the upper bound of each measure, so that the answer is as large as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.a: 13.5 ≤ a < 14.5
c: 9.45 ≤ c < 9.55 - Choose the upper bound of each measure. To make the answer as large as possible, use the bound that pushes it that way.a = 14.5 (upper bound), c = 9.45 (lower bound)
- Work out the answer with these values. Use the same order of working as the calculation.14.5 − 9.45 = 5.05
- Write the answer. 5.05 cm
Check. The other bound is 3.95, so the true value lies between 3.95 and 5.05.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the lower bound of each measure, so that the answer is as small as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.a: 13.5 ≤ a < 14.5
c: 9.45 ≤ c < 9.55 - Choose the lower bound of each measure. To make the answer as small as possible, use the bound that pushes it that way.a = 13.5 (lower bound), c = 9.55 (upper bound)
- Work out the answer with these values. Use the same order of working as the calculation.13.5 − 9.55 = 3.95
- Write the answer. 3.95 cm
Check. The other bound is 5.05, so the true value lies between 3.95 and 5.05.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the upper bound of each measure, so that the answer is as large as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.m: 3.55 ≤ m < 3.65
n: 2.845 ≤ n < 2.855 - Choose the upper bound of each measure. To make the answer as large as possible, use the bound that pushes it that way.m = 3.65 (upper bound), n = 2.845 (lower bound)
- Work out the answer with these values. Use the same order of working as the calculation.3.65 − 2.845 = 0.805
- Write the answer. 0.805 kg
Check. The other bound is 0.695, so the true value lies between 0.695 and 0.805.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the lower bound of each measure, so that the answer is as small as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.m: 3.55 ≤ m < 3.65
n: 2.845 ≤ n < 2.855 - Choose the lower bound of each measure. To make the answer as small as possible, use the bound that pushes it that way.m = 3.55 (lower bound), n = 2.855 (upper bound)
- Work out the answer with these values. Use the same order of working as the calculation.3.55 − 2.855 = 0.695
- Write the answer. 0.695 kg
Check. The other bound is 0.805, so the true value lies between 0.695 and 0.805.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the upper bound of each measure, so that the answer is as large as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.m: 235 ≤ m < 245
v: 47.5 ≤ v < 52.5 - Choose the upper bound of each measure. To make the answer as large as possible, use the bound that pushes it that way.m = 245 (upper bound), v = 47.5 (lower bound)
- Work out the answer with these values. Use the same order of working as the calculation.245 ÷ 47.5 = 5.1579 (to 4 d.p.)
- Round to 2 decimal places. The question asks for this many decimal places.5.1579 (to 4 d.p.) = 5.16 g/cm3
Check. The other bound is 4.48, so the true value lies between 4.48 and 5.16.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the lower bound of each measure, so that the answer is as small as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.m: 235 ≤ m < 245
v: 47.5 ≤ v < 52.5 - Choose the lower bound of each measure. To make the answer as small as possible, use the bound that pushes it that way.m = 235 (lower bound), v = 52.5 (upper bound)
- Work out the answer with these values. Use the same order of working as the calculation.235 ÷ 52.5 = 4.4762 (to 4 d.p.)
- Round to 2 decimal places. The question asks for this many decimal places.4.4762 (to 4 d.p.) = 4.48 g/cm3
Check. The other bound is 5.16, so the true value lies between 4.48 and 5.16.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the upper bound of each measure, so that the answer is as large as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.d: 59.5 ≤ d < 60.5
t: 4.15 ≤ t < 4.25 - Choose the upper bound of each measure. To make the answer as large as possible, use the bound that pushes it that way.d = 60.5 (upper bound), t = 4.15 (lower bound)
- Work out the answer with these values. Use the same order of working as the calculation.60.5 ÷ 4.15 = 14.5783 (to 4 d.p.)
- Round to 2 decimal places. The question asks for this many decimal places.14.5783 (to 4 d.p.) = 14.58 m/s
Check. The other bound is 14.00, so the true value lies between 14.00 and 14.58.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the lower bound of each measure, so that the answer is as small as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.d: 59.5 ≤ d < 60.5
t: 4.15 ≤ t < 4.25 - Choose the lower bound of each measure. To make the answer as small as possible, use the bound that pushes it that way.d = 59.5 (lower bound), t = 4.25 (upper bound)
- Work out the answer with these values. Use the same order of working as the calculation.59.5 ÷ 4.25 = 14
- Round to 2 decimal places. The question asks for this many decimal places.14 = 14.00 m/s
Check. The other bound is 14.58, so the true value lies between 14.00 and 14.58.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the upper bound of each measure, so that the answer is as large as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.x: 9.55 ≤ x < 9.65
y: 3.5 ≤ y < 4.5 - Choose the upper bound of each measure. To make the answer as large as possible, use the bound that pushes it that way.x = 9.65 (upper bound), y = 3.5 (lower bound)
- Work out the answer with these values. Use the same order of working as the calculation.9.65 ÷ 3.5 = 2.7571 (to 4 d.p.)
