"Is not equal to", for example 0.3 0.03.

N1, N2NumberFoundationGrades 1–3

Place Value & Ordering

Q: Is -0.5 bigger or smaller than -0.25?

Smaller. -0.5 is further from zero on the negative side, so -0.5 < -0.25.

Place value is the idea that a digit's value depends on its position: the 7 in 7,305 is worth 7,000, but in 0.07 it is worth seven hundredths. Everything in arithmetic — lining up columns, multiplying by 10, ordering decimals, rounding — rests on it.

30 min AQA GCSE Maths

The big picture

Our number system is built on tens. Each column is worth ten times the one to its right, so multiplying by 10 moves every digit one place to the left, and dividing by 10 moves every digit one place to the right. The decimal point never moves; the digits do.

Ordering is a place-value skill too. To compare 0.45 and 0.405, compare column by column from the left — not by counting digits. And on the negative side of zero, the order reverses: $-8$ is smaller than $-3$ .

What you'll learn

State the value of any digit in a whole number or decimal
Multiply and divide by 10, 100 and 1000, including decimals
Order integers, decimals and negative numbers
Use the symbols $=$ , $\ne$ , $<$ , $>$ , $\le$ , $\ge$

The value of a digit

Columns to the left of the decimal point are ones, tens, hundreds, thousands…; to the right are tenths, hundredths, thousandths…

In $52{,}836.47$ : the 5 is worth 50,000, the 8 is worth 800, the 4 is worth 4 tenths ( $0.4$ ) and the 7 is worth 7 hundredths ( $0.07$ ).

Example 1What is the value of the 6 in

3.068

1The columns after the point are tenths (0), hundredths (6), thousandths (8).

Answer

6 hundredths, or

0.06

Your turnSame method, new numbers — try it before moving on.

Write down the value of the 9 in $41.293$ as a decimal.

Your answer

Show worked solution

Solution

12 is in the tenths column, 9 in the hundredths, 3 in the thousandths.
2So the 9 is worth $0.09$ .

Answer: $0.09$

Multiplying and dividing by powers of 10

Multiplying by 10, 100 or 1000 moves every digit 1, 2 or 3 places to the left. Dividing moves them to the right. Fill empty places before the decimal point with zeros.

Multiplying by 0.1 is the same as dividing by 10, and dividing by 0.01 is the same as multiplying by 100.

Example 1Work out (a)

3.72 \times 100

(b)

45 \div 1000

(c)

0.6 \div 0.1

1(a) Digits move 2 places left:

372

2(b) Digits move 3 places right:

0.045

3(c) Dividing by 0.1 is multiplying by 10:

6

Answer

(a)

372

(b)

0.045

(c)

6

Your turnSame method, new numbers — try it before moving on.

Work out $8.4 \div 100$ .

Your answer

Show worked solution

Solution

1Dividing by 100 moves every digit 2 places right.
2 $8.4 \to 0.84 \to 0.084$

Answer: $0.084$

Example 2Given

23 \times 41 = 943

, work out

2.3 \times 4.1

2.3 = 23 \div 10

and

4.1 = 41 \div 10

2So the answer is

943 \div 100

Answer

9.43

Your turnSame method, new numbers — try it before moving on.

Given $37 \times 52 = 1924$ , work out $3.7 \times 520$ .

Your answer

Show hint

$3.7$ is 10 times smaller than 37, and 520 is 10 times bigger than 52.

Show worked solution

Solution

1 $3.7 = 37 \div 10$ and $520 = 52 \times 10$ .
2The two changes cancel out: $1924$ .

Answer: $1924$

Tip — Estimate to check: $2.3 \times 4.1$ is roughly $2 \times 4 = 8$ , so $9.43$ is sensible and $94.3$ is not.

Ordering numbers

For decimals, line up the decimal points and compare digits column by column from the left. Adding zeros to the end can help: $0.5 = 0.500$ , which is larger than $0.405$ .

For negative numbers, picture a number line: the further left, the smaller. $-12 < -5 < 0 < 3$ .

Symbols: $<$ less than, $>$ greater than, $\le$ less than or equal to, $\ge$ greater than or equal to, $\ne$ not equal to.

-8

is to the left of

-3

, so

-8 < -3

— even though 8 is bigger than 3.

Example 1Write in order, smallest first:

0.7, \ 0.07, \ 0.707, \ 0.77, \ 0.077

1Write with three decimal places:

0.700, 0.070, 0.707, 0.770, 0.077

2Compare:

70 < 77 < 700 < 707 < 770

(thousandths).

Answer

0.07, \ 0.077, \ 0.7, \ 0.707, \ 0.77

Your turnSame method, new numbers — try it before moving on.

Which is the smallest: $-2.5$ , $-2.05$ , $-0.25$ or $-25$ ?

Your answer

Show worked solution

Solution

1On a number line, $-25$ is furthest to the left.

