N14, N15 - Rounding, significant figures and estimation

Rounding to a place value

A group needs seats for $73$ people. Each minibus seats $16$. Will ordinary rounding always tell you how many minibuses to book? Keep that question in mind while we first learn what rounding actually does.

The number $4.23$ lies between $4.2$ and $4.3$. It is closer to $4.2$, so $4.23$ rounded to one decimal place is $4.2$. Their midpoint is $4.25$. Values below the midpoint round to $4.2$; for positive numbers, the midpoint and values above it round to $4.3$.

[DIAGRAM: asset_name: Rounding to the nearest tenth; asset_slug: aqa8300_n14_rounding_midpoint; caption: The midpoint 4.25 decides whether a positive number rounds to 4.2 or 4.3; description: A number line from 4.20 to 4.30 has a labelled midpoint at 4.25. A blue point marks 4.23 to the left of the midpoint. Pale blue and clay bands distinguish the two halves, with arrows indicating rounding towards 4.2 and 4.3 respectively. The labels show 4.23 rounds to 4.2 and 4.25 rounds to 4.3.]

To round, identify the last place you want to keep. Look at the next digit. A digit from $0$ to $4$ leaves the kept digit unchanged; a digit from $5$ to $9$ increases it by one. Remove digits to the right of the required decimal place. For whole numbers rounded to tens, hundreds or thousands, replace the discarded digits with zeros to preserve place value.

For instance, $684$ to the nearest hundred is $700$, not $7$. With negative numbers, think of the nearest point on the number line: $-3.26$ is closer to $-3.3$ than to $-3.2$.

Keep the hundredths column in the following example.

Now change the kept digit to $9$. Rounding up must carry into the next column.

The first discarded digit makes the rounding decision; the rest cannot reverse it.

Use the same carrying idea with a new number.

Rounding to significant figures

Significant figures count from the first non-zero digit, wherever the decimal point sits.

In $0.00407$, the first significant digit is $4$. The zeros before it only locate the number on the place-value scale. The zero between $4$ and $7$ counts, so this number has three significant figures. In $4.070$, the final zero also counts because it is written after the decimal point to show accuracy.

After finding the last significant digit you want, use the next digit to decide whether to round up. Keep place value: rounding $47\,261$ to two significant figures still gives a number in the tens of thousands. This rule is not on the formulae sheet: learn it.

The next example changes only the scale: the same digits appear in a small decimal. Start counting at the same $4$.

Decimal places and significant figures answer different questions.

For this check, the next digit now causes an increase.

Estimating and checking calculations

An estimate makes a calculation simple enough to check its size without a calculator.

For an exam question that says “By rounding each number to one significant figure”, show those rounded numbers. Keep the operations unchanged. In particular, rounding a number inside a power does not remove its exponent. This estimation method is not on the formulae sheet: learn it.

Change the square to a cube. That single symbol makes a large difference to the estimate.

An overestimate is greater than the exact value; an underestimate is smaller. For positive quantities, increasing a numerator increases a quotient, but increasing its denominator decreases it. When some rounding changes pull in opposite directions, do not guess which effect wins.

Estimation also checks technology. The exact first expression is $141.69$; a display of $1416.9$ is far too large compared with $149$. Check the entered expression, brackets and decimal points. A result near the estimate is plausible, but that alone does not prove every key was correct. A quick estimate for $38.7\div0.49$, for example, is $40\div0.5=80$, so a calculator answer near $8$ needs checking.

Choosing accuracy in context

The context decides whether an ordinary nearest-value answer is useful.

For money, a final price in pounds normally needs two decimal places. For whole objects, ask whether you need enough capacity or only complete groups. A fractional vehicle is not something you can book.

Now change the demand from enough capacity to complete groups. The quotient stays the same; the direction of rounding changes.

Even when ordinary rounding is appropriate, do it once at the end. A surprising trap is that $2.449$ rounds directly to $2.4$ at one decimal place, but rounding first to $2.45$, then to one decimal place, gives $2.5$. The first rounding has changed the later decision.

The opening minibus answer happens to agree with rounding $4.5625$ to the nearest integer. That is a coincidence: if the quotient had been $4.1$, enough seats would still require five vehicles. Read what the answer has to achieve.

Quick Recap

• Decimal places count from the decimal point; significant figures count from the first non-zero digit.
• Keep the number's place value and show final decimal zeros when needed for the requested accuracy.
• Estimate with simple inputs, keeping the original operations, then use the result to check the size of an answer.
• Keep intermediate values unrounded and round the final result once.
• In context, enough capacity may require rounding up; complete groups may require rounding down.

The group must book $5$ minibuses: ordinary rounding is not the rule that guarantees enough seats.