- Round to 2 decimal places. The question asks for this many decimal places.2.7571 (to 4 d.p.) = 2.76
Check. The other bound is 2.12, so the true value lies between 2.12 and 2.76.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the lower bound of each measure, so that the answer is as small as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.x: 9.55 ≤ x < 9.65
y: 3.5 ≤ y < 4.5 - Choose the lower bound of each measure. To make the answer as small as possible, use the bound that pushes it that way.x = 9.55 (lower bound), y = 4.5 (upper bound)
- Work out the answer with these values. Use the same order of working as the calculation.9.55 ÷ 4.5 = 2.1222 (to 4 d.p.)
- Round to 2 decimal places. The question asks for this many decimal places.2.1222 (to 4 d.p.) = 2.12
Check. The other bound is 2.76, so the true value lies between 2.12 and 2.76.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Think about which end makes the difference largest.
- Take an example. Let a be 10 and b be 9, both rounded.10 − 9 = 1
- Use the upper bound of b. If b is at its upper bound it is larger, so a − b gets smaller.10 − 9.5 = 0.5
- Use the lower bound of b. If b is at its lower bound it is smaller, so a − b gets larger.10 − 8.5 = 1.5
- Write the answer. The upper bound of a − b uses the lower bound of b.Lower bound of b
Check. The biggest difference uses the smallest number taken away.
Watch out. Subtracting a larger number gives a smaller answer.
Bounds for longer calculations
For a longer calculation, decide for each measurement whether a bigger value makes the answer bigger or smaller. Then choose the bound that gives the answer you want. Work out one bound at a time, and round only at the end.
The rule is the same for every calculation: an increase in a number that is added or multiplied makes the answer bigger, and an increase in a number that is subtracted or divides makes it smaller.
Key ideas
- Work out which bound of each measurement to use before you calculate.
- Use the upper bound of a measurement that increases the answer, and the lower bound of one that decreases it, for the upper bound.
- Round only at the end, and to the number of decimal places the question asks for.
- Check the other bound as well, so you know the true value lies between them.
Method. Use the upper bound of each measure, so that the answer is as large as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.l: 6.35 ≤ l < 6.45
w: 2.45 ≤ w < 2.55 - Choose the upper bound of each measure. To make the answer as large as possible, use the bound that pushes it that way.l = 6.45 (upper bound), w = 2.55 (upper bound)
- Work out the answer with these values. Use the same order of working as the calculation.2(6.45 + 2.55) = 18
- Write the answer. 18 cm
Check. The other bound is 17.6, so the true value lies between 17.6 and 18.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the lower bound of each measure, so that the answer is as small as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.x: 7.35 ≤ x < 7.45
y: 6.75 ≤ y < 6.85 - Choose the lower bound of each measure. To make the answer as small as possible, use the bound that pushes it that way.x = 7.35 (lower bound), y = 6.75 (lower bound)
- Work out the answer with these values. Use the same order of working as the calculation.(7.35 + 6.75) ÷ 2 = 7.05
- Write the answer. 7.05
Check. The other bound is 7.15, so the true value lies between 7.05 and 7.15.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the upper bound of each measure, so that the answer is as large as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.x: 3.15 ≤ x < 3.25
y: 1.45 ≤ y < 1.55
z: 3.5 ≤ z < 4.5 - Choose the upper bound of each measure. To make the answer as large as possible, use the bound that pushes it that way.x = 3.25 (upper bound), y = 1.55 (upper bound), z = 4.5 (upper bound)
- Work out the answer with these values. Use the same order of working as the calculation.(3.25 + 1.55) × 4.5 = 21.6
- Write the answer. 21.6
Check. The other bound is 16.1, so the true value lies between 16.1 and 21.6.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Watch out
Watch out: a bigger value in a denominator makes the answer smaller. For the upper bound of a fraction, use the lower bound on the bottom.
Exam tip
Exam tip. Write one line that says which bound you are using for each measurement. Examiners look for this.
