Grade 5
  1. Number Sense and Place Value  1.1. Understanding Large Numbers  1.2. Place Value up to Millions  1.3. Comparing and Ordering Numbers  1.4. Rounding Whole Numbers  1.5. Expanded Form and Standard Form  1.6. Estimation in Calculations
  2. Operations with Whole Numbers  2.1. Addition of Large Numbers  2.2. Subtraction of Large Numbers  2.3. Multiplication Concepts and Strategies  2.4. Division Concepts and Strategies  2.5. Multi-Digit Multiplication  2.6. Long Division  2.7. Order of Operations
  3. Fractions  3.1. Introduction to Fractions  3.2. Equivalent Fractions  3.3. Simplifying Fractions  3.4. Comparing and Ordering Fractions  3.5. Addition and Subtraction of Fractions  3.6. Multiplication of Fractions  3.7. Division of Fractions  3.8. Mixed Numbers and Improper Fractions  3.9. Converting Between Improper Fractions and Mixed Numbers
  4. Decimals  4.1. Understanding Decimals  4.2. Place Value in Decimals  4.3. Comparing and Ordering Decimals  4.4. Rounding Decimals  4.5. Addition and Subtraction of Decimals  4.6. Multiplication of Decimals  4.7. Division of Decimals  4.8. Converting Decimals to Fractions and Vice Versa
  5. Measurement  5.1. Length Measurement (Metric and Customary Units)  5.2. Weight and Mass Measurement  5.3. Volume and Capacity Measurement  5.4. Area of Squares and Rectangles  5.5. Perimeter of Polygons  5.6. Time and Elapsed Time  5.7. Temperature  5.8. Understanding Metric Conversions
  6. Geometry  6.1. Types of Lines and Angles  6.2. Measuring Angles  6.3. Classifying Triangles  6.4. Properties of Quadrilaterals  6.5. Symmetry and Reflection  6.6. Transformations (Translation, Rotation, Reflection)  6.7. Coordinate Plane and Plotting Points  6.8. 3D Shapes and Their Properties  6.9. Nets of 3D Shapes  6.10. Understanding Congruence and Similarity
  7. Data and Probability  7.1. Introduction to Data Collection  7.2. Organizing Data (Tables and Tally Charts)  7.3. Graphs (Bar Graphs, Line Plots, Pictographs)  7.4. Mean, Median, and Mode  7.5. Range of Data  7.6. Introduction to Probability  7.7. Simple Events and Outcomes  7.8. Probability of Independent Events
  8. Ratios and Proportions  8.1. Understanding Ratios  8.2. Writing and Simplifying Ratios  8.3. Equivalent Ratios  8.4. Introduction to Proportions  8.5. Solving Proportion Problems
  9. Algebraic Thinking  9.1. Understanding Variables and Expressions  9.2. Writing Algebraic Expressions  9.3. Simplifying Expressions  9.4. Solving One-Step Equations  9.5. Understanding Patterns and Sequences  9.6. Using the Distributive Property
  10. Financial Literacy  10.1. Introduction to Money and Currency  10.2. Budgeting Basics  10.3. Saving and Spending  10.4. Understanding Interest  10.5. Basic Financial Planning  10.6. Simple vs. Compound Interest

Grade 5 → Decimals ↓


Converting Decimals to Fractions and Vice Versa


Decimals and fractions are closely related, and understanding the conversion between the two is a basic concept in mathematics. Both represent numbers that are not whole, called rational numbers, but they have different ways of representing them. Decimals use the base ten system, and fractions have two numbers - a numerator and a denominator. Let's begin this journey to understand the relationship between decimals and fractions.

What is a decimal?

Decimals are a way of representing numbers that are not whole numbers. The decimal point separates the whole number part from the fractional part. For example, in the number 3.75, 3 is the whole number, and 75 is the decimal or fractional part.

Visual representation of decimals

Consider the decimal number 0.5:

0.5

In this SVG graphic, the full line represents a whole, and the blue area represents 0.5 on the number line from 0 to 1.

What is fraction?

A fraction has two numbers: the numerator (the top number) and the denominator (the bottom number). The numerator shows how many equal parts the whole is divided into. For example, in the fraction 3/4, 3 is the numerator, and 4 is the denominator.

Visual representation of fractions

Consider the fraction 1/2:

1/2

This visual shows what 1/2 looks like when a whole is divided into two equal parts, and one part is shaded.

Converting decimals to fractions

There are a few simple steps to convert a decimal to a fraction. Let's explore these steps using the example of 0.75.

  1. Write the decimal as a fraction without the decimal point: 75
  2. Find the denominator by counting the number of decimal places. There are two decimal places, so add two zeros to the 1. The denominator is 100.
  3. Write the fraction: 75/100
  4. Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor (GCD).
0.75 = 75/100 = 3/4

Example explained

To simplify 75/100, we divide both the numerator and denominator by 25, which is their GCD:

75 ÷ 25 = 3
100 ÷ 25 = 4

Thus, 75/100 = 3/4.

0.75 or 3/4

Special case: repeating decimals

Some decimals repeat indefinitely. For example, 0.333... repeats. Converting repeating decimals to fractions requires a slightly different process. Let's try 0.333...:

  1. Let x = 0.333...
  2. Multiply both sides by 10 to move the decimal point: 10x = 3.333...
  3. Subtract the original equation from the new equation: 10x - x = 3.333... - 0.333...
  4. Solve for x: 9x = 3
x = 3/9 = 1/3

Converting fractions to decimals

To convert a fraction to a decimal, divide the numerator by the denominator. Let's take the fraction 3/4 and convert it to a decimal.

Step-by-step long division

Divide 3 by 4:

  1. 4 goes into 30 (3 plus a decimal point and a zero) 7 times. Place the 7 in the decimal place.
  2. Multiply 7 by 4 and place the result under 30: 28.
  3. Subtracting 28 from 30 gives us 2. Subtracting the next 0 gives us 20.
  4. 4 goes into 20 exactly 5 times. Write 5 as the next decimal place.
3/4 = 0.75

After using long division, 3/4 equals 0.75.

Example explained

Here is a visual representation of converting 3/4 to 0.75:

3/4 = 0.75

Special case: whole numbers as fractions

Whole numbers can be written as fractions using 1 as the denominator. This is useful when performing operations with fractions. For example, the whole number 5 can be represented as:

5 = 5/1

While dividing, even if 5/1 is converted into decimal, we will still get 5.

Practice problems

To solidify your understanding, here are some practice problems:

  1. Convert 0.6 to a fraction.
  2. Convert the fraction 2/5 to a decimal.
  3. Convert the recurring decimal 0.444... to a fraction.

Practice solutions

1. Convert 0.6 to a fraction:

0.6 = 6/10 = 3/5

2. Convert 2/5 to decimal:

2/5 = 0.4

3. Convert 0.444... to a fraction:

Let x = 0.444... 10x = 4.444... 10x - x = 4.444... - 0.444... 9x = 4 x = 4/9

Conclusion

Understanding how to convert decimals to fractions and vice versa is a skill that adds depth to mathematical knowledge. Mastering these techniques makes it possible to manipulate numbers easily and gain a better understanding of the relationships between different numerical representations. These concepts are the building blocks that aid further learning in mathematics.


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Converting Decimals to Fractions and Vice Versa

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