Edexcel IGCSE

Decimals

A decimal writes whole units and parts of a unit in one string of digits, with a point marking where the whole units stop. It is section 1.3 of the Edexcel International GCSE Mathematics A specification, and every question on it turns on a single fact: each column is worth ten times the column to its right, and that rule does not stop at the point. Get the columns right and ordering, money, the power-of-ten shift and the conversion into a fraction all fall out of the same reading.

Edexcel IGCSE 4MA120 approved practice questions

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Before you start

What you need to be able to do already

If any of these is shaky, fix it first — every worked example below leans on all three.

Read the place value of a digit in a whole number, and add and subtract in columns
What is the digit $7$ worth in $3724$ ? It is $700$ — and $7$ , $70$ and $7000$ are all offered beside it as wrong answers on that exact question. If naming the column was not instant, fix integers first: every method step below is that same reading carried past the decimal point.
Cancel a fraction to its simplest form using the highest common factorFractions
Write $\frac{32}{100}$ in its simplest form. If $\frac{8}{25}$ did not come quickly, fix fractions first: every decimal-to-fraction question below ends in that cancel, and $\frac{32}{100}$ left uncancelled is stored in Learnly's bank as the wrong answer that question is built to catch.
Write a power of ten in index form and say what it is worthPowers and roots
What is $1 0^{3}$ as an ordinary number? It is $1000$ . The recurring-decimal method below multiplies by $1 0^{2}$ or $1 0^{3}$ depending on how long the repeating block is, and picking the wrong power turns a correct method into a wrong fraction.

The idea

What is decimals?

A decimal is a number written in place-value columns that simply carry on past a point. To the left of the point are units, tens and hundreds; to the right are tenths, hundredths and thousandths. So $0.75$ is $7$ tenths and $5$ hundredths, which is $\frac{75}{100}$ , which is $\frac{3}{4}$ . This is not a separate number system: a terminating decimal is a fraction whose denominator is a power of ten, written without the line. Read the columns and most of the work is already done.

The notation, and how to read it

Every worked solution below is written in these five symbols.

SymbolReads asMeans
$0.75$ nought point seven fiveThe point separates whole units from parts of a unit. Read the digits after it one at a time — saying 'nought point seventy-five' hides the columns that the whole topic depends on.
$0.001$ one thousandthThe third column after the point. Going right, each column is worth a tenth of the one before it: tenths, hundredths, thousandths.
$0. \dot{6}$ nought point six recurringOne dot over one digit: that digit repeats for ever. $0. \dot{6} = 0.6666 \dots$ , so it never terminates and has no last column to read.
$0. \dot{5} \dot{7}$ nought point five seven recurringDots on the first and last digit of a block: the whole block repeats. $0. \dot{5} \dot{7} = 0.575757 \dots$ , and the block length is what sets the power of ten you multiply by.
$\times 1 0^{3}$ times ten to the power threeMultiply by $1000$ . Every digit moves three columns to the left and the point stays put — $0.042 \times 1 0^{3} = 42$ .

The one idea underneath all of it

Every column is worth ten times the column to its right, and the decimal point is only a marker of where the units end — it is not a break in the rule. That one idea is the whole topic. It is why $0.3$ is bigger than $0.29$ (compare tenths first, and $3$ beats $2$ , so the extra digit never gets a vote); why multiplying by $10$ moves every digit one column left; and why a decimal with three digits after the point is a fraction over $1000$ . Move a digit into a different column and you have changed the number by a factor of ten.

The smallest possible example

$0.32$ is to be written as a fraction in its simplest form.

$0.32 = \frac{32}{100} = \frac{8}{25}$

There are two digits after the point, so the last one sits in the hundredths column and the whole number is $32$ hundredths. That gives $\frac{32}{100}$ with no working at all — just the column name. Then cancel: $32$ and $100$ share a factor of $4$ , so it becomes $\frac{8}{25}$ . Stopping at $\frac{32}{100}$ is the wrong answer recorded against this exact question in Learnly's bank, and $\frac{16}{50}$ — one cancel out of two — sits next to it. Every conversion on this page is these same two moves.

Key terms

The words a question will use

Exam wording is precise. These five carry the meaning a question depends on.

