Measures and accuracy — AQA GCSE Mathematics
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Measures and accuracy explained
When adding or comparing quantities, convert to a common unit first.
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Metric length, mass and capacity scale by powers of 10: 1 km = 1000 m, 1 kg = 1000 g, 1 litre = 1000 cm³. Area and volume conversions use squared and cubed factors, so 1 m² = 10 000 cm² and 1 m³ = 1 000 000 cm³; this matters for density and pressure. Time is not decimal: 2.5 hours is 2 hours 30 minutes, and 90 minutes is 1.5 hours. Compound measures divide different units, so speed = distance ÷ time, density = mass ÷ volume and pressure = force ÷ area. A car covering 150 km in 2.5 hours travels at 150 ÷ 2.5 = 60 km/h. Money is written to two decimal places, so £4.50, not £4.5.
estimate answers check calculations using approximation and estimation, including answers obtained using technology
An estimate is a deliberately rough answer you can produce in your head, and its purpose is to tell you whether a calculated value is believable. The standard method is to round every number to one significant figure and work with those instead. For 38.2 × 5.1 divided by 0.198, use 40 × 5 = 200, then 200 ÷ 0.2 = 1000, so the true answer should sit near a thousand; a display reading 98 or 10000 means a key went astray. Write the estimate with the ≈ sign rather than an equals sign. Dividing a positive number by a number between 0 and 1 makes the result larger, and that is the step most often taken backwards. A calculation can also be checked by reversing it: if 17 × 23 = 391, then 391 ÷ 23 must return 17.
round numbers and measures to an appropriate degree of accuracy (eg to a specified number of decimal places or significant figures)
Decimal places are counted after the point, while significant figures are counted from the first non-zero digit wherever it happens to sit. Look at the digit one place beyond where you are stopping: 5 or more sends the last kept digit up, anything less leaves it alone. So 3.0482 to two decimal places is 3.05, and 0.004561 to two significant figures is 0.0046, because the zeros after the point are holding place value rather than counting as figures. With whole numbers the place value must be kept by zeros, so 28471 to two significant figures is 28000. Choosing a sensible accuracy usually means matching the data you were given, and money is written to two decimal places. In a calculation with several steps, keep full accuracy on the calculator and round only the answer you write on the line.
use inequality notation to specify simple error intervals due to truncation or rounding
A rounded value stands for a whole range of true values, and the error interval is that range written with inequality signs. For rounding, go half a unit either side of the value you were given. A length recorded as 6.4 cm to one decimal place could really be anything from 6.35 cm up to, but not reaching, 6.45 cm, so the interval is 6.35 ≤ x < 6.45. The lower end takes the sign that includes it, because 6.35 rounds up to 6.4, and the upper end excludes its value, because 6.45 would round to 6.5. Truncation, which chops the extra digits off, behaves differently: a value truncated to 6.4 lies in 6.4 ≤ x < 6.5, since truncating never pushes a number up. Match the half unit to the accuracy stated, so a mass of 250 g to the nearest ten grams gives 245 ≤ m < 255.
apply and interpret limits of accuracy
Any measurement given to a stated accuracy is exact only within half of the unit it was rounded to, so it has a least possible value (lower bound) and a greatest possible value (upper bound). A shelf measured as 180 cm to the nearest centimetre is at least 179.5 cm and less than 180.5 cm. Applying bounds means asking what those extremes do to your answer. If three boxes are each 25 cm wide to the nearest centimetre, their total width has an upper bound of 3 × 25.5 = 76.5 cm, so a 76 cm gap might not take them. Interpreting means saying what the range shows: that something certainly fits, certainly does not, or may or may not. A result worked out from rounded measurements is itself only accurate within a range.
including upper and lower bounds (Higher tier only)
This is Higher tier only. The lower bound is the smallest value a rounded measurement could take and the upper bound is the value it stops short of, each half a unit away from the figure you were given. A time of 12.5 s to the nearest tenth of a second has bounds of 12.45 s and 12.55 s. The skill being tested is choosing which bound belongs in which position. For an addition or a multiplication the largest result uses the upper bound of both quantities. For a subtraction, the largest result is the upper bound of the first quantity minus the lower bound of the second. For a division, the largest result puts the upper bound on the top and the lower bound underneath. A distance of 100 m to the nearest metre run in 12.5 s gives a greatest speed of 100.5 ÷ 12.45 = 8.07 m/s and a least speed of 99.5 ÷ 12.55 = 7.93 m/s.