Answer: $-25$

Think like an examiner

In "use the fact that…" questions, you must use the given calculation; an answer found by long multiplication alone may not score.
Show a common number of decimal places when ordering — it makes your reasoning visible.
Answers like "7 hundredths" and " $0.07$ " are both accepted for the value of a digit; "7" on its own is not.

Place value

\times 10^n: \text{ digits move } n \text{ places left}

Dividing moves them right.

\times 0.1 = \div 10

Multiplying by a decimal can make a number smaller.

-8 < -3

Negatives: further from zero means smaller.

Watch out for these

0.45 > 0.5

because 45 is bigger than 5.

Compare tenths first: 4 tenths < 5 tenths, so

0.45 < 0.5

3.7 \times 10 = 3.70

The digits move, not the zero:

37

-8 > -3

because 8 > 3.

On a number line

-8

is further left:

-8 < -3

6 \div 0.1 = 0.6

Dividing by 0.1 multiplies by 10:

60

Stretch yourself

Using each of the digits 3, 0, 7 and 5 exactly once, and one decimal point, make the number closest to 50.

Hint — Try numbers just below and just above 50 in the form __.__ .

Reveal solution

SolutionTo be close to 50 the number should start with 5 then 0, giving

50.37

50.73

; the closer is

50.37

, just

0.37

away. Anything below 50 must start with 3 (the largest is

37.50

), which is far further away. So the answer is

50.37

\blacksquare

Questions students ask

Does the decimal point move when I multiply by 10?

It is clearer to think of the digits moving left while the point stays still. Either way you get the same answer.

-0.5

bigger or smaller than

-0.25

Smaller.

-0.5

is further from zero on the negative side, so

-0.5 < -0.25

What does

\ne

mean?

"Is not equal to", for example

0.3 \ne 0.03

Key takeaways

A digit's value depends on its column.
Multiply by $10^n$ : digits move $n$ places left; divide: right.
Compare decimals column by column from the left, padding with zeros.
For negatives, the further from zero, the smaller.
Use given facts and estimation to check decimal calculations.

Exam-style questions

Practise it the way it is examined

2 original questions · 6 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 13 marks

(a)Write down the value of the 4 in $6.842$ .[1]
Show mark scheme (a)
- B14 hundredths or $0.04$
Answer: $0.04$
(b)Work out $0.36 \times 1000$ .[1]
Show mark scheme (b)
- B1 $360$
Answer: $360$
(c)Write these numbers in order, starting with the smallest: $\quad -4, \ 2, \ -9, \ 0, \ -1$ [1]
Show mark scheme (c)
- B1 $-9, -4, -1, 0, 2$

Question 2Grade 33 marks

You are told that $148 \times 26 = 3848$ .

(a)Work out $14.8 \times 2.6$ .[1]
Show mark scheme (a)
- B1 $38.48$
Answer: $38.48$
(b)Work out $3848 \div 2.6$ .[2]
Show mark scheme (b)
- M1Recognises $3848 \div 26 = 148$
- A1 $1480$
Answer: $1480$

Examiner note

Dividing by a smaller number gives a bigger answer: $2.6$ is ten times smaller than 26, so the answer is ten times bigger than 148.

Unlimited practice

Skill drill

A new question every time, marked instantly, with a full worked solution. Aim for a streak of five, then move up a level.

0 / 0 first-time correct Streak 0Best at this level: 0

Generating a question…

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

What is the value of the 3 in $5.237$ ?
- $0.03$
- $3$
- $0.3$
- $0.003$
Show answer and explanation
Answer: $0.03$
3 is in the hundredths column.
Work out $4.5 \times 100$ .
- $450$
- $45$
- $4.500$
- $4500$
Show answer and explanation
Answer: $450$
Digits move 2 places left.
Work out $72 \div 1000$ .
- $0.072$
- $0.72$
- $7.2$
- $72000$
Show answer and explanation
Answer: $0.072$
Digits move 3 places right.
Which is the largest?
- $0.6$
- $0.59$
- $0.599$
- $0.066$
Show answer and explanation
Answer: $0.6$
$0.600$ beats $0.599$ in the thousandths.
Which statement is true?
- $-7 < -2$
- $-7 > -2$
- $-2 < -7$
- $0 < -2$
Show answer and explanation
Answer: $-7 < -2$
$-7$ is further left on a number line.
Work out $5 \div 0.01$ .
- $500$
- $0.05$
- $50$
- $0.5$
Show answer and explanation
Answer: $500$
Dividing by 0.01 multiplies by 100.
Given $19 \times 34 = 646$ , what is $1.9 \times 0.34$ ?
- $0.646$
- $6.46$
- $64.6$
- $0.0646$
Show answer and explanation
Answer: $0.646$
Three decimal places in total.
Put in order, smallest first: $-0.3, -0.33, -0.03$
- $-0.33, -0.3, -0.03$
- $-0.03, -0.3, -0.33$
- $-0.3, -0.33, -0.03$
- $-0.33, -0.03, -0.3$
Show answer and explanation
Answer: $-0.33, -0.3, -0.03$
The most negative is furthest from zero.

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