Solved Exercises on Representing Fractions as Sums of Unit Fractions in Grade 4

Master representing fractions as sums of unit fractions: rules for decomposing fractions into unit fraction additions through these 5 detailed exercises with visual representations.

Solution: Exercises 1 to 3
1 Represent fraction as sum of unit fractions
Exercise 1
Represent \(\frac{3}{5}\) as 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 representing fractions as sums of 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
\(\frac{3}{5}\)
Unit Fraction
\(\frac{1}{5}\)
Sum
\(\frac{1}{5} + \frac{1}{5} + \frac{1}{5}\)
Step 1: Identify the fraction

We have \(\frac{3}{5}\), where the numerator is 3 and the denominator is 5

Step 2: Identify the unit fraction

The unit fraction with denominator 5 is \(\frac{1}{5}\)

Step 3: Count how many unit fractions are needed

Since the numerator is 3, we need 3 copies of \(\frac{1}{5}\)

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

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

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 Represent fraction with larger numerator
Exercise 2
Represent \(\frac{7}{8}\) as a sum of unit fractions
Definition:

Unit fraction decomposition: Expressing a fraction as a sum of unit fractions with the same denominator

Original
\(\frac{7}{8}\)
Unit Fraction
\(\frac{1}{8}\)
Sum
7 copies of \(\frac{1}{8}\)
Step 1: Identify the fraction

We have \(\frac{7}{8}\), where the numerator is 7 and the denominator is 8

Step 2: Identify the unit fraction

The unit fraction with denominator 8 is \(\frac{1}{8}\)

Step 3: Count how many unit fractions are needed

Since the numerator is 7, we need 7 copies of \(\frac{1}{8}\)

\(\frac{1}{8}\)
+
\(\frac{1}{8}\)
+
\(\frac{1}{8}\)
+
\(\frac{1}{8}\)
+
\(\frac{1}{8}\)
+
\(\frac{1}{8}\)
+
\(\frac{1}{8}\)
=
\(\frac{7}{8}\)
1/8
1/8
1/8
1/8
1/8
1/8
1/8
\(\frac{7}{8} = \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8}\)
\(\frac{7}{8} = \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8}\)
Final answer:

\(\frac{7}{8} = \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8}\)

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

3 Represent unit fraction as sum
Exercise 3
Represent \(\frac{1}{4}\) as a sum of unit fractions
Definition:

Single unit fraction: A unit fraction is already a unit fraction, so it can be represented as itself

Original
\(\frac{1}{4}\)
Unit Fraction
\(\frac{1}{4}\)
Sum
1 copy of \(\frac{1}{4}\)
Step 1: Identify the fraction

We have \(\frac{1}{4}\), which is already a unit fraction

Step 2: Identify the unit fraction

The unit fraction is \(\frac{1}{4}\) itself

Step 3: Count how many unit fractions are needed

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

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

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

Applied rules:

• Unit fraction identity: A unit fraction equals itself

• Single unit: When numerator is 1, only one unit fraction is needed

• Self-representation: Unit fractions can be represented as themselves

Rules and methods, laws,...
\(\frac{n}{b} = \frac{1}{b} + \frac{1}{b} + \cdots + \frac{1}{b}\)
Decomposition into Unit Fractions
Rule 1
\(\frac{n}{b} = n \times \frac{1}{b}\)
Scaling unit fractions
Rule 2
\(\frac{n}{b} = \frac{1}{b} + \frac{1}{b} + \cdots + \frac{1}{b}\)
Sum of n unit fractions
Rule 3
\(\frac{1}{b} = \frac{1}{b}\)
Unit fraction identity
Key definitions:

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

Unit fraction decomposition: Expressing a fraction as a sum of unit fractions with the same denominator

Denominator: The bottom number in a fraction representing the total number of equal parts

Numerator: The top number in a fraction representing the number of parts taken

Non-unit fraction: A fraction with numerator greater than 1, such as \(\frac{2}{3}\), \(\frac{5}{8}\), etc.

Part-whole relationship: How unit 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 unit fraction: Create the unit fraction with the same denominator
  3. Count the copies: Use the numerator to determine how many unit fractions are needed
  4. Write the sum: Express the original fraction as a sum of unit fractions
  5. Verify the result: Check that adding the unit fractions gives back the original 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: Think of unit fractions as building blocks for all other fractions.
Tip 4: Use visual models to see how unit fractions combine to form larger fractions.
Common errors: Forgetting that unit fractions have numerator 1, not recognizing that the denominator stays the same, miscounting the number of unit fractions needed.
Memory aid: "The top number counts how many unit fractions you need, the bottom number stays the same."
Rules to know by heart:

• Decomposition rule: \(\frac{n}{b} = \frac{1}{b} + \frac{1}{b} + \cdots + \frac{1}{b}\) (n times)

