Solved Exercises on Decomposing Fractions into Sums in Grade 4

Master decomposing fractions into sums: rules for breaking down fractions into addition expressions through these 5 detailed exercises with visual representations.

Solution: Exercises 1 to 3
1 Decompose fraction into unit fractions
Exercise 1
Decompose \(\frac{4}{7}\) into a sum of unit fractions
Definition:

Unit fraction: A fraction with numerator 1, such as \(\frac{1}{2}\), \(\frac{1}{3}\), \(\frac{1}{4}\), etc.

Method for decomposing into unit fractions:
  1. Identify the denominator (the bottom number)
  2. Create unit fractions with the same denominator
  3. Add as many unit fractions as indicated by the numerator
Original
\(\frac{4}{7}\)
Unit Fraction
\(\frac{1}{7}\)
Sum
\(\frac{1}{7} + \frac{1}{7} + \frac{1}{7} + \frac{1}{7}\)
Step 1: Identify the fraction

We have \(\frac{4}{7}\), where the numerator is 4 and the denominator is 7

Step 2: Identify the unit fraction

The unit fraction with denominator 7 is \(\frac{1}{7}\)

Step 3: Count how many unit fractions are needed

Since the numerator is 4, we need 4 copies of \(\frac{1}{7}\)

\(\frac{1}{7}\)
+
\(\frac{1}{7}\)
+
\(\frac{1}{7}\)
+
\(\frac{1}{7}\)
=
\(\frac{4}{7}\)
1/7
1/7
1/7
1/7
\(\frac{4}{7} = \frac{1}{7} + \frac{1}{7} + \frac{1}{7} + \frac{1}{7}\)
\(\frac{4}{7} = \frac{1}{7} + \frac{1}{7} + \frac{1}{7} + \frac{1}{7}\)
Final answer:

\(\frac{4}{7} = \frac{1}{7} + \frac{1}{7} + \frac{1}{7} + \frac{1}{7}\)

Applied rules:

• Unit fraction decomposition: \(\frac{n}{b} = \frac{1}{b} + \frac{1}{b} + \cdots + \frac{1}{b}\) (n times)

• Counting principle: The numerator tells how many unit fractions are needed

• Denominator preservation: All unit fractions have the same denominator as the original fraction

2 Decompose fraction into different fractions
Exercise 2
Decompose \(\frac{5}{8}\) into a sum of two different fractions
Definition:

Different fraction decomposition: Expressing a fraction as a sum of two or more fractions with different numerators and denominators

Original
\(\frac{5}{8}\)
Option 1
\(\frac{2}{8} + \frac{3}{8}\)
Option 2
\(\frac{1}{8} + \frac{4}{8}\)
Step 1: Identify the fraction

We have \(\frac{5}{8}\), where the numerator is 5 and the denominator is 8

Step 2: Find combinations that sum to 5

We need two numbers that add up to 5: 1+4, 2+3, 3+2, 4+1

Step 3: Write the decomposition

For example: \(\frac{5}{8} = \frac{2}{8} + \frac{3}{8}\)

\(\frac{2}{8}\)
+
\(\frac{3}{8}\)
=
\(\frac{5}{8}\)
Other possible decompositions:

\(\frac{5}{8} = \frac{1}{8} + \frac{4}{8}\)

\(\frac{5}{8} = \frac{3}{8} + \frac{2}{8}\)

\(\frac{5}{8} = \frac{4}{8} + \frac{1}{8}\)

2/8
3/8
\(\frac{5}{8} = \frac{2}{8} + \frac{3}{8}\)
\(\frac{5}{8} = \frac{2}{8} + \frac{3}{8}\)
Final answer:

\(\frac{5}{8} = \frac{2}{8} + \frac{3}{8}\) (one possible decomposition)

Applied rules:

• Addition rule: \(\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}\)

• Flexibility principle: A fraction can be decomposed in multiple ways

• Denominator preservation: When adding like denominators, keep the denominator the same

3 Decompose fraction into mixed numbers
Exercise 3
Decompose \(\frac{7}{4}\) into a sum involving a whole number and a fraction
Definition:

Improper fraction decomposition: Expressing a fraction greater than 1 as a sum of a whole number and a proper fraction

Original
\(\frac{7}{4}\)
Unit Fractions
7 copies of \(\frac{1}{4}\)
Result
\(1 + \frac{3}{4}\)
Step 1: Identify the fraction

We have \(\frac{7}{4}\), where the numerator (7) is greater than the denominator (4)

