Common Mistakes With Bounds in GCSE Maths

Bounds questions often feel straightforward until one small decision changes the entire answer. A student may understand the calculation, enter everything correctly and still lose marks because they chose the wrong limit.

The most common mistakes with bounds in GCSE Maths come from four places: using the wrong rounding interval, including the upper bound, choosing limits without considering the operation, and rounding during the calculation. The reliable method is to write every error interval first, decide whether the required result must be as large or as small as possible, and only then calculate.

Imagine a student checking a marked paper. We will call them Sam. Their calculator work is accurate, but several bounds mistakes have cost marks. By following each error and correcting the reasoning behind it, Sam can turn bounds from a guessing exercise into a repeatable method.

A quick bounds checklist

Before beginning any bounds calculation, ask:

• What accuracy was each value rounded to?
• What is half of that rounding unit?
• Have I written the complete error interval?
• Does the question require a maximum or minimum?
• Which combination of bounds creates that result?
• Have I kept the full calculator value until the end?

You can review the complete method in the MathsGenie Bounds lesson or revisit the notation through the Error Intervals lesson.

Mistake: using the stated accuracy instead of half of it

Sam sees that a length is $8.4$ cm to the nearest $0.1$ cm. They add and subtract $0.1$, giving limits of $8.3$ cm and $8.5$ cm.

This happens because the stated accuracy looks like the natural amount to use. However, rounding boundaries sit halfway between consecutive rounded values. Half of $0.1$ is $0.05$.

Correct method

Start with the rounding unit:

$$0.1$$

Find half of it:

$$0.1÷2=0.05$$

Subtract and add $0.05$:

$$8.4-0.05=8.35$$ $$8.4+0.05=8.45$$

The correct error interval is:

$$8.35\le x<8.45$$

The same principle works for other accuracies. To the nearest whole number, use half of $1$, which is $0.5$. To the nearest $10$, use half of $10$, which is $5$.

Mistake: including the upper bound

Sam writes:

$$8.35\le x\le 8.45$$

The lower limit is included because $8.35$ rounds to $8.4$ to the nearest $0.1$. The upper limit is not included because $8.45$ rounds to $8.5$, not $8.4$.

The standard structure is therefore:

$$lower\; bound\le x<upper\; bound$$

It helps to read this aloud: the value can equal the lower boundary, but it must be less than the upper boundary. The MathsGenie Error Intervals questions provide focused practice with this notation.

A student discovers that bounds errors are caused by assumptions rather than calculator buttons

Mistake: choosing bounds by habit

Sam has learnt a shortcut: use upper bounds for the largest answer and lower bounds for the smallest answer. That works for addition and for multiplying positive measurements. It does not work unchanged for subtraction or division.

The better question is not simply, “Do I want an upper bound?” It is, “Which inputs make this expression as large or as small as possible?”

Use this sequence every time:

• Write the error interval for each value.
• Identify the required result.
• Consider how changing each input affects that result.
• Select the limits.
• Substitute and calculate.

Practise this sequence through the MathsGenie Bounds Lessons and Question Bank when you want to turn the checklist into an exam habit.

Mistake: using two upper bounds in division

A journey has a distance of $120$ km, correct to the nearest $10$ km, and a time of $3.0$ hours, correct to the nearest $0.1$ hour. Sam needs the upper bound for average speed and uses both upper bounds.

First write the intervals:

$$115\le d<125$$ $$2.95\le t<3.05$$

Average speed is:

$$v=\frac{d}{t}$$

To make a positive fraction as large as possible, make its numerator large and its denominator small. Therefore, use the upper bound for distance and the lower bound for time:

$$v_{upper}=\frac{125}{2.95}$$ $$v_{upper}\approx 42.372881$$

If the question asks for an appropriate rounded answer, this could be stated as $42.4$ km/h to $3$ significant figures. Keep the unrounded value available until you know the required accuracy.

For the lower bound, reverse the choices:

$$v_{lower}=\frac{115}{3.05}$$

The rule for division of positive quantities is:

$$maximum\; quotient=\frac{upper\; numerator}{lower\; denominator}$$ $$minimum\; quotient=\frac{lower\; numerator}{upper\; denominator}$$

Mistake: treating subtraction like addition

Suppose a large mass is $75$ kg to the nearest kilogram and a smaller mass is $28$ kg to the nearest kilogram. Sam wants the greatest possible difference and subtracts the two upper bounds.

The intervals are:

$$74.5\le a<75.5$$ $$27.5\le b<28.5$$

The difference is:

$$a-b$$

To make the difference as large as possible, make the first value large and the value being subtracted small:

$$75.5-27.5=48$$

To make it as small as possible, use the lower bound of the first value and the upper bound of the second:

$$74.5-28.5=46$$

Although $75.5$ is not included in the original interval, $48$ is still described as the upper bound of the difference. It is the boundary that possible results can approach but not exceed.

Mistake: rounding halfway through

Sam calculates one part of a bounds problem, rounds it to $2$ decimal places and uses that rounded answer in the next step. This can shift the final answer enough to lose an accuracy mark.

Keep full calculator values throughout. If a value must be reused, store it using the calculator memory or use the full expression in one calculation. Round only when the question tells you to, or when presenting the final answer to a sensible accuracy.

The Bounds revision guide can help you review these choices before attempting mixed questions.

Exam-style practice

Try these without looking back at the examples.

Question one

A length is $36$ cm, correct to the nearest centimetre. Write its error interval.

Question two

A rectangle has length $12.4$ cm and width $7.2$ cm, each correct to the nearest $0.1$ cm. Find the upper bound for its area.

Question three

A car travels $250$ km, correct to the nearest $10$ km, using $18$ litres of fuel, correct to the nearest litre. Find the lower bound for the distance travelled per litre.

Answers

For question one:

$$35.5\le x<36.5$$

For question two, use both upper bounds because both positive measurements are multiplied:

$$12.45\times 7.25=90.2625$$

The upper bound is $90.2625$ cm${}^2$.

For question three, make the numerator small and denominator large:

$$\frac{245}{18.5}\approx 13.243243$$

The lower bound is approximately $13.24$ km per litre. You can continue with further exam-style practice in the Edexcel GCSE Maths Question Bank. Students following Eduqas can use the corresponding Eduqas Bounds topic.

Turn the corrected method into a habit

Sam's important change is not a new calculator technique. It is the pause before calculating: write the intervals, identify the required extreme and choose the limits deliberately. Knowing that method is useful, but following it under exam pressure takes practice.

Start the MathsGenie Bounds Lessons, then answer the matching GCSE Maths Question Bank.