Place Value, Ordering and Rounding

This lesson covers the fundamental number skills required for AQA GCSE Mathematics. Understanding place value, ordering numbers correctly and rounding to appropriate degrees of accuracy underpins almost every other topic in the specification. These skills appear frequently in both non-calculator and calculator papers.

Place Value in Integers

Every digit in a whole number has a place value depending on its position. The place value system is based on powers of 10.

Millions	Hundred Thousands	Ten Thousands	Thousands	Hundreds	Tens	Units
1,000,000	100,000	10,000	1,000	100	10	1

For example, in the number 4,362,517:

The 4 is worth 4,000,000 (four million)
The 3 is worth 300,000 (three hundred thousand)
The 6 is worth 60,000 (sixty thousand)
The 2 is worth 2,000 (two thousand)
The 5 is worth 500 (five hundred)
The 1 is worth 10 (ten)
The 7 is worth 7 (seven)

Exam Tip: When a question asks "what is the value of the digit 5 in the number 358,201?", they want the full value (50,000), not just the column name (ten thousands).

Place Value in Decimals

The place value system extends to the right of the decimal point using negative powers of 10.

Units	.	Tenths	Hundredths	Thousandths
1	.	0.1	0.01	0.001

In the number 7.346:

The 7 is worth 7 units
The 3 is worth 3 tenths (0.3)
The 4 is worth 4 hundredths (0.04)
The 6 is worth 6 thousandths (0.006)

Ordering Numbers

To order numbers (including decimals and negative numbers), compare digit by digit from the largest place value.

Worked Example: Ordering Decimals

Put these numbers in ascending order: 0.45, 0.405, 0.5, 0.045

Step 1: Write each number with the same number of decimal places by adding trailing zeros:

Number	Rewritten
0.45	0.450
0.405	0.405
0.5	0.500
0.045	0.045

Step 2: Compare as whole numbers: 045, 405, 450, 500

Answer: 0.045, 0.405, 0.45, 0.5

Ordering Negative Numbers

Remember that negative numbers further from zero are smaller. On a number line, numbers increase from left to right.

Order these from smallest to largest: -3, 5, -7, 2, -1

Answer: -7, -3, -1, 2, 5

Exam Tip: Draw a quick number line if you are unsure about ordering negative numbers. It takes a few seconds and prevents silly mistakes.

Rounding to Decimal Places (d.p.)

To round to a given number of decimal places:

Count that many digits after the decimal point.
Look at the next digit (the "deciding digit").
If it is 5 or more, round up. If it is less than 5, round down.

Worked Example

Round 3.4572 to 2 decimal places.

The second decimal place digit is 5.
The deciding digit (third decimal place) is 7, which is 5 or more.
Round up: 3.46

Common Rounding to Decimal Places

Original	1 d.p.	2 d.p.	3 d.p.
4.6738	4.7	4.67	4.674
12.3451	12.3	12.35	12.345
0.9961	1.0	1.00	0.996

Exam Tip: If a question says "give your answer to 2 d.p.", always write two digits after the decimal point, even if the last is zero. Writing 3.10 is correct for 2 d.p.; writing 3.1 would lose the mark.

Rounding to Significant Figures (s.f.)

Significant figures count from the first non-zero digit.

Rules for Identifying Significant Figures

The first significant figure is the first non-zero digit reading from left to right.
All subsequent digits (including zeros) are significant until you stop counting.
Leading zeros are never significant.

Worked Example

Round 0.004629 to 2 significant figures.

The first significant figure is 4 (ignore the leading zeros).
The second significant figure is 6.
The deciding digit is 2, which is less than 5 — round down.
Answer: 0.0046

More Examples

Original	1 s.f.	2 s.f.	3 s.f.
3,482	3,000	3,500	3,480
0.07253	0.07	0.073	0.0725
45,678	50,000	46,000	45,700

Truncation

Truncation means cutting off digits without rounding. You simply remove the unwanted digits.

Worked Example

Truncate 7.836 to 2 decimal places.

Keep the first two decimal digits: 7.83
Do not round up, even though the next digit is 6.
Answer: 7.83

The key difference:

Rounding 7.836 to 2 d.p. gives 7.84 (round up because 6 >= 5)
Truncating 7.836 to 2 d.p. gives 7.83 (just cut off)

Exam Tip: Truncation is a less common topic but has appeared in recent AQA exams. Read the question carefully — if it says "truncate", do NOT round.