Step 2: Determine how many wholes

\(7 ÷ 4 = 1\) remainder \(3\), so we have 1 whole and \(\frac{3}{4}\) left

Step 3: Write the decomposition

\(\frac{7}{4} = \frac{4}{4} + \frac{3}{4} = 1 + \frac{3}{4}\)

\(1\)
+
\(\frac{3}{4}\)
=
\(\frac{7}{4}\)
Alternative decomposition:

\(\frac{7}{4} = \frac{4}{4} + \frac{3}{4} = \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4}\)

1 whole
3/4
\(\frac{7}{4} = 1 + \frac{3}{4}\)
\(\frac{7}{4} = 1 + \frac{3}{4}\)
Final answer:

\(\frac{7}{4} = 1 + \frac{3}{4}\)

Applied rules:

• Improper fraction rule: When numerator > denominator, decompose into whole + fraction

• Division method: Divide numerator by denominator to find wholes and remainder

• Mixed number formation: Quotient becomes whole number, remainder becomes numerator of fraction

Rules and methods, laws,...
\(\frac{n}{b} = \frac{1}{b} + \frac{1}{b} + \cdots + \frac{1}{b}\)
Unit Fraction Decomposition
Rule 1
\(\frac{n}{b} = n \times \frac{1}{b}\)
Scaling unit fractions
Rule 2
\(\frac{a+c}{b} = \frac{a}{b} + \frac{c}{b}\)
Addition of like denominators
Rule 3
\(\frac{n}{b} = q + \frac{r}{b}\)
Improper fraction decomposition
Key definitions:

Decompose: To break down a fraction into a sum of smaller fractions or numbers

Unit fraction: A fraction with numerator 1, such as \(\frac{1}{2}\), \(\frac{1}{3}\), \(\frac{1}{4}\), etc.

Like denominators: Fractions that have the same denominator, allowing direct addition of numerators

Unlike denominators: Fractions that have different denominators, requiring conversion before addition

Improper fraction: A fraction where the numerator is greater than or equal to the denominator

Proper fraction: A fraction where the numerator is less than the denominator

Mixed number: A combination of a whole number and a proper fraction

Complete methodology:
  1. Identify the fraction: Recognize the numerator and denominator of the fraction to decompose
  2. Determine the approach: Decide whether to decompose into unit fractions, like denominators, or mixed numbers
  3. Apply the appropriate rule: Use the correct decomposition method based on the approach
  4. Verify the result: Check that adding the decomposed parts gives back the original fraction
  5. Explore alternatives: Look for other possible ways to decompose the fraction
Tip 1: The numerator tells you how many unit fractions you need.
Tip 2: All unit fractions in the sum have the same denominator as the original fraction.
Tip 3: There are often multiple ways to decompose a fraction.
Tip 4: Use visual models to see how fractions can be broken down.
Common errors: Forgetting that unit fractions have numerator 1, adding denominators when adding fractions, not recognizing that there are multiple ways to decompose a fraction, miscounting the number of unit fractions needed.
Memory aid: "The top number counts how many parts you have, the bottom number tells what kind of parts they are."
Rules to know by heart:

• Unit fraction decomposition: \(\frac{n}{b} = \frac{1}{b} + \frac{1}{b} + \cdots + \frac{1}{b}\) (n times)

• Addition rule: \(\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}\)

• Improper fraction rule: When numerator > denominator, decompose into whole + fraction

• Multiple decompositions: Most fractions can be decomposed in several different ways

Solution: Exercises 4 to 5
4 Word problem with decomposition
Exercise 4
Sarah ate \(\frac{3}{5}\) of an apple. Show this as a sum of different fractions.
Definition:

Word problems with decomposition: Real-world scenarios where fractions need to be expressed as sums of other fractions

Original
\(\frac{3}{5}\)
Option 1
\(\frac{1}{5} + \frac{2}{5}\)
Option 2
\(\frac{1}{5} + \frac{1}{5} + \frac{1}{5}\)
Step 1: Identify the fraction

Sarah ate \(\frac{3}{5}\) of an apple, so we need to decompose this fraction

Step 2: Find possible decompositions

We need numbers that add up to 3: 1+2, 2+1, 1+1+1

Step 3: Write the decomposition

For example: \(\frac{3}{5} = \frac{1}{5} + \frac{2}{5}\)

\(\frac{1}{5}\)
+
\(\frac{2}{5}\)
=
\(\frac{3}{5}\)
Other possible decompositions:

\(\frac{3}{5} = \frac{2}{5} + \frac{1}{5}\)

\(\frac{3}{5} = \frac{1}{5} + \frac{1}{5} + \frac{1}{5}\)

1/5
2/5
\(\frac{3}{5} = \frac{1}{5} + \frac{2}{5}\)
\(\frac{3}{5} = \frac{1}{5} + \frac{2}{5}\)
Final answer:

\(\frac{3}{5} = \frac{1}{5} + \frac{2}{5}\) (one possible decomposition)

Applied rules:

• Like denominator addition: \(\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}\)

• Flexibility principle: A fraction can be decomposed in multiple ways

• Real-world application: Fraction decomposition occurs in everyday situations

5 Advanced fraction decomposition
Exercise 5
Decompose \(\frac{11}{6}\) in multiple ways and express as a mixed number.
Definition:

Advanced decomposition: Expressing fractions in multiple forms including unit fractions, like denominators, and mixed numbers

Original
\(\frac{11}{6}\)
Unit Fractions
11 copies of \(\frac{1}{6}\)
Mixed Number
\(1\frac{5}{6}\)
Step 1: Identify the fraction

We have \(\frac{11}{6}\), where the numerator (11) is greater than the denominator (6)

Step 2: Decompose into unit fractions

\(\frac{11}{6} = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6}\)

Step 3: Decompose into like denominators

\(\frac{11}{6} = \frac{6}{6} + \frac{5}{6} = 1 + \frac{5}{6}\)

Step 4: Express as mixed number

\(\frac{11}{6} = 1\frac{5}{6}\)

\(\frac{6}{6}\)
+
\(\frac{5}{6}\)
=
\(\frac{11}{6}\)
Other possible decompositions:

\(\frac{11}{6} = \frac{1}{6} + \frac{10}{6}\)

\(\frac{11}{6} = \frac{2}{6} + \frac{9}{6}\)

\(\frac{11}{6} = \frac{3}{6} + \frac{8}{6}\)

\(\frac{11}{6} = \frac{4}{6} + \frac{7}{6}\)

\(\frac{11}{6} = \frac{5}{6} + \frac{6}{6}\)

1 whole
5/6
\(\frac{11}{6} = 1 + \frac{5}{6} = 1\frac{5}{6}\)
\(\frac{11}{6} = 1\frac{5}{6}\)
Final answer:

\(\frac{11}{6} = 1\frac{5}{6}\) (expressed as a mixed number)

Applied rules:

• Improper fraction decomposition: \(\frac{n}{b} = q + \frac{r}{b}\) where \(n = q \times b + r\)

• Mixed number formation: Divide numerator by denominator to get whole number and remainder

• Multiple decompositions: Many different ways to express the same fraction as a sum

Complete Summary: Decompose Fractions into Sums
\(\frac{n}{b} = \frac{1}{b} + \frac{1}{b} + \cdots + \frac{1}{b}\)
Unit Fraction Decomposition
\(\frac{a+c}{b} = \frac{a}{b} + \frac{c}{b}\)
Like Denominator Addition
\(\frac{n}{b} = q + \frac{r}{b}\)
Improper Fraction Decomposition
Key definitions:

Decompose: To break down a fraction into a sum of smaller fractions or numbers.

Unit fraction: A fraction with numerator 1, such as \(\frac{1}{2}\), \(\frac{1}{3}\), \(\frac{1}{4}\), etc.

Like denominators: Fractions that have the same denominator, allowing direct addition of numerators.

Unlike denominators: Fractions that have different denominators, requiring conversion before addition.

Improper fraction: A fraction where the numerator is greater than or equal to the denominator.

Proper fraction: A fraction where the numerator is less than the denominator.

Mixed number: A combination of a whole number and a proper fraction.

Part-whole relationship: How smaller fractions combine to form larger fractions.

Complete methodology:
  1. Identify the fraction: Recognize the numerator and denominator of the fraction to decompose
  2. Determine the approach: Decide whether to decompose into unit fractions, like denominators, or mixed numbers
  3. Apply the appropriate rule: Use the correct decomposition method based on the approach
  4. Verify the result: Check that adding the decomposed parts gives back the original fraction
  5. Explore alternatives: Look for other possible ways to decompose the fraction
  6. Express in different forms: Convert between improper fractions, mixed numbers, and sums
Tip 1: The numerator tells you exactly how many unit fractions you need to add together.
Tip 2: All unit fractions in the sum have the same denominator as the original fraction.
Tip 3: Think of unit fractions as the building blocks that make up all other fractions.
Tip 4: Use visual models like circles or rectangles to see how fractions can be broken down.
Tip 5: Remember that there are often multiple ways to decompose a fraction.
Tip 6: When the numerator is larger than the denominator, think about how many wholes you can make.
Common errors: Forgetting that unit fractions must have numerator 1, adding denominators when adding fractions, confusing the role of numerator and denominator in the relationship, not recognizing that there are multiple ways to decompose a fraction, miscounting the number of unit fraction groups.
Memory aids: "The top number counts the parts from the bottom number of equal parts"; "Unit fractions are like building blocks that stack to make larger fractions".
Visualization: Draw circle models or bar models to see how fractions can be broken down into sums.
Real-world connection: Fraction decomposition occurs in many practical situations like sharing food items or dividing resources.
Rules to know by heart:

• Unit fraction decomposition: \(\frac{n}{b} = \frac{1}{b} + \frac{1}{b} + \cdots + \frac{1}{b}\) (n times)

• Addition rule: \(\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}\)

• Improper fraction rule: \(\frac{n}{b} = q + \frac{r}{b}\) where \(n = q \times b + r\)

• Counting principle: The numerator tells how many unit fractions are needed

• Denominator preservation: All unit fractions in the sum have the same denominator as the original fraction

• Multiple decompositions: Most fractions can be decomposed in several different ways

Fraction Decomposition Framework

📊
Unit Fraction Decomposition

\(\frac{3}{5} = \frac{1}{5} + \frac{1}{5} + \frac{1}{5}\)

3 groups of \(\frac{1}{5}\) make \(\frac{3}{5}\)

1
1
1
2
Like Denominator Decomposition

\(\frac{7}{8} = \frac{3}{8} + \frac{4}{8}\)

\(\frac{7}{8} = \frac{2}{8} + \frac{5}{8}\)

3
4
1
Improper Fraction Decomposition

\(\frac{5}{3} = \frac{3}{3} + \frac{2}{3} = 1 + \frac{2}{3}\)

\(\frac{5}{3} = 1\frac{2}{3}\)

3
2
Decomposition Process
1
Identify
2
Choose
3
Apply
4
Verify
5
Explore
Multiple Decomposition Options

Fractions can be decomposed in many different ways

Master Fraction Decomposition!

Questions & Answers

Question: Why do we need to decompose fractions? Isn't it just making things more complicated?

Answer: Actually, decomposing fractions makes things easier to understand and work with! It's like learning to break down big numbers to understand them better:

  • Understanding: Decomposing helps you understand what fractions really mean - parts of a whole
  • Operations: Adding and subtracting fractions becomes clearer when you see them as sums of smaller parts
  • Comparison: It's easier to compare fractions when you think about how they're composed
  • Flexibility: It helps you see that there are multiple ways to express the same value

For example, if you know that \(\frac{3}{4}\) can be thought of as \(\frac{1}{4} + \frac{1}{4} + \frac{1}{4}\), it's easier to add: \(\frac{3}{4} + \frac{1}{4} = \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} = \frac{4}{4} = 1\).

Decomposition helps you understand the structure of fractions!

Question: How many different ways can I decompose a fraction like \(\frac{5}{6}\)?

Answer: There are many different ways to decompose \(\frac{5}{6}\)! Here are some possibilities:

  • Into unit fractions: \(\frac{5}{6} = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6}\)
  • Into two parts: \(\frac{5}{6} = \frac{1}{6} + \frac{4}{6}\) or \(\frac{5}{6} = \frac{2}{6} + \frac{3}{6}\)
  • Into three parts: \(\frac{5}{6} = \frac{1}{6} + \frac{1}{6} + \frac{3}{6}\) or \(\frac{5}{6} = \frac{1}{6} + \frac{2}{6} + \frac{2}{6}\)

In general, for \(\frac{n}{b}\), you can decompose it in many ways by finding different combinations of numbers that add up to n. For \(\frac{5}{6}\), we need numbers that add to 5: 1+4, 2+3, 1+1+3, 1+2+2, etc.

The key is that the denominators stay the same, and the numerators must add up to the original numerator!

Question: What happens if the fraction is bigger than 1, like \(\frac{7}{4}\)? How do I decompose that?

Answer: When a fraction is greater than 1 (improper fraction), you have several options for decomposition:

  • Into unit fractions: \(\frac{7}{4} = \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4}\)
  • Into wholes and fractions: \(\frac{7}{4} = \frac{4}{4} + \frac{3}{4} = 1 + \frac{3}{4}\)
  • As a mixed number: \(\frac{7}{4} = 1\frac{3}{4}\)

To decompose into wholes and fractions:

  1. Divide the numerator by the denominator: \(7 ÷ 4 = 1\) remainder \(3\)
  2. The quotient (1) becomes the whole number
  3. The remainder (3) becomes the numerator of the fraction part
  4. The denominator stays the same (4)

So \(\frac{7}{4} = 1 + \frac{3}{4} = 1\frac{3}{4}\). This shows that \(\frac{7}{4}\) is more than one whole!