• Scaling rule: \(\frac{n}{b} = n \times \frac{1}{b}\)

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

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

Solution: Exercises 4 to 5
4 Word problem with unit fractions
Exercise 4
Sarah ate \(\frac{4}{6}\) of an apple. Represent this as a sum of unit fractions.
Definition:

Word problems with unit fractions: Real-world scenarios where fractions need to be represented as sums of unit fractions

Original
\(\frac{4}{6}\)
Unit Fraction
\(\frac{1}{6}\)
Sum
4 copies of \(\frac{1}{6}\)
Step 1: Identify the fraction

Sarah ate \(\frac{4}{6}\) of an apple, so the numerator is 4 and the denominator is 6

Step 2: Identify the unit fraction

The unit fraction with denominator 6 is \(\frac{1}{6}\)

Step 3: Count how many unit fractions are needed

Since the numerator is 4, we need 4 copies of \(\frac{1}{6}\)

\(\frac{1}{6}\)
+
\(\frac{1}{6}\)
+
\(\frac{1}{6}\)
+
\(\frac{1}{6}\)
=
\(\frac{4}{6}\)
1/6
1/6
1/6
1/6
\(\frac{4}{6} = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6}\)
\(\frac{4}{6} = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6}\)
Final answer:

\(\frac{4}{6} = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6}\)

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

5 Advanced unit fraction representation
Exercise 5
Represent \(\frac{9}{12}\) as a sum of unit fractions and simplify the result.
Definition:

Advanced decomposition: Representing fractions as sums of unit fractions and then simplifying the result

Original
\(\frac{9}{12}\)
Unit Fraction
\(\frac{1}{12}\)
Sum
9 copies of \(\frac{1}{12}\)
Step 1: Identify the fraction

We have \(\frac{9}{12}\), where the numerator is 9 and the denominator is 12

Step 2: Identify the unit fraction

The unit fraction with denominator 12 is \(\frac{1}{12}\)

Step 3: Count how many unit fractions are needed

Since the numerator is 9, we need 9 copies of \(\frac{1}{12}\)

Step 4: Write the sum

\(\frac{9}{12} = \frac{1}{12} + \frac{1}{12} + \frac{1}{12} + \frac{1}{12} + \frac{1}{12} + \frac{1}{12} + \frac{1}{12} + \frac{1}{12} + \frac{1}{12}\)

Step 5: Simplify the original fraction

\(\frac{9}{12} = \frac{3}{4}\) (dividing numerator and denominator by 3)

\(\frac{1}{12}\)
+
\(\frac{1}{12}\)
+
\(\frac{1}{12}\)
+
\(\frac{1}{12}\)
+
\(\frac{1}{12}\)
+
\(\frac{1}{12}\)
+
\(\frac{1}{12}\)
+
\(\frac{1}{12}\)
+
\(\frac{1}{12}\)
=
\(\frac{9}{12}\)
1/12
1/12
1/12
1/12
1/12
1/12
1/12
1/12
1/12
\(\frac{9}{12} = \frac{1}{12} + \frac{1}{12} + \frac{1}{12} + \frac{1}{12} + \frac{1}{12} + \frac{1}{12} + \frac{1}{12} + \frac{1}{12} + \frac{1}{12}\)
\(\frac{9}{12} = \frac{1}{12} + \frac{1}{12} + \frac{1}{12} + \frac{1}{12} + \frac{1}{12} + \frac{1}{12} + \frac{1}{12} + \frac{1}{12} + \frac{1}{12}\)
Final answer:

\(\frac{9}{12} = \frac{1}{12} + \frac{1}{12} + \frac{1}{12} + \frac{1}{12} + \frac{1}{12} + \frac{1}{12} + \frac{1}{12} + \frac{1}{12} + \frac{1}{12}\)

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

Complete Summary: Represent Fractions as Sums of Unit Fractions
\(\frac{n}{b} = \frac{1}{b} + \frac{1}{b} + \cdots + \frac{1}{b}\)
Basic Decomposition Rule
\(\frac{n}{b} = n \times \frac{1}{b}\)
Scaling Rule
\(\frac{1}{b} = \frac{1}{b}\)
Unit Fraction Identity
Key definitions:

Unit fraction: A fraction with numerator 1, such as \(\frac{1}{2}\), \(\frac{1}{3}\), \(\frac{1}{4}\), etc. It represents one part of a whole divided into equal parts.

Unit fraction decomposition: Expressing a fraction as a sum of unit fractions with the same denominator, showing how the fraction is built from unit parts.

Denominator: The bottom number in a fraction representing the total number of equal parts the whole is divided into.

Numerator: The top number in a fraction representing the number of parts taken from the equal divisions.