TermWhat it meansNotation
Decimal place
A column after the point, counted outwards from the point. $0.375$ has three decimal places. 'Give your answer to $2$ decimal places' means exactly two digits after the point — no more, and no fewer, so a trailing zero has to be written in. $2 d.p.$
Significant figures
Counted from the first non-zero digit, not from the point. In $0.00427$ the first significant figure is the $4$ , so $2$ significant figures gives $0.0043$ — which is four decimal places. Decimal places and significant figures are different instructions and they rarely land on the same answer. $0.0043$
Terminating decimal
A decimal that stops. Because it stops, its last digit names a power of ten, so it is always a fraction and you can read that fraction straight off the columns before cancelling it. $0.375 = \frac{3}{8}$
Recurring decimal
A decimal that never stops, because one digit or one block of digits repeats for ever. It is still a fraction, but there is no last column to read, so it has to be worked out with the multiply-and-subtract method rather than by place value. $0. \dot{5} \dot{7}$
Trailing zero
A zero at the right-hand end after the point. It changes nothing about the value, which is exactly why it is useful: padding numbers to the same length makes values comparable column by column. For currency, writing both decimal places — £540.80 rather than £540.8 — communicates the amount in standard money notation. $3.4 = 3.40$

The method

How to work out a decimals

Five steps, in this order. The worked examples below point back to these numbers.

Read the place value: name the column every digit after the point sits in.Tenths, hundredths, thousandths, working left to right. This step decides an ordering question before any comparing starts: $0.3$ is three tenths and $0.045$ is no tenths at all, so $0.045$ is smaller however many digits it carries. The habit of treating the longer number as the bigger one is what puts $0.303$ before $0.3$ , and that ordering is offered as a wrong answer on that question in the bank.
Pad every number with trailing zeros to the same number of decimal places, then line the points up in a column. $3.4$ becomes $3.40$ ; $0.3$ becomes $0.300$ . Now the numbers are the same shape, and you compare or add them column by column from the left, stopping at the first column where they differ. For money the same alignment does the arithmetic: keep the point in one column all the way down, so $50.00 - 37.15 = 12.85$ . Every worked money question on this subtopic is that one subtraction laid out in columns.
Multiply or divide by moving the digits one column for each power of ten. $\times 10$ moves every digit one column left, $\times 100$ moves them two, $\div 10$ moves them one to the right. For a multiplication that is not by a power of ten, ignore the points, multiply as whole numbers, then put back as many decimal places as the two numbers had between them: $60 \times 0.85$ is $60 \times 85 = 5100$ with two places restored, giving $51$ . For a division by a decimal, scale both numbers until the divisor is whole: $54 \div 0.6$ becomes $540 \div 6 = 90$ . The same shift drives the recurring-decimal method, where you multiply by $1 0^{n}$ and $n$ is the length of the repeating block.
Convert into the form the question named, then cancel.Decimal to fraction: the last decimal place names the denominator — two places is over $100$ , three places is over $1000$ — and then you simplify. $0.25 = \frac{25}{100} = \frac{1}{4}$ , and $0.375 = \frac{375}{1000} = \frac{3}{8}$ . Decimal to percentage is step 3 again: shift two columns. A recurring decimal needs the algebra instead — let $x$ be the decimal, multiply by $1 0^{n}$ , subtract the original so the endless tail cancels, and divide: $100 x - x = 57$ gives $x = \frac{57}{99} = \frac{19}{33}$ . The final cancel is not tidying up; on these questions it is the accuracy mark.
Round and format to the accuracy asked for, then check the size is sensible.'To $2$ decimal places' means exactly two digits after the point, so a money answer of $51.8$ is written £51.80. Carry full accuracy through the working and round once, at the very end. Then sanity-check against step 1: multiplying by a number below $1$ must make the answer smaller, dividing by a number below $1$ must make it bigger, and change from a £20 note cannot be more than £20 — on one of these questions £11.75, which is the total spent, is offered as the change for exactly that reason.

Use it when

The question is written in decimals, or asks you to move between a decimal and a fraction or a percentage: order a list of decimals, work out change from a note, write $0.32$ as a fraction, turn $0. \dot{5} \dot{7}$ into a fraction, or apply a decimal multiplier such as $\times 0.85$ or $\div 0.6$ . In every one of those the route is the same five steps — name the columns, line them up, shift them, convert, round.

Do not use it when

Do not compare decimals by counting digits. $3.044$ has more digits than $3.4$ but is smaller, and the ordering $3.044, 3.04, 3.4, 3.44$ — sorted by length rather than by value — is a wrong answer stored in Learnly's bank against that question with the note 'ordering by number of decimal places instead of value'. Do not read the digits after the point as a whole number either: $0.375$ is $375$ thousandths, so it is $\frac{375}{1000}$ and then $\frac{3}{8}$ , not $\frac{375}{100}$ and not $\frac{37}{100}$ , both of which are offered beside the correct answer. And do not reach for the place-value reading on a recurring decimal — $0. \dot{6}$ has no last column, so it needs the $\times 1 0^{n}$ subtraction instead.

Worked examples

6 IGCSE Maths practice questions on decimals, fully worked

One per objective, so the range is covered rather than the same question repeated. Every step is written out, and the answer is shown — this is study material, not a test.