Exercise 2D Bounds for longer calculations
Non-calculatorMethod. Use the upper bound of each measure, so that the answer is as large as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.l: 6.35 ≤ l < 6.45
w: 2.45 ≤ w < 2.55 - Choose the upper bound of each measure. To make the answer as large as possible, use the bound that pushes it that way.l = 6.45 (upper bound), w = 2.55 (upper bound)
- Work out the answer with these values. Use the same order of working as the calculation.2(6.45 + 2.55) = 18
- Write the answer. 18 cm
Check. The other bound is 17.6, so the true value lies between 17.6 and 18.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the lower bound of each measure, so that the answer is as small as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.l: 6.35 ≤ l < 6.45
w: 2.45 ≤ w < 2.55 - Choose the lower bound of each measure. To make the answer as small as possible, use the bound that pushes it that way.l = 6.35 (lower bound), w = 2.45 (lower bound)
- Work out the answer with these values. Use the same order of working as the calculation.2(6.35 + 2.45) = 17.6
- Write the answer. 17.6 cm
Check. The other bound is 18, so the true value lies between 17.6 and 18.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the upper bound of each measure, so that the answer is as large as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.x: 7.35 ≤ x < 7.45
y: 6.75 ≤ y < 6.85 - Choose the upper bound of each measure. To make the answer as large as possible, use the bound that pushes it that way.x = 7.45 (upper bound), y = 6.85 (upper bound)
- Work out the answer with these values. Use the same order of working as the calculation.(7.45 + 6.85) ÷ 2 = 7.15
- Write the answer. 7.15
Check. The other bound is 7.05, so the true value lies between 7.05 and 7.15.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the lower bound of each measure, so that the answer is as small as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.x: 7.35 ≤ x < 7.45
y: 6.75 ≤ y < 6.85 - Choose the lower bound of each measure. To make the answer as small as possible, use the bound that pushes it that way.x = 7.35 (lower bound), y = 6.75 (lower bound)
- Work out the answer with these values. Use the same order of working as the calculation.(7.35 + 6.75) ÷ 2 = 7.05
- Write the answer. 7.05
Check. The other bound is 7.15, so the true value lies between 7.05 and 7.15.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the upper bound of each measure, so that the answer is as large as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.x: 3.15 ≤ x < 3.25
y: 1.45 ≤ y < 1.55
z: 3.5 ≤ z < 4.5 - Choose the upper bound of each measure. To make the answer as large as possible, use the bound that pushes it that way.x = 3.25 (upper bound), y = 1.55 (upper bound), z = 4.5 (upper bound)
- Work out the answer with these values. Use the same order of working as the calculation.(3.25 + 1.55) × 4.5 = 21.6
- Write the answer. 21.6
Check. The other bound is 16.1, so the true value lies between 16.1 and 21.6.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the lower bound of each measure, so that the answer is as small as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.x: 3.15 ≤ x < 3.25
y: 1.45 ≤ y < 1.55
z: 3.5 ≤ z < 4.5 - Choose the lower bound of each measure. To make the answer as small as possible, use the bound that pushes it that way.x = 3.15 (lower bound), y = 1.45 (lower bound), z = 3.5 (lower bound)
- Work out the answer with these values. Use the same order of working as the calculation.(3.15 + 1.45) × 3.5 = 16.1
- Write the answer. 16.1
Check. The other bound is 21.6, so the true value lies between 16.1 and 21.6.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the upper bound of each measure, so that the answer is as large as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.x: 19.5 ≤ x < 20.5
y: 8.35 ≤ y < 8.45
z: 3.5 ≤ z < 4.5 - Choose the upper bound of each measure. To make the answer as large as possible, use the bound that pushes it that way.x = 20.5 (upper bound), y = 8.35 (lower bound), z = 3.5 (lower bound)
- Work out the answer with these values. Use the same order of working as the calculation.(20.5 − 8.35) ÷ 3.5 = 3.4714 (to 4 d.p.)
- Round to 2 decimal places. The question asks for this many decimal places.3.4714 (to 4 d.p.) = 3.47
Check. The other bound is 2.46, so the true value lies between 2.46 and 3.47.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the lower bound of each measure, so that the answer is as small as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.x: 19.5 ≤ x < 20.5
y: 8.35 ≤ y < 8.45
z: 3.5 ≤ z < 4.5 - Choose the lower bound of each measure. To make the answer as small as possible, use the bound that pushes it that way.x = 19.5 (lower bound), y = 8.45 (upper bound), z = 4.5 (upper bound)
- Work out the answer with these values. Use the same order of working as the calculation.(19.5 − 8.45) ÷ 4.5 = 2.4556 (to 4 d.p.)
- Round to 2 decimal places. The question asks for this many decimal places.2.4556 (to 4 d.p.) = 2.46
Check. The other bound is 3.47, so the true value lies between 2.46 and 3.47.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the upper bound of each measure, so that the answer is as large as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.l: 1.95 ≤ l < 2.05
w: 2.95 ≤ w < 3.05
h: 3.95 ≤ h < 4.05 - Choose the upper bound of each measure. To make the answer as large as possible, use the bound that pushes it that way.l = 2.05 (upper bound), w = 3.05 (upper bound), h = 4.05 (upper bound)
- Work out the answer with these values. Use the same order of working as the calculation.2.05 × 3.05 × 4.05 = 25.322625
- Round to 2 decimal places. The question asks for this many decimal places.25.322625 = 25.32 cm3
Check. The other bound is 22.72, so the true value lies between 22.72 and 25.32.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the lower bound of each measure, so that the answer is as small as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.l: 1.95 ≤ l < 2.05
w: 2.95 ≤ w < 3.05
h: 3.95 ≤ h < 4.05 - Choose the lower bound of each measure. To make the answer as small as possible, use the bound that pushes it that way.l = 1.95 (lower bound), w = 2.95 (lower bound), h = 3.95 (lower bound)
- Work out the answer with these values. Use the same order of working as the calculation.1.95 × 2.95 × 3.95 = 22.722375
- Round to 2 decimal places. The question asks for this many decimal places.22.722375 = 22.72 cm3
Check. The other bound is 25.32, so the true value lies between 22.72 and 25.32.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the upper bound of each measure, so that the answer is as large as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.b: 7.5 ≤ b < 8.5
h: 4.5 ≤ h < 5.5 - Choose the upper bound of each measure. To make the answer as large as possible, use the bound that pushes it that way.b = 8.5 (upper bound), h = 5.5 (upper bound)
- Work out the answer with these values. Use the same order of working as the calculation.½ × 8.5 × 5.5 = 23.375
- Write the answer. 23.375 cm2
Check. The other bound is 16.875, so the true value lies between 16.875 and 23.375.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the lower bound of each measure, so that the answer is as small as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.b: 7.5 ≤ b < 8.5
h: 4.5 ≤ h < 5.5 - Choose the lower bound of each measure. To make the answer as small as possible, use the bound that pushes it that way.b = 7.5 (lower bound), h = 4.5 (lower bound)
- Work out the answer with these values. Use the same order of working as the calculation.½ × 7.5 × 4.5 = 16.875
- Write the answer. 16.875 cm2
Check. The other bound is 23.375, so the true value lies between 16.875 and 23.375.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the lower bound of each measure, so that the answer is as small as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.x: 3.15 ≤ x < 3.25
y: 0.55 ≤ y < 0.65 - Choose the lower bound of each measure. To make the answer as small as possible, use the bound that pushes it that way.x = 3.15 (lower bound), y = 0.55 (lower bound)
- Work out the answer with these values. Use the same order of working as the calculation.3.15 + 4 × 0.55 = 5.35
- Write the answer. 5.35 kg
Check. The other bound is 5.85, so the true value lies between 5.35 and 5.85.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Using bounds to judge accuracy
The bounds tell us the smallest and largest the true value could be. If both bounds are on the same side of a target, the answer is certain. If the target falls between the two bounds, we cannot be sure.
