Decimals : Basics and Operations

🔢 Decimal Operations

Master decimal basics, place value, conversions, and operations

📚

Decimal Basics

Understanding what decimals are and how they work

📍

Decimal Place Value

Learn about tenths, hundredths, thousandths, and beyond

🔄

Fraction to Decimal

Convert fractions to decimals and understand the relationship

➕➖✖️➗

Decimal Operations

Addition, subtraction, multiplication, and division with decimals

🏆

Mastery Quiz

Test your knowledge with 15 multiple choice questions

📚 Decimal Basics

What Are Decimals?

🎯 Understanding Decimals

Decimals are another way to represent parts of a whole, just like fractions. They use a decimal point (.) to separate the whole number part from the fractional part.

3 . 75

3.75 means "3 and 75 hundredths"

This is the same as 3¾ or 3 75/100

The Decimal Point

🔴 The Magic Dot

The decimal point is the most important part of a decimal number. It separates:

• Left side: Whole numbers (ones, tens, hundreds...)

• Right side: Parts of one (tenths, hundredths, thousandths...)

42 . 68

42 = whole part

68 = decimal part (68 hundredths)

📊 Visual Representation

Think of decimals as parts of a square divided into 10 or 100 equal parts:

0.3 = 3 out of 10 parts shaded

0.25 = 25 out of 100 parts shaded

0.7 = 7 out of 10 parts shaded

💡 Memory Tip

The decimal point is like a fence that separates whole numbers from parts of numbers!

Types of Decimals

🔢 Terminating Decimals

These decimals end after a certain number of digits.

0.5 (ends after 1 digit)

0.75 (ends after 2 digits)

0.125 (ends after 3 digits)

2.4375 (ends after 4 digits)

🔄 Repeating Decimals

These decimals have digits that repeat forever.

0.333... (3 repeats forever)

0.666... (6 repeats forever)

0.142857142857... (pattern repeats)

Note: We often write repeating decimals with a bar over the repeating part: 0.3̄ means 0.333...

Reading Decimals

📖 How to Say Decimal Numbers

There are two main ways to read decimal numbers:

1

Method 1 - Place Value Method:
3.47 = "Three and forty-seven hundredths"
0.6 = "Six tenths"
12.009 = "Twelve and nine thousandths"

2

Method 2 - Digit by Digit:
3.47 = "Three point four seven"
0.6 = "Zero point six"
12.009 = "Twelve point zero zero nine"

Common Decimal Examples

💰 Money

Money is the most common use of decimals in daily life:

$3.50 = 3 dollars and 50 cents

$0.25 = 25 cents (quarter)

$12.99 = 12 dollars and 99 cents

📏 Measurements

Decimals are used in measurements:

5.5 feet = 5 feet and 6 inches

2.3 meters = 2 meters and 30 centimeters

98.6°F = normal body temperature

⚖️ Weight

Weight measurements often use decimals:

2.5 pounds = 2 pounds and 8 ounces

1.2 kilograms

0.75 tons

📊 Statistics & Sports

Decimals show precise measurements:

Batting average: 0.325

Race time: 9.58 seconds

Grade: 87.5%

Interactive Practice

🎯 Decimal Recognition Practice

4.25

How do you read this decimal using the place value method?

📍 Decimal Place Value

Understanding Place Value in Decimals

🎯 The Place Value System

Just like whole numbers, each digit in a decimal has a specific place value. The decimal point separates whole number places from decimal places.

Hundreds	Tens	Ones	.	Tenths	Hundredths	Thousandths
100	10	1	.	0.1	0.01	0.001
2	4	7	.	3	6	5

247.365 = 2 hundreds + 4 tens + 7 ones + 3 tenths + 6 hundredths + 5 thousandths

Decimal Place Values Explained

🥇 Tenths Place (0.1)

The first digit after the decimal point represents tenths.

0 . 7

• 0.7 = 7 tenths = 7/10

• Think: 7 out of 10 equal parts

• Like cutting a pizza into 10 slices and taking 7

🥈 Hundredths Place (0.01)

The second digit after the decimal point represents hundredths.

0 . 2 5

• 0.25 = 25 hundredths = 25/100

• Think: 25 out of 100 equal parts

• Like a dollar divided into 100 pennies

🥉 Thousandths Place (0.001)

The third digit after the decimal point represents thousandths.