Your focus
- Write down the conversions between km and m, kg and g, litres and cm³, and between hours and minutes.
- Calculate a speed, density or pressure from its defining division and rearrange it to find a missing quantity with its unit.
- Explain why both quantities must be in matching units before any calculation, using 150 km in 2.5 hours giving 60 km/h.
Show all 20 objectives
- Estimate the answer to a calculation by rounding every number to one significant figure and writing the estimate with ≈.
- Work out an estimate such as 40 × 5 ÷ 0.2 = 1000 and say what a display reading 98 tells you about the keying.
- Show that a result is right by reversing it, so 17 × 23 = 391 means 391 ÷ 23 must return 17.
- Explain whether an estimate sits above or below the true value, giving a reason drawn from how each number was rounded.
- Explain the difference between decimal places counted after the point and significant figures counted from the first non-zero digit.
- Work out a value rounded as asked, so 3.0482 to two decimal places is 3.05 and 28471 to two significant figures is 28000.
- Explain why an unrounded value is carried through a multi-step calculation and only the written answer is rounded.
- Write down the half unit that matches a stated accuracy, such as 0.005 either side of a value given to two decimal places.
- Work out the error interval for a rounded measurement as one statement, so 6.4 cm to one decimal place gives 6.35 ≤ x < 6.45.
- Explain why a truncated 6.4 gives 6.4 ≤ x < 6.5, since truncating chops digits off and never pushes a number up.
- Write down the least and greatest possible values of a measurement given to a stated accuracy, such as 180 cm to the nearest cm.
- Work out the upper bound of a total from rounded measurements, so three boxes of 25 cm give 3 × 25.5 = 76.5 cm.
- Explain in context whether something certainly fits, certainly does not, or may or may not, referring back to the bounds found.
- Write down both bounds of a measurement as values, so 12.5 s to the nearest tenth gives 12.45 s and 12.55 s.
- Explain which bound belongs on top and which underneath for a greatest quotient, and the reverse for a least one.
- Calculate the greatest and least speed for 100 m to the nearest metre run in 12.5 s, giving 8.07 m/s and 7.93 m/s.
- Justify quoting a final answer only to the digits that the upper and lower bounds agree on.
Measures and accuracy exam tips
Quick Revision Summary (Key Takeaway)
Measures and accuracy covers reading scales, converting between metric units, and applying bounds of measurement to calculate maximum and minimum possible values. In AQA GCSE Mathematics, you must round appropriately, use inequality notation for error intervals, and calculate upper and lower bounds for calculations involving measured quantities.
Topic Overview
Measures and accuracy is a fundamental topic in GCSE Mathematics that deals with how we quantify the physical world and the inherent uncertainties in measurement. It covers reading and interpreting scales, converting between units of length, mass, capacity, and time, and understanding the precision of measuring instruments. You will learn to express the accuracy of a measurement using error intervals and to calculate upper and lower bounds for calculations involving measured values.
This topic is essential for real-world problem-solving and is heavily examined in AQA GCSE Mathematics, often appearing in both calculator and non-calculator papers. It links to many other areas such as ratio, proportion, geometry, and statistics, and is crucial for science subjects. Mastering measures and accuracy ensures you can handle data responsibly and make informed judgements about the reliability of results.
Key Concepts
- →Units of measurement: Know the metric conversions (e.g., 1 km = 1000 m, 1 m = 100 cm, 1 kg = 1000 g, 1 litre = 1000 ml) and be able to convert between them confidently.
- →Reading scales: Understand how to read values from a variety of scales, including those with different intervals, and estimate values between marked points.