Key ideas
- Work out the upper and lower bounds of the whole calculation.
- If both bounds are less than the target, the answer is always less than the target: answer Yes.
- If both bounds are at least the target, the answer is never less than the target: answer No.
- If the target lies between the two bounds, the answer is Not necessarily.
Method. Find both bounds of the calculation, then compare them with the target.
- Write the bounds of each measure. These are the smallest and largest values each measurement could really be.p: 4.65 ≤ p < 4.75
q: 3.15 ≤ q < 3.25 - Work out the upper bound. Use the values that make the total length as large as possible.4.75 + 3.25 = 8
- Work out the lower bound. Use the values that make the total length as small as possible.4.65 + 3.15 = 7.8
- Compare with the target. The upper bound 8 is less than 8.1, so the total length is always less than 8.1.
- Write the answer. Yes
Check. If the two bounds are on the same side of the target, the answer is certain. Otherwise it is not.
Watch out. Use the bounds that go in the right direction. Do not use the middle value alone.
Method. Find both bounds of the calculation, then compare them with the target.
- Write the bounds of each measure. These are the smallest and largest values each measurement could really be.l: 2.45 ≤ l < 2.55
w: 3.5 ≤ w < 4.5 - Work out the upper bound. Use the values that make the area as large as possible.2.55 × 4.5 = 11.475
- Work out the lower bound. Use the values that make the area as small as possible.2.45 × 3.5 = 8.575
- Compare with the target. The bounds 8.575 and 11.475 are on opposite sides of 9.7, so we cannot be sure.
- Write the answer. Not necessarily
Check. If the two bounds are on the same side of the target, the answer is certain. Otherwise it is not.
Watch out. Use the bounds that go in the right direction. Do not use the middle value alone.
Method. Find both bounds of the calculation, then compare them with the target.
- Write the bounds of each measure. These are the smallest and largest values each measurement could really be.x: 11.5 ≤ x < 12.5
y: 7.35 ≤ y < 7.45 - Work out the upper bound. Use the values that make the difference as large as possible.12.5 − 7.35 = 5.15
- Work out the lower bound. Use the values that make the difference as small as possible.11.5 − 7.45 = 4.05
- Compare with the target. The bounds 4.05 and 5.15 are on opposite sides of 5, so we cannot be sure.
- Write the answer. Not necessarily
Check. If the two bounds are on the same side of the target, the answer is certain. Otherwise it is not.
Watch out. Use the bounds that go in the right direction. Do not use the middle value alone.
Watch out
Watch out: a single rounded value can be right on the boundary. Always check both bounds before you answer Yes or No.
Exam tip
Exam tip. Write both bounds with a clear conclusion, such as "The lower bound is more than 9.7, so the area is always more than 9.7."
Exercise 2E Is the estimate accurate enough?
Non-calculatorMethod. Find both bounds of the calculation, then compare them with the target.
- Write the bounds of each measure. These are the smallest and largest values each measurement could really be.p: 4.65 ≤ p < 4.75
q: 3.15 ≤ q < 3.25 - Work out the upper bound. Use the values that make the total length as large as possible.4.75 + 3.25 = 8
- Work out the lower bound. Use the values that make the total length as small as possible.4.65 + 3.15 = 7.8
- Compare with the target. The upper bound 8 is less than 8.1, so the total length is always less than 8.1.
- Write the answer. Yes
Check. If the two bounds are on the same side of the target, the answer is certain. Otherwise it is not.
Watch out. Use the bounds that go in the right direction. Do not use the middle value alone.
Method. Find both bounds of the calculation, then compare them with the target.
- Write the bounds of each measure. These are the smallest and largest values each measurement could really be.p: 4.65 ≤ p < 4.75
q: 3.15 ≤ q < 3.25 - Work out the upper bound. Use the values that make the total length as large as possible.4.75 + 3.25 = 8
- Work out the lower bound. Use the values that make the total length as small as possible.4.65 + 3.15 = 7.8
- Compare with the target. The bounds 7.8 and 8 are on opposite sides of 7.9, so we cannot be sure.