Example 1

Jamie buys a bottle of water and a sandwich. The bottle of water costs £1.85. The total cost is £4.70. Work out the cost of the sandwich. [2 marks]

Use decimal notation

Worked solution

Step 1: Write down the total cost: £4.70.
Step 2: Write down the cost of the water: £1.85.
Step 3: Subtract the water cost from the total: £4.70 − £1.85.
Step 4: Work out the subtraction: 4.70 − $1.85 = 2.85$ .
Step 5: So the sandwich costs £2.85.

Answer£2.85

Example 2

What is the value of the digit 7 in the number 3 724? [1 mark]

Understand place value

Worked solution

Step 1: The number is 3 724. We look at the place value of each digit from right to left: units, tens, hundreds, thousands.
Step 2: The digit 7 is in the third position from the right, which is the hundreds place.
Step 3: 7 in the hundreds place means $7 \times 100 = 700$ .

Answer700

Example 3

Which of the following lists the numbers 3.4, 3.04, 3.44, 3.044 in order from smallest to largest? [1 mark]

Order decimals

Worked solution

Step 1: Write the numbers in a column, aligning the decimal points.
Step 2: For 3.4, add a zero: 3.40. Now all have two decimal places except 3.044 which has three.
Step 3: Look at the digits after the decimal point one by one for each number.
Step 4: For the first digit after the decimal: 3.4 is 4, 3.04 is 0, 3.44 is 4, 3.044 is 0.
Step 5: Smallest first digit is 0, so 3.04 and 3.044 come before 3.4 and 3.44.
Step 6: Compare 3.04 and 3.044: the second digit after the decimal is 4 for 3.04 and 4 for 3.044. The third digit is 0 for 3.04 and 4 for 3.044. So 3.04 is smaller.
Step 7: Compare 3.4 and 3.44: 3.4 has one decimal digit (4), 3.44 has two (44). 3.4 is smaller.
Step 8: The full order is: 3.04, 3.044, 3.4, 3.44.

Answer3.04, 3.044, 3.4, 3.44

Show 3 more worked examples

Example 4

Write 0.25 as a fraction in its simplest form. [1 mark]

Convert a decimal to a fraction or a percentage (terminating decimals only at Foundation)

Worked solution

Step 1: The decimal 0.25 means 25 hundredths, so we write it as $\frac{25}{100}$ .
Step 2: Find the highest common factor of 25 and 100, which is 25.
Step 3: Divide both numerator and denominator by 25: $25 \div 25 = 1$ and $100 \div 25 = 4$ .
Step 4: The fraction simplifies to $\frac{1}{4}$ .

Answer

\frac{1}{4}

Example 5

Which of the following is the correct fraction representation of the terminating decimal 0.375? [1 mark]

Recognise that a terminating decimal is a fraction

Worked solution

Step 1: Write 0.375 as a fraction with denominator 1000, since there are three decimal places: $\frac{375}{1000}$ .
Step 2: Simplify the fraction by dividing numerator and denominator by 125, because 125 is the highest common factor of 375 and 1000.
Step 3: $375 \div 125 = 3$ and $1000 \div 125 = 8$ , so the simplified fraction is $\frac{3}{8}$ .

Answer

\frac{3}{8}

Example 6

Show that $0.1 \dot{4} \dot{2}$ (meaning 0.142142142...) can be written as $\frac{142}{999}$ . Deduce the fraction for $0.0 \dot{4} \dot{2}$ (meaning 0.042042042...). Write this fraction in its simplest form. Give your answer as a fraction in its simplest form, e.g. $\frac{3}{10}$ (not a decimal). [3 marks]

Convert recurring decimals into fractions

Worked solution

Step 1: Let $x = 0. \dot{0} 4 \dot{2} = 0.042042042...$ .
Step 2: Multiply by 1000: $1000 x = 42.042042042...$ .
Step 3: Subtract the original: $1000 x - x = 42.042... - 0.042... = 42$ .
Step 4: So $999 x = 42$ . Divide: $x = \frac{42}{999}$ .
Step 5: Simplify by dividing numerator and denominator by 3: $\frac{42}{999} = \frac{14}{333}$ .

Answer

\frac{14}{333}

Exam technique

How the marks are actually awarded

Method marks are given for working a marker can follow. This is what to put on the page.