Estimation

Estimation means rounding each number to one significant figure and then performing the calculation. This gives an approximate answer.

Worked Example

Estimate the value of (4.87 x 21.3) / 0.246

Step 1: Round each number to 1 significant figure:

4.87 rounds to 5
21.3 rounds to 20
0.246 rounds to 0.2

Step 2: Calculate:

(5 x 20) / 0.2
= 100 / 0.2
= 500

When to Use Estimation

Checking whether your calculator answer is reasonable.
Non-calculator paper questions that say "estimate".
Questions involving complex expressions where exact calculation is not required.

graph TD
    A[Estimation Steps] --> B[Round each value to 1 s.f.]
    B --> C[Perform the calculation]
    C --> D[State your approximate answer]
    D --> E[Check: is your answer reasonable?]

Using Inequality Symbols

You need to know and use these symbols confidently:

Symbol	Meaning
<	Less than
>	Greater than
<=	Less than or equal to
>=	Greater than or equal to

Example: Write the correct symbol between 0.67 and 2/3.

Since 2/3 = 0.6666... (recurring), and 0.67 > 0.6666..., the answer is: 0.67 > 2/3

Summary

Place value determines the value of each digit based on its position.
To order decimals, write them with the same number of decimal places and compare.
Rounding to d.p. — count decimal places, then check the deciding digit.
Rounding to s.f. — count from the first non-zero digit.
Truncation cuts off digits without rounding.
Estimation uses 1 s.f. rounding to give approximate answers.
Always show your rounded values when estimating — marks are given for the method.

Exam Tip: In estimation questions, always show your rounded values before calculating. The marks are for the rounding process, not just the final answer.

Extended Worked Examples

Worked Example A: Place Value in a Large Decimal

State the value of each underlined digit in $3{,}084.9\underline{6}2$ and $\underline{7}8{,}501.34$ .

In $3{,}084.962$ , the underlined $6$ sits in the hundredths column. Its value is:

$6 \times \frac{1}{100} = 0.06$

In $78{,}501.34$ , the underlined $7$ sits in the ten thousands column. Its value is:

$7 \times 10{,}000 = 70{,}000$

A very common trap is to answer "hundredths" or "ten thousands"; AQA awards the mark only for the numerical value ( $0.06$ or $70{,}000$ ).

Worked Example B: Rounding to a Given Number of Significant Figures

Round $0.0498721$ to (i) 1 s.f., (ii) 2 s.f., (iii) 3 s.f.

The first significant figure is the $4$ (the leading zeros do not count).

(i) 1 s.f. Look at the next digit, $9$ . Since $9 \geq 5$ we round up. $0.04 \to 0.05$ .

$0.0498721 \approx 0.05 \ \text{(1 s.f.)}$

(ii) 2 s.f. Keep $4$ and $9$ . The deciding digit is $8$ , so round up. But $9 + 1 = 10$ , which carries:

$0.049 + 0.001 = 0.050$

$0.0498721 \approx 0.050 \ \text{(2 s.f.)}$

Note the trailing zero must be kept; writing $0.05$ would be only 1 s.f.

(iii) 3 s.f. Keep $4$ , $9$ , $8$ . The deciding digit is $7$ , so round up: $498 \to 499$ .

$0.0498721 \approx 0.0499 \ \text{(3 s.f.)}$

Worked Example C: Estimation in a Calculator Check

Estimate the value of $\dfrac{38.7 \times 6.19}{0.512}$ and use your estimate to decide whether a calculator answer of $467.9$ is reasonable.

Round every value to 1 s.f.:

$\frac{38.7 \times 6.19}{0.512} \approx \frac{40 \times 6}{0.5} = \frac{240}{0.5} = 480$

The true answer should be near $480$ . The calculator answer $467.9$ is close to this estimate, so it is reasonable. If the calculator had given $46.79$ or $4679$ , the estimate would reveal a slipped decimal point.

Worked Example D: Truncation vs Rounding

A runner's lap time is recorded as $72.847$ seconds. Give this value (i) rounded to 2 d.p. and (ii) truncated to 2 d.p.

(i) Rounding to 2 d.p.: the deciding digit is $7 \geq 5$ , so round up:

$72.847 \to 72.85 \ \text{s}$

(ii) Truncating to 2 d.p.: simply chop off the digits after the second decimal place:

$72.847 \to 72.84 \ \text{s}$

This $0.01$ second difference matters in sprint finishes.