- →Error intervals: For a value rounded to a given degree of accuracy, the error interval describes the range of possible original values. Use inequality notation: lower bound <= x < upper bound.
- →Upper and lower bounds: When a quantity is rounded, its true value lies within a range. The upper bound is the largest possible value, and the lower bound is the smallest possible value.
- →Calculations with bounds: When adding, subtracting, multiplying, or dividing measured quantities, use the appropriate combination of upper and lower bounds to find the maximum or minimum possible result.
Marking Points
- converting quantities to a common unit before adding, comparing or substituting
- correctly converting area or volume units using squared or cubed factors, such as 1 m³ = 1 000 000 cm³
- a correct compound measure calculation, such as distance ÷ time, even if the unit is then omitted
- giving the answer in the unit the question asks for, including a change such as m/s into km/h
- expressing a time correctly in hours and minutes where a duration or clock time is required
- every number in the calculation rounded to one significant figure, written down before the working
- the rounded calculation evaluated correctly, even if the comparison drawn from it is wrong
- a comparison with the given answer when the question asks whether that answer is sensible
- saying whether the estimate is above or below the true value, with a reason drawn from the rounding
- identifying the correct digit to look at, especially where leading zeros are not significant
- the rounded value written with its place value intact, so a rounded whole number keeps its zeros
- an answer given to the accuracy the question demands, since the accuracy mark is lost when the working is right but the rounding is not
- on a multi-step question, carry unrounded values forward and round the final answer to the requested accuracy
- the correct half unit, such as 0.005 either side of a value given to two decimal places
- the two end values, even where the inequality signs are the wrong way round
- the interval written as one statement, with the variable the question names in the middle
- for a positive truncated value, give an interval whose included lower endpoint is the truncated value itself
- the least possible value (lower bound) of a measurement given to a stated accuracy
- the greatest possible value (upper bound), whether written as a value the measurement stops short of or with a strict inequality
- using those extreme values in the calculation, rather than the rounded measurement itself
- a conclusion in context that refers back to the values found
- for a division, choosing the extreme that makes the answer largest, which is not always the largest measurement
- both bounds of each measurement, written as values rather than as the rounded figure
- selecting the bounds that produce the extreme the question asks for, such as the upper bound over the lower bound for a greatest quotient
- the calculation carried out with those bounds, even if the rounding of the final answer is wrong
- quoting only the digits that both bounds agree on, where the question asks for a suitable degree of accuracy
Examiner Tips
- 💡Write the unit beside every number; the units show whether to multiply or divide.
- 💡Convert minutes to hours by dividing by 60 before using a speed in km/h.
- 💡Check the answer against everyday sizes: a walking speed near 5 km/h is believable, 500 km/h is not.
- 💡Round sensibly, show the approximate calculation, then evaluate it so your estimation method is clear.
- 💡If asked to estimate, show a sensible approximation rather than giving only an exact calculator result.
- 💡Run a one significant figure check over any calculator answer before you write it down.
- 💡Underline the digit you are keeping and look only at the one immediately to its right.
- 💡Keep the full value in the calculator memory and round once, at the very end.
- 💡Where no accuracy is stated, three significant figures is normally accepted for a decimal answer that does not work out exactly.
- 💡Write the unit of accuracy down first, then halve it; everything else follows from that half unit.
- 💡Keep the variable between the two values so the answer reads as one statement rather than two.
- 💡Look for the word truncated in the question, because it moves both ends of the interval.
- 💡List the least and greatest value of every measurement in the question before you calculate anything.
- 💡Ask which extreme makes the answer largest; with a division it is not always the largest measurement.
- 💡Finish with a sentence that answers the question asked and quotes the numbers you found.
- 💡Set the four bounds out in a small table before you calculate; the marks are easier to pick up from a clear list.
- 💡If you are unsure which combination gives the maximum, work out both and take the larger answer.
- 💡For a suitable degree of accuracy, compare the two bounds digit by digit and quote only what they share.
- 💡Always write down the error interval using correct inequality notation. Remember that the lower bound is included (<=) and the upper bound is excluded (<) when rounding to the nearest unit.