Where the marks are: Questions on this section run from $1$ to $4$ marks. The one-mark questions are readings and conversions — the value of a digit, an ordering, $0.25$ as a fraction — and they carry no method mark, so the answer simply has to be right. Two-mark questions give a method mark for the set-up, which means the column subtraction written out with the points aligned, or the unsimplified fraction $\frac{32}{100}$ on the page before you cancel it; put that line down and the mark is banked even if the cancelling afterwards goes wrong. The 'show that' recurring-decimal questions are the exception, because they print the answer: every mark lives in the working — $x = 0.575757 \dots$ , then $100 x = 57.575757 \dots$ , then the subtraction, then the cancel — and a bare $\frac{19}{33}$ scores nothing.
Command words to expect: Work outCalculateWrite down the value ofWrite these numbers in order of sizeShow thatHence, or otherwiseGive your answer as a fraction in its simplest formGive your answer in pounds to 2 decimal placesYou must show all your working
Accuracy and rounding: 'To $2$ decimal places' means exactly two digits after the point, trailing zero included: a money answer of $51.8$ is conventionally written £51.80. Decimal places are counted from the point; significant figures are counted from the first non-zero digit, so $0.00427$ to $2$ significant figures is $0.0043$ — four decimal places, not two — and the two instructions are not interchangeable. When the question gives no accuracy instruction, prefer an exact fraction for a recurring value; if a decimal approximation is required, follow the paper's stated accuracy convention. Keep full accuracy inside the working and round once at the end. A fraction asked for 'in its simplest form' must be fully cancelled: $\frac{8}{25}$ , not $\frac{32}{100}$ and not $\frac{16}{50}$ ; $\frac{19}{33}$ , not $\frac{57}{99}$ ; $\frac{14}{333}$ , not $\frac{42}{999}$ . Each of those uncancelled forms is a wrong answer offered on the question it belongs to.
Calculator: A calculator is allowed in every paper of this qualification, on both Foundation and Higher tiers, and most of these questions are one keystroke sequence on it. That is precisely the trap. It will hand you $0.375$ and it will hand you $12.85$ , but it will not write the column subtraction that carries the method mark, it will not put the second decimal place back onto a money answer, and its fraction key will not produce the line-by-line working that a 'show that' recurring-decimal question is paying for. Use it to check the arithmetic and to convert, and write the set-up down anyway.

Check yourself

You should now be able to:

Read a decimal in place-value columns, naming tenths, hundredths and thousandths, and say what a digit after the point is worth.
Put a list of decimals in order of size by padding them to the same number of decimal places and comparing column by column.
Explain why $0.3$ is larger than $0.29$ , and why having more digits does not make a decimal bigger.
Add and subtract decimals in columns with the points aligned, and lay a money calculation out to two decimal places.
Multiply and divide by $10$ , $100$ and $1000$ by moving the digits one column for each power of ten.
Multiply and divide by a decimal, turning $60 \times 0.85$ into a whole-number multiplication and $54 \div 0.6$ into $540 \div 6$ .
Convert a terminating decimal into a fraction by reading the last decimal place as the denominator, then cancel to the simplest form.
Convert a decimal to a percentage and a percentage back to a decimal by shifting two columns.
Recognise that every terminating decimal is a fraction, and tell a terminating decimal apart from a recurring one.
Convert a recurring decimal into a fraction with the $\times 1 0^{n}$ subtraction, choosing the power from the length of the repeating block.
Round a decimal to a given number of decimal places or significant figures, and write a money answer with both decimal places present.

Specification coverage

The 6 things 4MA1 asks you to do

The assessable objectives for numbers and the number system — decimals, in the order the specification lists them.

Use decimal notation
Understand place value
Order decimals
Convert a decimal to a fraction or a percentage (terminating decimals only at Foundation)
Recognise that a terminating decimal is a fraction
Convert recurring decimals into fractions

See who is stuck on decimals before you teach it.

Create a class, share the 8-character code, and set decimals as practice. Your students work through all 20 approved practice questions — not just the 6 on this page — and you see which objectives the class has not got yet. Free for individual students and teachers.

How Learnly works for teachers

Set up your class — free Practise on your own

What you need to be able to do already

Read the place value of a digit in a whole number, and add and subtract in columns

Cancel a fraction to its simplest form using the highest common factorFractions

Write a power of ten in index form and say what it is worthPowers and roots

What is decimals?

The notation, and how to read it

The words a question will use

Decimal place

Significant figures

Terminating decimal

Recurring decimal

Trailing zero

How to work out a decimals

6 IGCSE Maths practice questions on decimals, fully worked

Jamie buys a bottle of water and a sandwich. The bottle of water costs £1.85. The total cost is £4.70. Work out the cost of the sandwich. [2 marks]

What is the value of the digit 7 in the number 3 724? [1 mark]

Which of the following lists the numbers 3.4, 3.04, 3.44, 3.044 in order from smallest to largest? [1 mark]

Write 0.25 as a fraction in its simplest form. [1 mark]

Which of the following is the correct fraction representation of the terminating decimal 0.375? [1 mark]

Show that 0.14˙2˙ (meaning 0.142142142...) can be written as 999142​. Deduce the fraction for 0.04˙2˙ (meaning 0.042042042...). Write this fraction in its simplest form. Give your answer as a fraction in its simplest form, e.g. 103​ (not a decimal). [3 marks]

How the marks are actually awarded

You should now be able to:

The 6 things 4MA1 asks you to do

See who is stuck on decimals before you teach it.