Solved Exercises on Adding Rational Numbers in Grade 7

Master adding rational numbers: fractions, decimals, mixed numbers, and their applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Adding Fractions with Same Denominator
Exercise 1
Add: \(\frac{3}{8} + \frac{5}{8}\)
Definition:

Rational Numbers: Numbers that can be expressed as \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b \neq 0\)

Addition method for same denominators:
  1. Keep the common denominator
  2. Add the numerators together
  3. Simplify the fraction if possible
Expression
\(\frac{3}{8} + \frac{5}{8}\)
Same Denominator
\(\frac{3+5}{8}\)
Result
\(\frac{8}{8}\)
Simplified
\(1\)
Step 1: Identify the common denominator

Both fractions have denominator 8

Step 2: Add the numerators

\(3 + 5 = 8\)

Step 3: Write the sum over the common denominator

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

Step 4: Simplify the fraction

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

\(\frac{3}{8} + \frac{5}{8} = 1\)
Final answer:

\(\frac{3}{8} + \frac{5}{8} = 1\)

Applied rules:

• Same Denominator Rule: \(\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}\)

• Simplification: Reduce to lowest terms when possible

2 Adding Fractions with Different Denominators
Exercise 2
Add: \(\frac{2}{3} + \frac{1}{4}\)
Definition:

Least Common Denominator (LCD): The smallest number that is a multiple of all denominators involved

Expression
\(\frac{2}{3} + \frac{1}{4}\)
Find LCD
\(12\)
Convert
\(\frac{8}{12} + \frac{3}{12}\)
Result
\(\frac{11}{12}\)
Step 1: Find the least common denominator (LCD)

Factors of 3: 3, 6, 9, 12, 15...

Factors of 4: 4, 8, 12, 16...

LCD = 12

Step 2: Convert each fraction to have the LCD

\(\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}\)

\(\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}\)

Step 3: Add the fractions with common denominators

\(\frac{8}{12} + \frac{3}{12} = \frac{11}{12}\)

Step 4: Check if simplification is needed

GCD of 11 and 12 is 1, so \(\frac{11}{12}\) is already in simplest form

\(\frac{2}{3} + \frac{1}{4} = \frac{11}{12}\)
Final answer:

\(\frac{2}{3} + \frac{1}{4} = \frac{11}{12}\)

Applied rules:

• Find LCD: Smallest common multiple of denominators

• Equivalent Fractions: Multiply numerator and denominator by the same number

• Add Numerators: With common denominators

3 Adding Mixed Numbers
Exercise 3
Add: \(2\frac{1}{3} + 1\frac{2}{5}\)
Definition:

Mixed Number: A combination of a whole number and a proper fraction, e.g., \(a\frac{b}{c} = a + \frac{b}{c}\)

Mixed Numbers
\(2\frac{1}{3} + 1\frac{2}{5}\)
Convert to Improper
\(\frac{7}{3} + \frac{7}{5}\)
Find LCD
\(15\)
Convert
\(\frac{35}{15} + \frac{21}{15}\)
Result
\(\frac{56}{15}\)
Back to Mixed
\(3\frac{11}{15}\)
Step 1: Convert mixed numbers to improper fractions

\(2\frac{1}{3} = \frac{(2 \times 3) + 1}{3} = \frac{7}{3}\)

\(1\frac{2}{5} = \frac{(1 \times 5) + 2}{5} = \frac{7}{5}\)

Step 2: Find the LCD

Denominators are 3 and 5

LCD = 15

Step 3: Convert to equivalent fractions with LCD

\(\frac{7}{3} = \frac{7 \times 5}{3 \times 5} = \frac{35}{15}\)

\(\frac{7}{5} = \frac{7 \times 3}{5 \times 3} = \frac{21}{15}\)

Step 4: Add the fractions

\(\frac{35}{15} + \frac{21}{15} = \frac{56}{15}\)

Step 5: Convert back to mixed number

\(56 \div 15 = 3\) remainder \(11\)

\(\frac{56}{15} = 3\frac{11}{15}\)

\(2\frac{1}{3} + 1\frac{2}{5} = 3\frac{11}{15}\)
Final answer:

\(2\frac{1}{3} + 1\frac{2}{5} = 3\frac{11}{15}\)

Applied rules:

• Convert Mixed to Improper: \(a\frac{b}{c} = \frac{ac + b}{c}\)

• Find LCD: For unlike denominators

• Convert Back: Divide numerator by denominator for mixed number

Adding Rational Numbers: Concepts and Methods
\(\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}\)
Adding Fractions with Same Denominator
Rule 1
\(\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}\)
Cross-Multiplication Method
Rule 2
\(a\frac{b}{c} = \frac{ac + b}{c}\)
Mixed to Improper
Rule 3
\(\frac{a}{b} + \frac{c}{d} = \frac{a \cdot \frac{LCD}{b} + c \cdot \frac{LCD}{d}}{LCD}\)
LCD Method
Key definitions:

Rational Number: Any number that can be expressed as a fraction \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b \neq 0\)

