Decimals Explained Step by Step

🎯 By the end of this lesson...

You should be able to:

Explain what a decimal represents.
Identify decimal place values.
Write decimals in expanded form.
Compare and order decimals.
Round decimals to a required place value.
Add and subtract decimals.
Multiply and divide decimals.
Convert fractions to decimals.
Recognise terminating and recurring decimals.
Solve real-life problems involving decimals.

🤔 Think About This...

🛒

Which shop is cheaper?

Two shops sell the same bottle of cooking oil.

Shop A charges US$3.85.
Shop B charges US$3.79.

Which shop offers the lower price?

We compare decimals every day when dealing with prices, distances, measurements, time and money.

Understanding decimal place value helps us compare numbers accurately.

1️⃣ What Is a Decimal?

A decimal is a way of writing numbers that include parts of a whole.

A decimal uses a decimal point (.) to separate the whole-number part from the fractional part.

3/10 = 0.3

75/100 = 0.75

In the number 24.37, 24 is the whole-number part, while the digits after the decimal point represent parts of one whole.

Whole-number part

24

.

Decimal part

37

In the number 24.37, 24 is the whole-number part and .37 represents thirty-seven hundredths.

2️⃣ Decimal Place Value

Each digit has a value determined by its position.

Thousands	Hundreds	Tens	Ones	Tenths	Hundredths	Thousandths
2	5	7	3	4	8	9

The number represented is: 2573.489

2573.489

The first four digits form the whole-number part. The remaining digits form the decimal part.

2 represents 2000.
5 represents 500.
7 represents 70.
3 represents 3.
4 represents 4 tenths.
8 represents 8 hundredths.
9 represents 9 thousandths.

Remember: The decimal point is not a digit. It simply separates the whole-number part from the decimal part.

🔍 Reading Decimal Place Values

Tenths

0.1 = 1/10

Hundredths

0.01 = 1/100

Thousandths

0.001 = 1/1000

Example 1: What is the value of 6 in 14.263?

The digit 6 is in the hundredths position.

Value of 6 = 6/100 = 0.06

The value of the digit 6 is 0.06.

3️⃣ Expanded Form

Expanded form shows the value of every digit separately.

Example 2: Write 352.407 in expanded form

352.407

= 300 + 50 + 2 + 0.4 + 0.007

The zero in the hundredths position holds the place between tenths and thousandths.

💡 Did You Know?

The decimal system is called a base-10 system because it uses ten digits:

0, 1, 2, 3, 4, 5, 6, 7, 8 and 9

Moving one place to the left makes a digit's value ten times larger.

Moving one place to the right makes its value ten times smaller.

3 → 0.3 → 0.03 → 0.003

4️⃣ Comparing Decimals

Compare digits from left to right, beginning with the greatest place value.

Example 3: Compare 0.7 and 0.65

Add a zero so both numbers have the same number of decimal places.

0.70

0.65

Both numbers have 0 ones.

Compare the tenths:

7 tenths > 6 tenths

0.7 > 0.65

Do not compare decimals by counting digits.

Although 0.65 has more digits, 0.7 is greater.

🧠 StudyNest Method: Comparing Decimals

Step 1: Line up the decimal points.

Step 2: Add zeros where necessary so the numbers have the same number of decimal places.

Step 3: Compare the whole-number parts.

Step 4: If they are equal, compare tenths.

Step 5: If the tenths are equal, compare hundredths, then thousandths, and so on.

5️⃣ Ordering Decimals

Example 4: Arrange in ascending order

2.75, 2.7, 2.705, 2.68

Rewrite each number using three decimal places.

2.750

2.700

2.705

2.680

Now compare from left to right.

Ascending order:

2.68, 2.7, 2.705, 2.75

Ascending order means smallest to largest. Descending order means largest to smallest.

6️⃣ Rounding Decimals

Rounding gives an approximate value with fewer decimal places.

Next digit is 0, 1, 2, 3 or 4

Keep the rounding digit unchanged.

Next digit is 5, 6, 7, 8 or 9

Increase the rounding digit by 1.

Example 5: Round 7.364 to one decimal place

The tenths digit is 3.

Look at the next digit, which is 6.

Since 6 is at least 5, increase 3 to 4.

7.364 rounded to one decimal place is 7.4.

Example 6: Round 5.7826 to two decimal places

The hundredths digit is 8.

Look at the next digit, which is 2.

Since 2 is less than 5, keep 8 unchanged.

5.7826 rounded to two decimal places is 5.78.

🌍 Real-Life Example: Race Time

Example 7

A runner completes a race in 12.67 seconds.

The results board displays times to one decimal place.

The tenths digit is 6 and the next digit is 7.

Round the 6 upwards.