- Write the answer. Not necessarily
Check. If the two bounds are on the same side of the target, the answer is certain. Otherwise it is not.
Watch out. Use the bounds that go in the right direction. Do not use the middle value alone.
Method. Find both bounds of the calculation, then compare them with the target.
- Write the bounds of each measure. These are the smallest and largest values each measurement could really be.p: 4.65 ≤ p < 4.75
q: 3.15 ≤ q < 3.25 - Work out the upper bound. Use the values that make the total length as large as possible.4.75 + 3.25 = 8
- Work out the lower bound. Use the values that make the total length as small as possible.4.65 + 3.15 = 7.8
- Compare with the target. The lower bound 7.8 is at least 7.8, so the total length is never less than 7.8.
- Write the answer. No
Check. If the two bounds are on the same side of the target, the answer is certain. Otherwise it is not.
Watch out. Use the bounds that go in the right direction. Do not use the middle value alone.
Method. Find both bounds of the calculation, then compare them with the target.
- Write the bounds of each measure. These are the smallest and largest values each measurement could really be.l: 6.35 ≤ l < 6.45
w: 2.45 ≤ w < 2.55 - Work out the upper bound. Use the values that make the area as large as possible.6.45 × 2.55 = 16.4475
- Work out the lower bound. Use the values that make the area as small as possible.6.35 × 2.45 = 15.5575
- Compare with the target. The lower bound 15.5575 is more than 15.5, so the area is always more than 15.5.
- Write the answer. Yes
Check. If the two bounds are on the same side of the target, the answer is certain. Otherwise it is not.
Watch out. Use the bounds that go in the right direction. Do not use the middle value alone.
Method. Find both bounds of the calculation, then compare them with the target.
- Write the bounds of each measure. These are the smallest and largest values each measurement could really be.l: 6.35 ≤ l < 6.45
w: 2.45 ≤ w < 2.55 - Work out the upper bound. Use the values that make the area as large as possible.6.45 × 2.55 = 16.4475
- Work out the lower bound. Use the values that make the area as small as possible.6.35 × 2.45 = 15.5575
- Compare with the target. The upper bound 16.4475 is less than 16.5, so the area is always less than 16.5.
- Write the answer. Yes
Check. If the two bounds are on the same side of the target, the answer is certain. Otherwise it is not.
Watch out. Use the bounds that go in the right direction. Do not use the middle value alone.
Method. Find both bounds of the calculation, then compare them with the target.
- Write the bounds of each measure. These are the smallest and largest values each measurement could really be.a: 11.5 ≤ a < 12.5
b: 7.5 ≤ b < 8.5 - Work out the upper bound. Use the values that make the area as large as possible.12.5 × 8.5 = 106.25
- Work out the lower bound. Use the values that make the area as small as possible.11.5 × 7.5 = 86.25
- Compare with the target. The bounds 86.25 and 106.25 are on opposite sides of 100, so we cannot be sure.
- Write the answer. Not necessarily
Check. If the two bounds are on the same side of the target, the answer is certain. Otherwise it is not.
Watch out. Use the bounds that go in the right direction. Do not use the middle value alone.
Method. Find both bounds of the calculation, then compare them with the target.
- Write the bounds of each measure. These are the smallest and largest values each measurement could really be.a: 11.5 ≤ a < 12.5
b: 7.5 ≤ b < 8.5 - Work out the upper bound. Use the values that make the area as large as possible.12.5 × 8.5 = 106.25
- Work out the lower bound. Use the values that make the area as small as possible.11.5 × 7.5 = 86.25
- Compare with the target. The lower bound 86.25 is more than 80, so the area is always more than 80.
- Write the answer. Yes
Check. If the two bounds are on the same side of the target, the answer is certain. Otherwise it is not.
Watch out. Use the bounds that go in the right direction. Do not use the middle value alone.
Method. Find both bounds of the calculation, then compare them with the target.
- Write the bounds of each measure. These are the smallest and largest values each measurement could really be.x: 11.5 ≤ x < 12.5
y: 7.35 ≤ y < 7.45 - Work out the upper bound. Use the values that make the difference as large as possible.12.5 − 7.35 = 5.15
- Work out the lower bound. Use the values that make the difference as small as possible.11.5 − 7.45 = 4.05
- Compare with the target. The bounds 4.05 and 5.15 are on opposite sides of 5, so we cannot be sure.
- Write the answer. Not necessarily
Check. If the two bounds are on the same side of the target, the answer is certain. Otherwise it is not.
Watch out. Use the bounds that go in the right direction. Do not use the middle value alone.
Method. Find both bounds of the calculation, then compare them with the target.
- Write the bounds of each measure. These are the smallest and largest values each measurement could really be.x: 11.5 ≤ x < 12.5
y: 7.35 ≤ y < 7.45 - Work out the upper bound. Use the values that make the difference as large as possible.12.5 − 7.35 = 5.15
- Work out the lower bound. Use the values that make the difference as small as possible.11.5 − 7.45 = 4.05
- Compare with the target. The lower bound 4.05 is more than 4, so the difference is always more than 4.
- Write the answer. Yes
Check. If the two bounds are on the same side of the target, the answer is certain. Otherwise it is not.
