Solved Exercises on Addition and Subtraction of Fractions with Like Denominators in Grade 4

Master addition and subtraction of fractions with like denominators: rules for adding and subtracting fractions with same denominators through these 5 detailed exercises with visual representations.

Solution: Exercises 1 to 3
1 Adding fractions with like denominators
Exercise 1
Calculate: \(\frac{2}{7} + \frac{3}{7}\)
Definition:

Like denominators: Fractions that have the same denominator (bottom number), allowing direct addition or subtraction of numerators

Method for adding like denominators:
  1. Keep the same denominator
  2. Add the numerators together
  3. Write the result as a new fraction
Expression
\(\frac{2}{7} + \frac{3}{7}\)
Denominator
Keep 7
Result
\(\frac{5}{7}\)
Step 1: Verify like denominators

Both fractions have denominator 7, so we can add directly

Step 2: Add the numerators

Add numerators: \(2 + 3 = 5\)

Step 3: Keep the denominator

The denominator remains 7

\(\frac{2}{7}\)
+
\(\frac{3}{7}\)
=
\(\frac{5}{7}\)
2/7
First Fraction
+
3/7
Second Fraction
=
5/7
Result
\(\frac{2}{7} + \frac{3}{7} = \frac{2+3}{7} = \frac{5}{7}\)
\(\frac{2}{7} + \frac{3}{7} = \frac{5}{7}\)
Final answer:

\(\frac{2}{7} + \frac{3}{7} = \frac{5}{7}\)

Applied rules:

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

• Like denominators: When denominators are the same, add numerators directly

• Denominator preservation: The denominator remains unchanged in the result

2 Subtracting fractions with like denominators
Exercise 2
Calculate: \(\frac{5}{8} - \frac{2}{8}\)
Definition:

Subtraction with like denominators: When fractions have the same denominator, subtract the numerators while keeping the denominator unchanged

Expression
\(\frac{5}{8} - \frac{2}{8}\)
Denominator
Keep 8
Result
\(\frac{3}{8}\)
Step 1: Verify like denominators

Both fractions have denominator 8, so we can subtract directly

Step 2: Subtract the numerators

Subtract numerators: \(5 - 2 = 3\)

Step 3: Keep the denominator

The denominator remains 8

\(\frac{5}{8}\)
-
\(\frac{2}{8}\)
=
\(\frac{3}{8}\)
5/8
Starting Amount
-
2/8
Amount Removed
=
3/8
Remaining
\(\frac{5}{8} - \frac{2}{8} = \frac{5-2}{8} = \frac{3}{8}\)
\(\frac{5}{8} - \frac{2}{8} = \frac{3}{8}\)
Final answer:

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

Applied rules:

• Subtraction rule: \(\frac{a}{c} - \frac{b}{c} = \frac{a-b}{c}\)

• Like denominators: When denominators are the same, subtract numerators directly

• Denominator preservation: The denominator remains unchanged in the result

3 Mixed operations with like denominators
Exercise 3
Calculate: \(\frac{4}{9} + \frac{3}{9} - \frac{2}{9}\)
Definition:

Mixed operations: Performing both addition and subtraction with fractions that have like denominators, following order of operations from left to right

Expression
\(\frac{4}{9} + \frac{3}{9} - \frac{2}{9}\)
Process
Left to right
Result
\(\frac{5}{9}\)
Step 1: Verify like denominators

All fractions have denominator 9, so we can perform operations directly

Step 2: Perform addition first

\(\frac{4}{9} + \frac{3}{9} = \frac{7}{9}\)

Step 3: Perform subtraction

\(\frac{7}{9} - \frac{2}{9} = \frac{5}{9}\)

\(\frac{4}{9}\)
+
\(\frac{3}{9}\)
-
\(\frac{2}{9}\)
=
\(\frac{5}{9}\)
4/9
Start
+
3/9
Add
-
2/9
Subtract
=
5/9
Final
\(\frac{4}{9} + \frac{3}{9} - \frac{2}{9} = \frac{4+3-2}{9} = \frac{5}{9}\)
\(\frac{4}{9} + \frac{3}{9} - \frac{2}{9} = \frac{5}{9}\)
Final answer:

\(\frac{4}{9} + \frac{3}{9} - \frac{2}{9} = \frac{5}{9}\)

Applied rules:

• Order of operations: Perform operations from left to right

• Like denominators: When denominators are the same, operate on numerators directly

• Denominator preservation: The denominator remains unchanged throughout operations

Rules and methods, laws,...
\(\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}\)
Addition with Like Denominators
\(\frac{a}{c} - \frac{b}{c} = \frac{a-b}{c}\)
Subtraction with Like Denominators
Rule 1
\(\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}\)
Add fractions with same denominator
Rule 2
\(\frac{a}{c} - \frac{b}{c} = \frac{a-b}{c}\)
Subtract fractions with same denominator
Rule 3
\(\frac{a}{c} + \frac{b}{c} - \frac{d}{c} = \frac{a+b-d}{c}\)
Mixed operations with same denominator
Key definitions:

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

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

Like denominators: Fractions that have the same denominator (same-sized parts)

Unlike denominators: Fractions that have different denominators (different-sized parts)

Addition of fractions: Combining fractional amounts to find a total

Subtraction of fractions: Taking away one fractional amount from another

Like denominator operations: Operations that can be performed by operating on numerators only

Complete methodology:
  1. Verify like denominators: Check that all fractions have the same denominator
  2. Identify operation: Determine if adding or subtracting
  3. Operate on numerators: Add or subtract the numerators accordingly
  4. Keep denominator: The denominator remains unchanged in the result
  5. Simplify if needed: Reduce the fraction to lowest terms if possible
Tip 1: Only add or subtract numerators when denominators are the same.
Tip 2: The denominator stays the same in the answer.
Tip 3: Think of fractions as parts of the same whole when denominators match.
Tip 4: For mixed operations, work from left to right.
Common errors: Adding denominators along with numerators, subtracting the wrong way (numerator - denominator), forgetting that denominators must be the same before operating on numerators.
Memory aid: "Bottom stays the same, top changes" - only operate on numerators when denominators match.
Rules to know by heart:

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

• Subtraction rule: \(\frac{a}{c} - \frac{b}{c} = \frac{a-b}{c}\)

• Denominator preservation: When denominators are alike, they remain unchanged

• Like denominator requirement: Operations can only be performed directly when denominators match

Solution: Exercises 4 to 5
4 Word problem with addition
Exercise 4
Tom ate \(\frac{3}{10}\) of a pizza and Jerry ate \(\frac{4}{10}\) of the same pizza. How much of the pizza did they eat together?
Definition:

Word problems with like denominators: Real-world scenarios where fractions with the same denominator need to be added or subtracted

Tom's portion
\(\frac{3}{10}\)
Jerry's portion
\(\frac{4}{10}\)
Total eaten
\(\frac{7}{10}\)
Step 1: Identify the fractions

Tom ate \(\frac{3}{10}\) and Jerry ate \(\frac{4}{10}\) of the same pizza

Step 2: Verify like denominators

Both fractions have denominator 10, so we can add directly

Step 3: Add the numerators

Total: \(\frac{3}{10} + \frac{4}{10} = \frac{7}{10}\)

\(\frac{3}{10}\)
+
\(\frac{4}{10}\)
=
\(\frac{7}{10}\)
Tom: 3/10
Tom's Share
+
Jerry: 4/10
Jerry's Share
=
Total: 7/10
Combined
\(\frac{3}{10} + \frac{4}{10} = \frac{7}{10}\) of the pizza
Tom and Jerry ate \(\frac{7}{10}\) of the pizza together
Final answer:

Tom and Jerry ate \(\frac{7}{10}\) of the pizza together

Applied rules:

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

• Like denominators: When denominators match, add numerators directly

• Real-world application: Fractions with like denominators can represent parts of the same whole

5 Word problem with subtraction
Exercise 5
A baker had \(\frac{9}{12}\) of a cake. She sold \(\frac{5}{12}\) of the cake. How much cake does she have left?
Definition:

Subtraction word problems: Real-world scenarios requiring subtraction of fractions with like denominators to find remaining amounts

Starting amount
\(\frac{9}{12}\)
Amount sold
\(\frac{5}{12}\)
Remaining
\(\frac{4}{12}\)
Step 1: Identify the fractions

Starting with \(\frac{9}{12}\) and selling \(\frac{5}{12}\) of the cake

Step 2: Verify like denominators

Both fractions have denominator 12, so we can subtract directly

Step 3: Subtract the numerators

Remaining: \(\frac{9}{12} - \frac{5}{12} = \frac{4}{12}\)

\(\frac{9}{12}\)
-
\(\frac{5}{12}\)
=
\(\frac{4}{12}\)
Start: 9/12
Initial Amount
-
Sold: 5/12
Amount Sold
=
Left: 4/12
Remaining
\(\frac{9}{12} - \frac{5}{12} = \frac{4}{12}\) of the cake
The baker has \(\frac{4}{12}\) of the cake left
Final answer:

The baker has \(\frac{4}{12}\) of the cake left

Applied rules:

• Subtraction rule: \(\frac{a}{c} - \frac{b}{c} = \frac{a-b}{c}\)

• Like denominators: When denominators match, subtract numerators directly

• Real-world application: Subtraction of like fractions finds remaining amounts

Complete Summary: Addition and Subtraction of Fractions with Like Denominators
\(\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}\)
Addition Rule
\(\frac{a}{c} - \frac{b}{c} = \frac{a-b}{c}\)
Subtraction Rule
\(\frac{a}{c} + \frac{b}{c} - \frac{d}{c} = \frac{a+b-d}{c}\)
Mixed Operations Rule
Key definitions:

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

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

Like denominators: Fractions that have the same denominator, meaning they represent parts of the same-sized whole.

Unlike denominators: Fractions that have different denominators, requiring conversion before operations can be performed.

Addition of fractions: Combining fractional amounts to find a total, possible directly only when denominators match.

Subtraction of fractions: Taking away one fractional amount from another, possible directly only when denominators match.