The displayed time is 12.7 seconds.

7️⃣ Adding Decimals

Always line up the decimal points before adding.

Example 8: Calculate 12.45 + 3.8

Write 3.8 as 3.80.

12.45

+ 3.80

16.25

12.45 + 3.8 = 16.25

Adding zeros at the end of a decimal does not change its value.

3.8 = 3.80

8️⃣ Subtracting Decimals

Example 9: Calculate 20 − 7.36

Write 20 as 20.00.

20.00

− 7.36

12.64

20 − 7.36 = 12.64

If decimal points are not aligned, digits with different place values may be incorrectly added or subtracted.

🧠 StudyNest Method: Adding and Subtracting Decimals

Step 1: Write the numbers vertically.

Step 2: Line up the decimal points.

Step 3: Add zeros where needed.

Step 4: Add or subtract as you would with whole numbers.

Step 5: Place the decimal point directly below the others.

9️⃣ Multiplying Decimals

Multiply the numbers as if they were whole numbers first.

Then count the total number of decimal places in the original factors.

Example 10: Calculate 2.4 × 0.3

Ignore the decimal points temporarily.

24 × 3 = 72

2.4 has one decimal place.

0.3 has one decimal place.

Altogether, the answer needs two decimal places.

2.4 × 0.3 = 0.72

Example 11: Calculate 1.25 × 0.4

125 × 4 = 500

There are three decimal places altogether.

500 → 0.500

1.25 × 0.4 = 0.5

🧠 StudyNest Method: Multiplying Decimals

Step 1: Ignore the decimal points.

Step 2: Multiply the numbers as whole numbers.

Step 3: Count all decimal places in the original factors.

Step 4: Place the decimal point in the answer so it has the same total number of decimal places.

Step 5: Check whether the size of the answer is reasonable.

🔟 Dividing Decimals

Dividing by a Whole Number

Example 12: Calculate 8.4 ÷ 2

84 tenths ÷ 2 = 42 tenths

8.4 ÷ 2 = 4.2

Dividing by a Decimal

Example 13: Calculate 8.4 ÷ 0.2

Make the divisor a whole number by multiplying both numbers by 10.

8.4 ÷ 0.2

= 84 ÷ 2

= 42

8.4 ÷ 0.2 = 42

Multiplying both the dividend and divisor by the same power of 10 does not change the value of the division.

🧠 StudyNest Method: Dividing by a Decimal

Step 1: Look at the divisor.

Step 2: Move its decimal point until it becomes a whole number.

Step 3: Move the decimal point in the dividend by the same number of places.

Step 4: Divide normally.

Step 5: Check whether the answer is reasonable.

1️⃣1️⃣ Converting Fractions to Decimals

A fraction means division.

numerator ÷ denominator

1/2

→

1 ÷ 2 = 0.5

3/4

→

3 ÷ 4 = 0.75

7/10

→

7 ÷ 10 = 0.7

13/20

→

13 ÷ 20 = 0.65

1️⃣2️⃣ Terminating and Recurring Decimals

Terminating Decimal

A decimal that ends.

0.25

0.875

Terminating decimals are rational.

Recurring Decimal

A decimal that continues but repeats a pattern.

0.333…

0.727272…

Recurring decimals are also rational.

A decimal is rational if it ends or repeats.

🌍 Real-Life Example: Shopping

Example 14

A customer buys:

bread for US$1.35;
milk for US$2.80;
fruit for US$3.45.

Find the total cost.

1.35

2.80

+ 3.45

7.60

The total cost is US$7.60.

🌍 Real-Life Example: Fuel Cost

Example 15

Fuel costs US$1.62 per litre.

A driver buys 5.5 litres.

1.62 × 5.5

Multiply as whole numbers:

162 × 55 = 8910

There are three decimal places altogether.

8.910

The fuel costs US$8.91.

🧠 The StudyNest Decimal Strategy

Step 1: Identify the operation or skill required.

Step 2: Check the place values carefully.

Step 3: For addition or subtraction, line up decimal points.

Step 4: For multiplication, calculate first and then replace the decimal point.

Step 5: For division by a decimal, make the divisor a whole number first.

Step 6: Round only when the question asks you to.

Step 7: Estimate whether the answer is sensible.

🎯 Exam Tips

Line up decimal points before adding or subtracting.

Add zeros where necessary to make place values easier to compare.

When multiplying decimals, count the decimal places in both original numbers carefully.

Do not round during intermediate steps unless the question instructs you to do so.

Include units such as dollars, metres, kilograms or seconds in your final answer.

⚠ Common Mistakes

Mistake 1:

Thinking 0.65 is greater than 0.7 because 65 is greater than 7.

Write 0.7 as 0.70 before comparing.