Worked Example E: Ordering Mixed Forms

Place $\tfrac{3}{8}$ , $0.38$ , $37\%$ , $\tfrac{7}{20}$ in ascending order.

Convert every value to a decimal:

Value	Working	Decimal
$\frac{3}{8}$	$3 \div 8$	$0.375$
$0.38$	already a decimal	$0.380$
$37\%$	$37 \div 100$	$0.370$
$\frac{7}{20}$	$7 \div 20$	$0.350$

Ascending order: $\tfrac{7}{20},\ 37\%,\ \tfrac{3}{8},\ 0.38$ .

Answering at different grade levels

Exam-style question: The number $5.0492$ is to be rounded. Find its value rounded to (a) 1 d.p., (b) 3 s.f., (c) the nearest integer.

Grades 3–4 answer. (a) $5.0492 \approx 5.0$ (1 d.p.). (b) The first three significant figures are $5$ , $0$ , $4$ . Deciding digit is $9$ , so round up: $5.05$ . (c) The integer part rounds to $5$ .
Grades 5–6 answer. The same as above, but I justify the zero in $5.0$ : it is kept as a placeholder because the question asks for 1 d.p., and the trailing zero shows the degree of accuracy. The digit $5$ is a rational integer so it is exact; it is neither recurring decimal nor surd.
Grades 7–9 answer. In 3 s.f. the value is $5.05$ because the deciding $4$ th s.f. is $9$ , forcing a carry from $4 \to 5$ . I note $5.0492$ lies strictly between the upper and lower bound of the rounded value $5.05$ ; specifically, any true value in the interval $5.045 \leq x < 5.055$ rounds to $5.05$ (3 s.f.). Writing $5.0492$ in standard form gives $5.0492 \times 10^{0}$ , which makes the leading $5$ the first significant figure explicit and confirms that removing the decimal information does not change the integer part.

Worked Example F: Comparing Rounded and Truncated Totals

A shop totals three items priced $\pounds 2.47$ , $\pounds 3.82$ and $\pounds 5.69$ . A customer estimates the bill by (i) rounding each to the nearest pound, and (ii) truncating each to the nearest pound. Compare both estimates with the true total.

True total: $2.47 + 3.82 + 5.69 = 11.98$ .

(i) Rounding to the nearest pound: $2 + 4 + 6 = 12$ . Very close.

(ii) Truncating to the nearest pound: $2 + 3 + 5 = 10$ . Noticeably low.

This illustrates why rounding usually gives a better estimate than truncation: truncation always underestimates positive values, while rounding can go either way and averages out.

Worked Example G: Inequality Between a Fraction and a Decimal

Insert the correct symbol ( $<$ , $=$ or $>$ ) between $\tfrac{3}{7}$ and $0.43$ .

Convert $\tfrac{3}{7}$ to a decimal:

$\frac{3}{7} = 0.428571\ldots$

This is a recurring decimal with the block $428571$ . Compare with $0.430000$ :

$0.428571\ldots < 0.430000$

So:

$\frac{3}{7} < 0.43$

AQA alignment: This content is aligned with AQA GCSE Mathematics (8300) specification — specifically Topic N1 Structure and calculation, N2 Fractions/decimals/percentages, N3 Measures and accuracy, and N15 Error intervals and bounds [H]. Assessed on Papers 1 (non-calculator), 2 and 3 (calculator).

Place Value, Ordering and Rounding

Place Value, Ordering and Rounding

Place Value in Integers

Place Value in Decimals

Ordering Numbers

Worked Example: Ordering Decimals

Ordering Negative Numbers

Rounding to Decimal Places (d.p.)

Worked Example

Common Rounding to Decimal Places

Rounding to Significant Figures (s.f.)

Rules for Identifying Significant Figures

Worked Example

More Examples

Truncation

Worked Example

Estimation

Worked Example

When to Use Estimation

Using Inequality Symbols

Summary

Extended Worked Examples

Worked Example A: Place Value in a Large Decimal

Worked Example B: Rounding to a Given Number of Significant Figures

Worked Example C: Estimation in a Calculator Check

Worked Example D: Truncation vs Rounding

Worked Example E: Ordering Mixed Forms

Answering at different grade levels

Worked Example F: Comparing Rounded and Truncated Totals

Worked Example G: Inequality Between a Fraction and a Decimal

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