- 💡When a calculation involves multiple steps, keep track of the bounds at each stage and avoid premature rounding. Use exact values or keep extra decimal places until the final answer.
- 💡For 'maximum' or 'minimum' questions, clearly state which bounds you are using and why. Show all steps of your working to gain method marks even if the final answer is incorrect.
Common Mistakes
- reading 2.75 hours as 2 hours 75 minutes instead of 2 hours 45 minutes; multiply the decimal part by 60
- dividing time by distance when finding a speed; speed is distance ÷ time
- using 100 rather than 1 000 000 to convert m³ to cm³; volume factors are cubed
- writing money as £4.5 or mixing pounds and pence in one figure; use two decimal places
- rounding 0.198 to 0 rather than 0.2, which makes the division impossible
- rounding to one decimal place instead of one significant figure, so 0.0482 becomes 0.0 rather than 0.05
- using an equals sign between the original calculation and the rounded version
- assuming that dividing by 0.2 makes the answer smaller
- counting the zeros in 0.00456 as significant, so quoting 0.00 to two significant figures
- rounding 28471 to 28 and dropping the zeros that hold the place value
- rounding twice, taking 4.348 to 4.35 and then to 4.4 rather than straight to 4.3
- rounding down when the next digit is exactly 5
- using the same type of sign at both ends, so the upper value is wrongly included
- halving the measurement rather than the unit of accuracy
- treating a truncated value like a rounded one and going half a unit below it
- writing the interval with the larger number on the left
- using the rounded measurement when the question asks what could happen at the extremes; correct by substituting the lower and upper bounds instead
- halving the measurement instead of halving the unit of accuracy; correct by halving the unit, so 180 cm to the nearest cm gives ±0.5 cm
- giving the greatest possible value of 180 cm to the nearest centimetre as 180.4 or 180.49; correct by using 180.5 cm as the value it stops short of
- answering yes or no with no numbers offered in support; correct by quoting the bounds found
- putting the lower bound underneath when the minimum is wanted, which produces the maximum instead
- giving the upper bound of 12.5 s as 12.54 or 12.549 rather than 12.55
- subtracting the two upper bounds when the greatest possible difference is wanted
- rounding the bounds themselves before feeding them into the calculation
- Students often think that if a length is given as 5 cm to the nearest cm, the lower bound is 4.5 cm and the upper bound is 5.5 cm, but they might incorrectly include the upper bound in the error interval. The correct interval is 4.5 <= x < 5.5 because 5.5 would round up to 6 cm.
- When calculating with bounds, students may use the upper bound for both quantities in a division to find the maximum, but actually for division, maximum = UB(a)/LB(b). Similarly, for subtraction, maximum = UB(a) - LB(b).
- Students sometimes forget to convert units before performing calculations, leading to incorrect bounds. Always ensure all measurements are in the same unit before calculating.
Revision Plan
- 1Week 1: Start by revising unit conversions and reading scales. Practice converting between metric units and reading values from different types of scales, including those with fractional intervals.
- 2Week 1: Learn how to write error intervals for values rounded to the nearest whole number, 1 decimal place, 2 decimal places, and significant figures. Practice with a variety of examples.
- 3Week 2: Move on to calculating upper and lower bounds for simple calculations (addition, subtraction, multiplication, division). Focus on understanding which combination of bounds gives the maximum and minimum results.
- 4Week 2: Attempt past paper questions on measures and accuracy, including multi-step problems. Review your mistakes and ensure you understand the mark scheme.
- 5Ongoing: Create a summary sheet with key formulas and rules for bounds calculations, and test yourself regularly using active recall.
Exam Question Types
- 📋Error interval questions: Given a value rounded to a certain degree of accuracy, write down the error interval. Advice: Pay attention to whether the lower bound is included and the upper bound excluded.
- 📋Upper and lower bounds calculations: Calculate the maximum or minimum possible value of an expression involving measured quantities. Advice: Identify the correct combination of bounds for the operation and show your working clearly.