Mistake 2:

Failing to line up decimal points when adding or subtracting.

Mistake 3:

Miscounting the decimal places after multiplication.

Mistake 4:

Moving the decimal point in only one number when dividing by a decimal.

Move it the same number of places in both numbers.

Mistake 5:

Assuming that adding a zero changes a decimal's value.

0.6 = 0.60 = 0.600

Mistake 6:

Rounding too early and reducing the accuracy of the final answer.

🎮 Your Turn!

State the value of the digit 7 in 23.748.
Write 406.305 in expanded form.
Which is greater: 0.8 or 0.79?
Arrange in ascending order: 3.6, 3.06, 3.66, 3.606
Round 8.47 to one decimal place.
Round 12.836 to two decimal places.
Calculate 7.45 + 3.8.
Calculate 20 − 6.78.
Calculate 3.4 × 0.6.
Calculate 1.25 × 2.4.
Calculate 9.6 ÷ 3.
Calculate 7.2 ÷ 0.3.
Convert 7/8 to a decimal.
State whether 0.454545… is terminating or recurring.
A notebook costs US$2.75 and a pen costs US$1.40. Find the total cost of 3 notebooks and 2 pens.

✅ Show Answers

7 tenths = 0.7
400 + 6 + 0.3 + 0.005
0.8 = 0.80, so 0.8 is greater.
3.06, 3.6, 3.606, 3.66
8.5
12.84
11.25
13.22
2.04
3.00 = 3
3.2
24
7 ÷ 8 = 0.875
Recurring
3 × 2.75 + 2 × 1.40

= 8.25 + 2.80

= US$11.05

🧠 Remember This

Decimal place value begins with tenths after the decimal point.

Compare decimals from left to right.

Line up decimal points when adding or subtracting.

Count decimal places carefully when multiplying.

Make the divisor a whole number before dividing by a decimal.

A terminating or recurring decimal is rational.

🎯 Before Moving On...

✅ Can I identify decimal place values?
✅ Can I write a decimal in expanded form?
✅ Can I compare and order decimals?
✅ Can I round a decimal correctly?
✅ Can I add and subtract decimals?
✅ Can I multiply and divide decimals?
✅ Can I convert a fraction to a decimal?
✅ Can I recognise terminating and recurring decimals?

🎉 Well Done!

You can now understand decimal place value and calculate confidently with decimals.

Remember:

Check place value → Choose the correct method → Keep decimal points accurate → Check the answer

📈 Your Progress

Lesson 7 of 17

Real Numbers Progress 41%

Decimals

🎬 Watch place value shift

🎯 By the end of this lesson...

🤔 Think About This...

Which shop is cheaper?

1️⃣ What Is a Decimal?

Whole-number part

Decimal part

2️⃣ Decimal Place Value

🔍 Reading Decimal Place Values

Tenths

Hundredths

Thousandths

Example 1: What is the value of 6 in 14.263?

3️⃣ Expanded Form

Example 2: Write 352.407 in expanded form

💡 Did You Know?

4️⃣ Comparing Decimals

Example 3: Compare 0.7 and 0.65

🧠 StudyNest Method: Comparing Decimals

5️⃣ Ordering Decimals

Example 4: Arrange in ascending order

6️⃣ Rounding Decimals

Next digit is 0, 1, 2, 3 or 4

Next digit is 5, 6, 7, 8 or 9

Example 5: Round 7.364 to one decimal place

Example 6: Round 5.7826 to two decimal places

🌍 Real-Life Example: Race Time

Example 7

7️⃣ Adding Decimals

Example 8: Calculate 12.45 + 3.8

8️⃣ Subtracting Decimals

Example 9: Calculate 20 − 7.36

🧠 StudyNest Method: Adding and Subtracting Decimals

9️⃣ Multiplying Decimals

Example 10: Calculate 2.4 × 0.3

Example 11: Calculate 1.25 × 0.4

🧠 StudyNest Method: Multiplying Decimals

🔟 Dividing Decimals

Dividing by a Whole Number

Example 12: Calculate 8.4 ÷ 2

Dividing by a Decimal

Example 13: Calculate 8.4 ÷ 0.2

🧠 StudyNest Method: Dividing by a Decimal

1️⃣1️⃣ Converting Fractions to Decimals

1️⃣2️⃣ Terminating and Recurring Decimals

Terminating Decimal

Recurring Decimal

🌍 Real-Life Example: Shopping

Example 14

🌍 Real-Life Example: Fuel Cost

Example 15

🧠 The StudyNest Decimal Strategy

🎯 Exam Tips

⚠ Common Mistakes

🎮 Your Turn!

🧠 Remember This

🎯 Before Moving On...

🎉 Well Done!

📈 Your Progress