0 . 1 2 8

• 0.128 = 128 thousandths = 128/1000

• Think: 128 out of 1000 equal parts

• Very precise measurements

🔢 Beyond Thousandths

The pattern continues for even more precise measurements:

• Ten-thousandths (0.0001)

• Hundred-thousandths (0.00001)

• Millionths (0.000001)

• And so on...

Expanded Form

🔍 Breaking Down Decimal Numbers

Expanded form shows the value of each digit in a decimal number.

1

Example: 5.247
5 = 5 ones = 5.000
2 = 2 tenths = 0.200
4 = 4 hundredths = 0.040
7 = 7 thousandths = 0.007

2

Expanded Form:
5.247 = 5.000 + 0.200 + 0.040 + 0.007
or
5.247 = 5 + 0.2 + 0.04 + 0.007

Comparing Decimal Place Values

⚖️ Which is Larger?

To compare decimals, look at each place value from left to right:

Compare: 0.6 vs 0.58

0.6 = 0.60 (6 tenths, 0 hundredths)

0.58 = 0.58 (5 tenths, 8 hundredths)

Since 6 tenths > 5 tenths, 0.6 > 0.58

🎯 Key Rules for Comparing

• Start from the leftmost digit after the decimal

• Compare digit by digit

• The first different digit determines which is larger

• Adding zeros to the right doesn't change the value

• 0.5 = 0.50 = 0.500

Rounding Decimals

🎯 Rounding to Different Place Values

Rounding decimals follows the same rules as rounding whole numbers.

1

Find the place you're rounding to
Example: Round 3.476 to the nearest tenth
The tenths place has 4

2

Look at the digit to the right
The hundredths place has 7
Since 7 ≥ 5, round up

3

Round and drop the rest
3.476 rounded to the nearest tenth = 3.5

Interactive Practice

🎯 Place Value Practice

5.247

What is the value of the digit 4 in this number?

🔄 Fraction to Decimal Conversion

Converting Fractions to Decimals

🎯 The Division Method

To convert a fraction to a decimal, divide the numerator by the denominator.

3 4 = 3 ÷ 4 = 0.75

1

Set up the division: 3 ÷ 4

2

Add decimal point and zeros: 3.000 ÷ 4

3

Divide: 30 ÷ 4 = 7 remainder 2, so 0.7...

4

Continue: 20 ÷ 4 = 5, so 0.75

Common Fraction-Decimal Equivalents

🔢 Halves, Fourths, and Eighths

1/2 = 0.5

1/4 = 0.25

3/4 = 0.75

1/8 = 0.125

3/8 = 0.375

5/8 = 0.625

7/8 = 0.875

🔢 Thirds and Sixths

1/3 = 0.333... (0.3̄)

2/3 = 0.666... (0.6̄)

1/6 = 0.1666... (0.16̄)

5/6 = 0.8333... (0.83̄)

Note: The bar over a digit means it repeats forever.

🔢 Fifths and Tenths

1/5 = 0.2

2/5 = 0.4

3/5 = 0.6

4/5 = 0.8

1/10 = 0.1

3/10 = 0.3

7/10 = 0.7

9/10 = 0.9

🔢 Other Common Fractions

1/7 = 0.142857... (repeating)

1/9 = 0.111... (0.1̄)

1/11 = 0.090909... (0.09̄)

1/12 = 0.08333... (0.083̄)

1/16 = 0.0625

1/20 = 0.05

1/25 = 0.04

1/100 = 0.01

Types of Decimal Results

✅ Terminating Decimals

These fractions convert to decimals that end:

When does this happen?

When the denominator has only factors of 2 and/or 5.

1/2 = 0.5 (denominator: 2)

1/4 = 0.25 (denominator: 2²)

1/5 = 0.2 (denominator: 5)

1/8 = 0.125 (denominator: 2³)

3/20 = 0.15 (denominator: 2² × 5)

🔄 Repeating Decimals

These fractions convert to decimals that repeat:

When does this happen?

When the denominator has factors other than 2 and 5.

1/3 = 0.333... (denominator: 3)

1/6 = 0.1666... (denominator: 2 × 3)

1/7 = 0.142857... (denominator: 7)

1/9 = 0.111... (denominator: 3²)

Converting Mixed Numbers

🔄 Mixed Numbers to Decimals

For mixed numbers, convert the fraction part and add it to the whole number.