- 📋Unit conversion problems: Convert between different metric units and perform calculations. Advice: Always convert to the same unit before calculating and check your answer for reasonableness.
- 📋Problem-solving with bounds: Multi-step problems that require you to use bounds to determine whether a statement is true or to find a maximum/minimum quantity. Advice: Break the problem into steps, calculate bounds at each stage, and interpret your result in context.
Command Word Expectations (AQA)
You must work out a numerical answer using the given information. For bounds questions, this means performing the correct operation with the appropriate upper or lower bounds. Show all steps of your working.
You are expected to state a value or interval without detailed calculation. For error intervals, write the inequality correctly, e.g., 4.5 <= x < 5.5.
You must give a reason or justification for your answer. For example, explain why a certain combination of bounds gives the maximum value, referring to the effect of rounding.
How Students Lose Marks (Examiner Pitfalls)
Step-by-Step Worked Solutions
Question: A rectangle has length 8.4 cm and width 5.6 cm, both measured to 1 decimal place. Calculate the upper bound for the area of the rectangle.
- 1.Step 1: Identify the bounds for each measurement. Length = 8.4 cm to 1 d.p., so bounds are 8.35 <= L < 8.45. Width = 5.6 cm to 1 d.p., so bounds are 5.55 <= W < 5.65.
- 2.Step 2: For the upper bound of the area, use the upper bound of both length and width: UB(Area) = UB(L) * UB(W) = 8.45 * 5.65.
- 3.Step 3: Calculate: 8.45 * 5.65 = 47.7425. Since the original measurements are given to 1 d.p., the area should be given to an appropriate degree of accuracy, typically 2 d.p. So, upper bound = 47.74 cm^2 (to 2 d.p.).
Question: The mass of a coin is 8.2 g to the nearest 0.1 g. The mass of a bag is 1.5 kg to the nearest 0.1 kg. Calculate the maximum possible number of coins that could be in the bag if the total mass of the bag and coins is 1.5 kg to the nearest 0.1 kg.
- 1.Step 1: Convert all units to grams. Bag mass = 1.5 kg = 1500 g to the nearest 100 g? Wait, 1.5 kg to the nearest 0.1 kg means to the nearest 100 g. So bounds for bag mass: 1450 g <= B < 1550 g. Coin mass = 8.2 g to nearest 0.1 g, so bounds: 8.15 g <= C < 8.25 g.
- 2.Step 2: To find the maximum number of coins, we need the maximum total mass of coins and the minimum mass of each coin. The total mass of coins = total mass - bag mass. To maximise the number of coins, we want the total mass of coins to be as large as possible and each coin to be as light as possible. So, maximum total mass of coins = upper bound of total mass - lower bound of bag mass. But the total mass is given as 1.5 kg to the nearest 0.1 kg, so its bounds are 1450 g <= T < 1550 g. So max total mass of coins = 1550 - 1450 = 100 g? That seems too small. Actually, the total mass is the mass of the bag with coins, which is 1.5 kg to the nearest 0.1 kg. So the total mass T has bounds 1450 <= T < 1550. The bag alone has mass B with bounds 1450 <= B < 1550. The mass of coins = T - B. To maximise the number of coins, we need to maximise (T - B)/C. To maximise T - B, we use max T and min B: 1550 - 1450 = 100 g. To minimise C, we use lower bound of C: 8.15 g. So max number = 100 / 8.15 = 12.269... So maximum number of coins is 12 (since we can't have a fraction of a coin).
- 3.Step 3: Check if 12 coins is possible: 12 coins of mass 8.15 g each = 97.8 g. Then total mass = bag mass + 97.8 g. If bag mass is 1450 g, total = 1547.8 g, which rounds to 1.5 kg to the nearest 0.1 kg (since 1547.8 g is 1.5478 kg, which rounds to 1.5 kg). So 12 coins is possible. 13 coins would be 13*8.15=105.95 g, total = 1450+105.95=1555.95 g, which rounds to 1.6 kg, not 1.5 kg. So maximum is 12.