Watch out. Use the bounds that go in the right direction. Do not use the middle value alone.
Method. Find both bounds of the calculation, then compare them with the target.
- Write the bounds of each measure. These are the smallest and largest values each measurement could really be.m: 3.55 ≤ m < 3.65
n: 2.845 ≤ n < 2.855 - Work out the upper bound. Use the values that make the total mass as large as possible.3.65 + 2.855 = 6.505
- Work out the lower bound. Use the values that make the total mass as small as possible.3.55 + 2.845 = 6.395
- Compare with the target. The upper bound 6.505 is less than 6.6, so the total mass is always less than 6.6.
- Write the answer. Yes
Check. If the two bounds are on the same side of the target, the answer is certain. Otherwise it is not.
Watch out. Use the bounds that go in the right direction. Do not use the middle value alone.
Method. Find both bounds of the calculation, then compare them with the target.
- Write the bounds of each measure. These are the smallest and largest values each measurement could really be.d: 145 ≤ d < 155
t: 12.45 ≤ t < 12.55 - Work out the upper bound. Use the values that make the speed as large as possible.155 ÷ 12.45 = 12.4498 (to 4 d.p.)
- Work out the lower bound. Use the values that make the speed as small as possible.145 ÷ 12.55 = 11.5538 (to 4 d.p.)
- Compare with the target. The lower bound 11.5538 (to 4 d.p.) is more than 11, so the speed is always more than 11.
- Write the answer. Yes
Check. If the two bounds are on the same side of the target, the answer is certain. Otherwise it is not.
Watch out. Use the bounds that go in the right direction. Do not use the middle value alone.
Method. Find both bounds of the calculation, then compare them with the target.
- Write the bounds of each measure. These are the smallest and largest values each measurement could really be.x: 0.55 ≤ x < 0.65
y: 0.55 ≤ y < 0.65 - Work out the upper bound. Use the values that make the total width as large as possible.0.65 + 0.65 = 1.3
- Work out the lower bound. Use the values that make the total width as small as possible.0.55 + 0.55 = 1.1
- Compare with the target. The bounds 1.1 and 1.3 are on opposite sides of 1.25, so we cannot be sure.
- Write the answer. Not necessarily
Check. If the two bounds are on the same side of the target, the answer is certain. Otherwise it is not.
Watch out. Use the bounds that go in the right direction. Do not use the middle value alone.
Chapter summary
| Skill | Method |
|---|---|
| Multiply standard form | Multiply the front numbers and add the powers of 10. Then tidy to 1 ≤ A < 10. |
| Divide standard form | Divide the front numbers and subtract the powers of 10. Then tidy. |
| Add standard form | Write both with the same power of 10, add the front numbers, then tidy. |
| Bounds of a rounded value | Lower = value − half the unit. Upper = value + half the unit. |
| Bounds of a sum or product | Upper uses the two upper bounds. Lower uses the two lower bounds. |
| Bounds of a difference or quotient | Use the upper bound of the first number and the lower bound of the second for the upper bound. |
| Accuracy | Compare both bounds with the target: if both are on one side, the answer is certain. |
Can you…?
- I can multiply, divide and add numbers in standard form
- I can read a calculator display in standard form and write the number in full
- I can find the upper and lower bounds of a sum or product of rounded values
- I can find the bounds of a difference or quotient, using the correct end of each bound
- I can find bounds for a longer calculation with two or more measures
- I can decide whether an estimate is accurate enough, using both bounds
Key terms
- standard form
- a number written as A × 10n, where 1 ≤ A < 10 and n is a whole number
- upper bound
- the largest value a rounded measurement could have
- lower bound
- the smallest value a rounded measurement could have
- error interval
- the range of values from the lower bound up to, but not including, the upper bound
- truncate
- cut a number off at a given place, without rounding
Exercise 2Z Check-up
Non-calculatorMethod. Multiply the front numbers, add the powers of 10, then write the answer in standard form.
- Multiply the numbers at the front. Multiply the two numbers that are between 1 and 10.2 × 4 = 8
- Add the powers of 10. When powers of 10 are multiplied, their indices are added.105 × 10−2 = 103
- Put the front number between 1 and 10. Standard form needs one digit before the decimal point. Move the point and change the index to match.8 × 103 = 8 × 103
- Write the answer. 8 × 103
Check. Check: 8,000 is the same as 8,000.
Watch out. If the front number is 10 or more, or less than 1, adjust it. Do not leave a front number of 12.
Method. Divide the front numbers, subtract the powers of 10, then tidy the front number.
- Divide the numbers at the front. Divide the first front number by the second.3.6 ÷ 1.2 = 3
- Subtract the powers of 10. When powers of 10 are divided, subtract the indices.109 ÷ 105 = 104
- Put the front number between 1 and 10. Move the decimal point, and change the index to keep the same value.3 × 104 = 3 × 104
- Write the answer. 3 × 104
Check. Check: 30,000 × 1.2 × 105 is 3,600,000,000, so the division is correct.
Watch out. Subtract the index of the bottom number from the index of the top number: k1 − k2.
Method. Write both numbers with the same power of 10, then add or subtract the front numbers.