Like denominator operations: Operations that can be performed by operating on numerators only while keeping the denominator unchanged.

Complete methodology:
  1. Verify like denominators: Check that all fractions involved have the same denominator
  2. Identify operation: Determine if adding or subtracting (or both in mixed operations)
  3. Operate on numerators: Add or subtract the numerators according to the operation
  4. Keep denominator: The denominator remains unchanged in the result
  5. Simplify if needed: Reduce the fraction to lowest terms if possible
  6. Verify the result: Check that the numerator makes sense in the context of the problem
Tip 1: Only add or subtract numerators when denominators are the same - never add denominators.
Tip 2: The denominator stays the same in the answer - it doesn't change.
Tip 3: Think of fractions as parts of the same whole when denominators match - this makes addition/subtraction intuitive.
Tip 4: For mixed operations, work from left to right, just like with regular numbers.
Tip 5: Use visual models like pie charts or rectangles to verify your results.
Tip 6: Remember that you can only perform these operations directly when denominators are alike.
Common errors: Adding denominators along with numerators, subtracting the wrong way (subtracting denominator from numerator), forgetting that denominators must be the same before operating on numerators, subtracting in the wrong order (numerator - denominator instead of minuend - subtrahend).
Memory aids: "Bottom stays the same, top changes" - only operate on numerators when denominators match; "Same bottom, change top" for like denominators.
Visualization: Draw circle models or bar models to see how fractions combine or separate.
Real-world connection: Like denominator operations occur frequently in cooking, construction, and sharing scenarios.
Rules to know by heart:

• Addition rule: \(\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}\) - add numerators, keep denominator

• Subtraction rule: \(\frac{a}{c} - \frac{b}{c} = \frac{a-b}{c}\) - subtract numerators, keep denominator

• Denominator preservation: When denominators are alike, they remain unchanged in the result

• Like denominator requirement: Operations can only be performed directly when denominators match

• Order of operations: For mixed operations, work from left to right

• Verification: The result should make sense in the context of the problem

Fraction Operations Framework

📊
Addition with Like Denominators

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

Add numerators, keep denominator

2
1
2
Subtraction with Like Denominators

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

Subtract numerators, keep denominator

4
4
Mixed Operations

\(\frac{6}{9} + \frac{2}{9} - \frac{3}{9} = \frac{5}{9}\)

Work from left to right

5
4
Operation Process
1
Check
2
Operate
3
Keep
4
Verify
Like Denominator Operations

Operations performed only on numerators when denominators match

Master Like Denominator Operations!

Questions & Answers

Question: Why can't I just add the denominators too? Why do we only add the numerators when the denominators are the same?

Answer: Great question! The denominator tells us how many equal parts the whole is divided into. When we add fractions with like denominators, we're combining parts of the SAME size:

  • \(\frac{2}{7}\) means 2 parts of size 1/7 each
  • \(\frac{3}{7}\) means 3 parts of size 1/7 each
  • When we add them: 2 parts + 3 parts = 5 parts of the same size (1/7)

Think of it like adding apples: 2 apples + 3 apples = 5 apples. The "type" (apples) stays the same, only the count changes.

If we added denominators too: \(\frac{2}{7} + \frac{3}{7} = \frac{5}{14}\), this would mean 5 parts of size 1/14 each, which is actually smaller than our original fractions! That doesn't make sense when we're combining amounts.

The denominator stays the same because we're working with the same-sized pieces.

Question: What happens if the denominators are different? Can I still add or subtract the fractions?

Answer: When denominators are different, you cannot add or subtract directly. You must first convert the fractions to have the same denominator:

  • You need to find a common denominator (usually the Least Common Multiple)
  • Convert each fraction to an equivalent fraction with the common denominator
  • Then you can add or subtract the numerators

Example: \(\frac{1}{2} + \frac{1}{3}\)

  • Find common denominator: LCM of 2 and 3 is 6
  • Convert: \(\frac{1}{2} = \frac{3}{6}\) and \(\frac{1}{3} = \frac{2}{6}\)
  • Add: \(\frac{3}{6} + \frac{2}{6} = \frac{5}{6}\)

This topic (adding/subtracting unlike denominators) comes later in your studies. For now, focus on mastering like denominators!

The key is that you can only add or subtract parts that are the same size (same denominator).

Question: How do I know if I should add or subtract in word problems?

Answer: Look for clue words that indicate the operation:

Addition clues: total, altogether, combined, in all, sum, both, together, increase, more than, plus

  • "Tom ate 1/4 of a pizza and Jerry ate 1/4 of a pizza. How much did they eat altogether?" → ADDITION

Subtraction clues: left, remaining, difference, take away, less than, decreased by, how many more, how much more

  • "I had 3/4 of a cake and ate 1/4 of it. How much cake is left?" → SUBTRACTION

Also consider the logic:

  • Are we combining amounts? → ADDITION
  • Are we finding what remains after taking away? → SUBTRACTION
  • Are we finding the difference between amounts? → SUBTRACTION

Always read the problem carefully and think about what makes sense in the real-world situation described!