Non-unit fraction: A fraction with numerator greater than 1, such as \(\frac{2}{3}\), \(\frac{5}{8}\), \(\frac{7}{10}\), etc.

Part-whole relationship: The fundamental connection between unit fractions and the larger fractions they compose.

Unit fraction representation: Expressing any fraction as a sum of equal unit fractions based on its numerator and denominator.

Complete methodology:
  1. Identify the fraction: Recognize the numerator and denominator of the fraction to decompose
  2. Determine the unit fraction: Create the unit fraction with the same denominator as the original fraction
  3. Count the copies: Use the numerator to determine how many unit fractions are needed
  4. Write the sum: Express the original fraction as a sum of unit fractions
  5. Verify the result: Check that adding the unit fractions gives back the original fraction
  6. Apply to word problems: Use the concept in real-world contexts where fractional amounts are discussed
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 unit fractions combine.
Tip 5: Remember that unit fractions always have numerator 1 - this is their defining characteristic.
Tip 6: When solving word problems, look for fractional amounts that can be broken down into unit fractions.
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, 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 unit fractions combine to form larger fractions.
Real-world connection: Unit fraction decomposition occurs in many practical situations like eating portions over multiple days or sharing items equally.
Rules to know by heart:

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

• Scaling rule: \(\frac{n}{b} = n \times \frac{1}{b}\)

• 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

• Unit fraction identity: Unit fractions can be represented as themselves

• Verification method: Adding all unit fractions should give back the original fraction

Unit Fraction Decomposition

📊
Basic 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
Larger Numerator

\(\frac{7}{8} = 7 \times \frac{1}{8}\)

\(\frac{7}{8} = \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8}\)

7
1
Unit Fraction Identity

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

Single unit fraction equals itself

1
3
Decomposition Process
1
Identify
2
Determine
3
Count
4
Write
5
Verify
Unit Fraction Formation

Unit fractions combine to form all other fractions

Master Unit Fraction Decomposition!

Questions & Answers

Question: Why do we need to represent fractions as sums of unit fractions? Isn't it just making things more complicated?

Answer: Actually, representing fractions as sums of unit fractions makes things easier to understand! It's like learning the alphabet before reading words:

  • Understanding: Unit fractions help you understand what fractions really mean - parts of a whole
  • Operations: Adding fractions becomes clearer when you see them as groups of unit fractions
  • Comparison: It's easier to compare fractions when you think about how many unit fraction parts each has

For example, if you know that \(\frac{3}{5}\) means 3 parts of size \(\frac{1}{5}\) each, it's easier to add: \(\frac{3}{5} + \frac{2}{5} = 3\) parts of \(\frac{1}{5}\) plus 2 parts of \(\frac{1}{5}\) equals 5 parts of \(\frac{1}{5}\), which is \(\frac{5}{5} = 1\).

Unit fractions are the "building blocks" of all fractions!

Question: How do I know how many unit fractions to write when I decompose a fraction like \(\frac{7}{9}\)?

Answer: The numerator (top number) of any fraction tells you exactly how many unit fractions it contains!

  • In \(\frac{7}{9}\), the numerator is 7, so there are 7 unit fractions of \(\frac{1}{9}\)
  • You can write: \(\frac{7}{9} = \frac{1}{9} + \frac{1}{9} + \frac{1}{9} + \frac{1}{9} + \frac{1}{9} + \frac{1}{9} + \frac{1}{9}\)

The denominator (bottom number) tells you what kind of unit fraction it is. In this case, the denominator is 9, so we're working with ninths.

More examples:

  • \(\frac{3}{7}\) needs 3 unit fractions of \(\frac{1}{7}\)
  • \(\frac{12}{15}\) needs 12 unit fractions of \(\frac{1}{15}\)
  • \(\frac{1}{5}\) needs 1 unit fraction of \(\frac{1}{5}\) (since it's already a unit fraction)

Just remember: the top number tells you how many!

Question: What happens if the fraction is bigger than 1, like \(\frac{5}{4}\)? Can I still represent it as unit fractions?

Answer: Yes, absolutely! Even fractions greater than 1 can be represented as sums of unit fractions:

  • \(\frac{5}{4}\) means 5 parts of size \(\frac{1}{4}\) each
  • So: \(\frac{5}{4} = \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4}\)

Even though \(\frac{5}{4}\) is greater than 1 whole (since 5 > 4), it's still made up of 5 unit fractions of \(\frac{1}{4}\).

Think of it like this: if you have 5 pieces of a pizza that was cut into 4 equal pieces, you have more than one whole pizza. You have 1 whole pizza plus \(\frac{1}{4}\) of another pizza.

The rule works for ANY fraction: \(\frac{n}{b} = n\) copies of \(\frac{1}{b}\), regardless of whether the fraction is greater than or less than 1!