- Write both numbers with the same power of 10. Use the larger power, 106, for both. Change the other front number to match.5 × 106 = 5 × 106
2 × 105 = 0.2 × 106 - Add or subtract the front numbers. The powers of 10 are now the same, so only the front numbers change.5 + 0.2 = 5.2
- Put the answer into standard form. The front number must be between 1 and 10.5.2 × 106 = 5.2 × 106
- Write the answer. 5.2 × 106
Check. Check: 5,000,000 + 200,000 = 5,200,000.
Watch out. You cannot add the front numbers before the powers of 10 are the same. Do not add the indices.
Method. Use the power of 10 to move the decimal point.
- Read the calculator display. The E means times 10 to the power of the number after it.7E−4 means 7 × 10−4
- Move the decimal point. A positive index makes the number bigger, so move the point to the right. A negative index makes it smaller, so move it to the left.7 × 10−4 = 0.0007
- Write the answer. 0.0007
Check. Write 0.0007 in standard form to get back to 7 × 10−4.
Watch out. Count the places carefully. Each place moved is one step in the index.
Method. Use the lower bound of each measure, so that the answer is as small as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.l: 9.45 ≤ l < 9.55
w: 3.5 ≤ w < 4.5 - Choose the lower bound of each measure. To make the answer as small as possible, use the bound that pushes it that way.l = 9.45 (lower bound), w = 3.5 (lower bound)
- Work out the answer with these values. Use the same order of working as the calculation.9.45 × 3.5 = 33.075
- Write the answer. 33.075 cm2
Check. The other bound is 42.975, so the true value lies between 33.075 and 42.975.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the upper bound of each measure, so that the answer is as large as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.l: 9.45 ≤ l < 9.55
w: 3.5 ≤ w < 4.5 - Choose the upper bound of each measure. To make the answer as large as possible, use the bound that pushes it that way.l = 9.55 (upper bound), w = 4.5 (upper bound)
- Work out the answer with these values. Use the same order of working as the calculation.9.55 × 4.5 = 42.975
- Write the answer. 42.975 cm2
Check. The other bound is 33.075, so the true value lies between 33.075 and 42.975.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the upper bound of each measure, so that the answer is as large as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.x: 11.5 ≤ x < 12.5
y: 7.245 ≤ y < 7.255 - Choose the upper bound of each measure. To make the answer as large as possible, use the bound that pushes it that way.x = 12.5 (upper bound), y = 7.255 (upper bound)
- Work out the answer with these values. Use the same order of working as the calculation.12.5 + 7.255 = 19.755
- Write the answer. 19.755 cm
Check. The other bound is 18.745, so the true value lies between 18.745 and 19.755.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the lower bound of each measure, so that the answer is as small as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.x: 15.35 ≤ x < 15.45
y: 5.5 ≤ y < 6.5 - Choose the lower bound of each measure. To make the answer as small as possible, use the bound that pushes it that way.x = 15.35 (lower bound), y = 6.5 (upper bound)
- Work out the answer with these values. Use the same order of working as the calculation.15.35 − 6.5 = 8.85
- Write the answer. 8.85 cm
Check. The other bound is 9.95, so the true value lies between 8.85 and 9.95.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the upper bound of each measure, so that the answer is as large as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.x: 83.5 ≤ x < 84.5
y: 5.5 ≤ y < 6.5 - Choose the upper bound of each measure. To make the answer as large as possible, use the bound that pushes it that way.x = 84.5 (upper bound), y = 5.5 (lower bound)
- Work out the answer with these values. Use the same order of working as the calculation.84.5 ÷ 5.5 = 15.3636 (to 4 d.p.)
- Round to 2 decimal places. The question asks for this many decimal places.15.3636 (to 4 d.p.) = 15.36
Check. The other bound is 12.85, so the true value lies between 12.85 and 15.36.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the lower bound of each measure, so that the answer is as small as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.x: 5.55 ≤ x < 5.65
y: 2.445 ≤ y < 2.455 - Choose the lower bound of each measure. To make the answer as small as possible, use the bound that pushes it that way.x = 5.55 (lower bound), y = 2.455 (upper bound)
- Work out the answer with these values. Use the same order of working as the calculation.5.55 − 2.455 = 3.095
- Write the answer. 3.095
Check. The other bound is 3.205, so the true value lies between 3.095 and 3.205.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Total 22 marks. Allow about 22 minutes.
Exercise 2X Practice test questions
Non-calculatorMethod. Multiply the front numbers, add the powers of 10, then write the answer in standard form.
- Multiply the numbers at the front. Multiply the two numbers that are between 1 and 10.3 × 2 = 6
- Add the powers of 10. When powers of 10 are multiplied, their indices are added.106 × 102 = 108
- Put the front number between 1 and 10. Standard form needs one digit before the decimal point. Move the point and change the index to match.6 × 108 = 6 × 108
- Write the answer. 6 × 108
Check. Check: 600,000,000 is the same as 600,000,000.
Watch out. If the front number is 10 or more, or less than 1, adjust it. Do not leave a front number of 12.
Method. Use the power of 10 to move the decimal point.
- Read the calculator display. The E means times 10 to the power of the number after it.3.6E−6 means 3.6 × 10−6
- Move the decimal point. A positive index makes the number bigger, so move the point to the right. A negative index makes it smaller, so move it to the left.3.6 × 10−6 = 0.0000036
- Write the answer. 0.0000036
Check. Write 0.0000036 in standard form to get back to 3.6 × 10−6.