1

Example: Convert 2¾ to a decimal

2

Keep the whole number: 2

3

Convert the fraction: ¾ = 3 ÷ 4 = 0.75

4

Add them together: 2 + 0.75 = 2.75

Converting Decimals to Fractions

🔄 The Reverse Process

To convert a decimal to a fraction, use the place value of the last digit.

1

Example: Convert 0.375 to a fraction

2

Identify place value: 5 is in the thousandths place

3

Write as fraction: 375/1000

4

Simplify: 375/1000 = 3/8 (divide by 125)

Interactive Practice

🎯 Fraction-Decimal Conversion Practice

3 8 = ?

➕➖✖️➗ Decimal Operations

Decimal Operations Overview

🎯 Key Principles

Decimal operations follow similar rules to whole number operations, but we need to pay special attention to the decimal point placement.

• Addition & Subtraction: Line up the decimal points

• Multiplication: Count total decimal places

• Division: Move decimal points to make divisor whole

Addition with Decimals

➕ Adding Decimal Numbers

The key to adding decimals is to line up the decimal points vertically.

3.45 + 2.7 = 6.15

1

Line up decimal points:

  3.45
+ 2.70  (add zero to make same length)
------

2

Add from right to left:

  3.45
+ 2.70
------
  6.15

3

Place decimal point: The decimal point in the answer goes directly below the decimal points in the problem.

Subtraction with Decimals

➖ Subtracting Decimal Numbers

Subtraction follows the same alignment rule as addition.

5.6 - 2.38 = 3.22

1

Line up decimal points:

  5.60  (add zero)
- 2.38
------

2

Subtract from right to left:

  5.60
- 2.38
------
  3.22

3

Borrowing works the same: If you need to borrow, treat it like whole numbers but keep the decimal point aligned.

Multiplication with Decimals

✖️ Multiplying Decimal Numbers

Multiply as if they were whole numbers, then place the decimal point.

2.3 × 1.4 = 3.22

1

Multiply ignoring decimals:

   23
×  14
-----
   92  (23 × 4)
+ 230  (23 × 10)
-----
  322

2

Count decimal places:
2.3 has 1 decimal place
1.4 has 1 decimal place
Total: 1 + 1 = 2 decimal places

3

Place decimal point: 322 becomes 3.22 (2 places from right)

Division with Decimals

➗ Dividing by Whole Numbers

When dividing a decimal by a whole number, place the decimal point directly above.

Example: 6.84 ÷ 4

    1.71
   ------
4 ) 6.84
    4
    --
    28
    28
    --
     04
      4
     --
      0

➗ Dividing by Decimals

Move the decimal point in both numbers to make the divisor a whole number.

Example: 8.4 ÷ 2.1

Move decimal 1 place right in both:

84 ÷ 21 = 4

So 8.4 ÷ 2.1 = 4

Order of Operations with Decimals

📋 PEMDAS Still Applies

The order of operations works the same way with decimals as with whole numbers.

2.5 + 3.2 × 1.5 = ?

1

Multiply first: 3.2 × 1.5 = 4.8

2

Then add: 2.5 + 4.8 = 7.3

Estimating with Decimals

🎯 Quick Estimation Strategies

Rounding decimals can help you estimate answers quickly.

Example: 4.7 × 3.2

Round: 5 × 3 = 15

Actual: 4.7 × 3.2 = 15.04

The estimate is very close!

💡 Estimation Tips

• Round to the nearest whole number for quick estimates

• Use estimation to check if your answer is reasonable

• For money problems, round to the nearest dollar

Interactive Practice

🎯 Decimal Operations Practice

3.4 + 2.7 = ?

🏆 Decimal Mastery Quiz

Test Your Decimal Knowledge

Ready to test your understanding of decimals? This comprehensive quiz covers all the topics we've learned:

• 15 multiple choice questions

• Covers: Decimal basics, place value, conversions, and operations

• Immediate feedback after each question

• Final score and performance breakdown

Question 1 of 15

🎉 Quiz Complete!

85%

13 out of 15 correct

Excellent work! You have a strong understanding of decimals!

4/4

Decimal Basics

3/4

Place Value

3/3

Conversions

3/4

Operations