Watch out. Count the places carefully. Each place moved is one step in the index.
Method. Divide the front numbers, subtract the powers of 10, then tidy the front number.
- Divide the numbers at the front. Divide the first front number by the second.4.8 ÷ 1.6 = 3
- Subtract the powers of 10. When powers of 10 are divided, subtract the indices.109 ÷ 104 = 105
- Put the front number between 1 and 10. Move the decimal point, and change the index to keep the same value.3 × 105 = 3 × 105
- Write the answer. 3 × 105
Check. Check: 300,000 × 1.6 × 104 is 4,800,000,000, so the division is correct.
Watch out. Subtract the index of the bottom number from the index of the top number: k1 − k2.
Method. Write both numbers with the same power of 10, then add or subtract the front numbers.
- Write both numbers with the same power of 10. Use the larger power, 10−2, for both. Change the other front number to match.7 × 10−2 = 7 × 10−2
3 × 10−3 = 0.3 × 10−2 - Add or subtract the front numbers. The powers of 10 are now the same, so only the front numbers change.7 + 0.3 = 7.3
- Put the answer into standard form. The front number must be between 1 and 10.7.3 × 10−2 = 7.3 × 10−2
- Write the answer. 7.3 × 10−2
Check. Check: 0.07 + 0.003 = 0.073.
Watch out. You cannot add the front numbers before the powers of 10 are the same. Do not add the indices.
Method. Use the lower bound of each measure, so that the answer is as small as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.l: 4.5 ≤ l < 5.5
w: 2.5 ≤ w < 3.5 - Choose the lower bound of each measure. To make the answer as small as possible, use the bound that pushes it that way.l = 4.5 (lower bound), w = 2.5 (lower bound)
- Work out the answer with these values. Use the same order of working as the calculation.4.5 × 2.5 = 11.25
- Write the answer. 11.25 cm2
Check. The other bound is 19.25, so the true value lies between 11.25 and 19.25.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the upper bound of each measure, so that the answer is as large as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.x: 20.45 ≤ x < 20.55
y: 11.5 ≤ y < 12.5 - Choose the upper bound of each measure. To make the answer as large as possible, use the bound that pushes it that way.x = 20.55 (upper bound), y = 11.5 (lower bound)
- Work out the answer with these values. Use the same order of working as the calculation.20.55 − 11.5 = 9.05
- Write the answer. 9.05 cm
Check. The other bound is 7.95, so the true value lies between 7.95 and 9.05.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the upper bound of each measure, so that the answer is as large as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.d: 59.5 ≤ d < 60.5
t: 4.15 ≤ t < 4.25 - Choose the upper bound of each measure. To make the answer as large as possible, use the bound that pushes it that way.d = 60.5 (upper bound), t = 4.15 (lower bound)
- Work out the answer with these values. Use the same order of working as the calculation.60.5 ÷ 4.15 = 14.5783 (to 4 d.p.)
- Round to 2 decimal places. The question asks for this many decimal places.14.5783 (to 4 d.p.) = 14.58 m/s
Check. The other bound is 14.00, so the true value lies between 14.00 and 14.58.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Use the upper bound of each measure, so that the answer is as large as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.l: 5.25 ≤ l < 5.35
w: 2.75 ≤ w < 2.85 - Choose the upper bound of each measure. To make the answer as large as possible, use the bound that pushes it that way.l = 5.35 (upper bound), w = 2.85 (upper bound)
- Work out the answer with these values. Use the same order of working as the calculation.2(5.35 + 2.85) = 16.4
- Write the answer. 16.4 cm
Check. The other bound is 16, so the true value lies between 16 and 16.4.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.
Method. Find both bounds of the calculation, then compare them with the target.
- Write the bounds of each measure. These are the smallest and largest values each measurement could really be.x: 2.45 ≤ x < 2.55
y: 3.5 ≤ y < 4.5 - Work out the upper bound. Use the values that make the product as large as possible.2.55 × 4.5 = 11.475
- Work out the lower bound. Use the values that make the product as small as possible.2.45 × 3.5 = 8.575
- Compare with the target. The bounds 8.575 and 11.475 are on opposite sides of 9.7, so we cannot be sure.
- Write the answer. Not necessarily
Check. If the two bounds are on the same side of the target, the answer is certain. Otherwise it is not.
Watch out. Use the bounds that go in the right direction. Do not use the middle value alone.
Method. Use the lower bound of each measure, so that the answer is as small as possible.
- Write the bounds of each measure. A rounded value could be up to half a unit above or below the value shown.l: 8.45 ≤ l < 8.55
w: 5.95 ≤ w < 6.05 - Choose the lower bound of each measure. To make the answer as small as possible, use the bound that pushes it that way.l = 8.45 (lower bound), w = 5.95 (lower bound)
- Work out the answer with these values. Use the same order of working as the calculation.8.45 × 5.95 = 50.2775
- Write the answer. 50.2775 m2
Check. The other bound is 51.7275, so the true value lies between 50.2775 and 51.7275.
Watch out. Use the upper bounds to find the upper bound, and the lower bounds to find the lower bound, unless a measure is subtracted or divides.