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

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

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

Complete methodology:
  1. Identify the type of rational numbers: Fractions, decimals, or mixed numbers
  2. Check denominators: Are they the same or different?
  3. Find LCD if needed: For fractions with different denominators
  4. Convert to equivalent fractions: With common denominators
  5. Add numerators: Keep the common denominator
  6. Simplify: Reduce to lowest terms if possible
Tip 1: Always find the LCD before adding fractions with different denominators.
Tip 2: To find LCD, list multiples of each denominator and find the smallest common one.
Tip 3: When converting to equivalent fractions, multiply both numerator and denominator by the same number.
Tip 4: Always check if your final answer can be simplified.
Common errors: Adding numerators and denominators separately, not finding LCD, forgetting to convert mixed numbers properly.
Exam preparation: Practice converting between mixed and improper fractions, master LCD finding, and simplify all answers.
Key rules to remember:

• Same denominators: Add numerators only

• Different denominators: Find LCD first

• Mixed numbers: Convert to improper fractions first

• Always simplify final answers

Solution: Exercises 4 to 5
4 Adding Decimals
Exercise 4
Add: \(2.75 + 1.48\)
Definition:

Decimal Numbers: Another way to represent rational numbers using base-10 place values

Align Decimals
\(\begin{align} & 2.75 \\ + & 1.48 \\ \hline & 4.23 \end{align}\)
Step 1: Align decimal points vertically

\(\begin{align} & 2.75 \\ + & 1.48 \end{align}\)

Step 2: Add digits in each column from right to left

Hundredths: \(5 + 8 = 13\), write 3, carry 1

Tenths: \(7 + 4 + 1 = 12\), write 2, carry 1

Ones: \(2 + 1 + 1 = 4\)

Step 3: Place the decimal point in the answer

Line up with the original decimal points

\(2.75 + 1.48 = 4.23\)
Final answer:

\(2.75 + 1.48 = 4.23\)

Applied rules:

• Align Decimal Points: Essential for accurate addition

• Carry Over: When sum exceeds 9 in any column

• Place Decimal Point: Directly below original points

5 Adding Mixed Forms
Exercise 5
Add: \(\frac{3}{4} + 0.6\)
Definition:

Mixed Forms: Problems containing both fractional and decimal representations of rational numbers

Convert to Same Form
\(\frac{3}{4} + \frac{6}{10}\)
Find LCD
\(20\)
Convert
\(\frac{15}{20} + \frac{12}{20}\)
Result
\(\frac{27}{20}\)
Or Decimal
\(1.35\)
Step 1: Convert to the same form (fractions)

\(0.6 = \frac{6}{10} = \frac{3}{5}\)

Step 2: Find the LCD of \(\frac{3}{4}\) and \(\frac{3}{5}\)

Denominators are 4 and 5

LCD = 20

Step 3: Convert to equivalent fractions

\(\frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20}\)

\(\frac{3}{5} = \frac{3 \times 4}{5 \times 4} = \frac{12}{20}\)

Step 4: Add the fractions

\(\frac{15}{20} + \frac{12}{20} = \frac{27}{20}\)

Step 5: Convert to mixed number or decimal if needed

\(\frac{27}{20} = 1\frac{7}{20} = 1.35\)

\(\frac{3}{4} + 0.6 = \frac{27}{20} = 1.35\)
Final answer:

\(\frac{3}{4} + 0.6 = \frac{27}{20}\) or \(1.35\)

Applied rules:

• Convert to Same Form: Either all fractions or all decimals

• Find LCD: For fraction addition

• Multiple Representations: Answer can be in different forms

Adding Rational Numbers: Comprehensive Guide
\(\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}\)
Cross-Multiplication Method
Key definitions:

Rational Numbers: Numbers that can be expressed as \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q \neq 0\)

Equivalent Fractions: Fractions that represent the same value despite having different numerators and denominators

Least Common Denominator (LCD): The smallest number that is a multiple of all denominators in the problem

Complete methodology:
  1. Analyze the problem: Identify the types of rational numbers involved
  2. Plan the approach: Decide whether to work with fractions or convert to decimals
  3. Execute the conversion: If necessary, convert to a common form
  4. Find LCD if needed: For fraction addition with different denominators
  5. Perform the addition: Following the appropriate method
  6. Simplify the result: Reduce fractions to lowest terms
Tip 1: When adding mixed numbers, you can add whole numbers and fractions separately.
Tip 2: To quickly find LCD, multiply the denominators if they have no common factors.
Tip 3: When adding decimals, annex zeros to make columns align properly.
Tip 4: Always check if your answer is reasonable by estimating.
Common errors: Adding numerators and denominators separately, misaligning decimal points, not finding LCD, incorrect conversions between forms.
Exam preparation: Master all conversion techniques, practice with various problem types, and verify answers by estimation.
Key rules to remember:

• Same denominators: Add numerators only

• Different denominators: Find LCD first

• Mixed numbers: Convert or add parts separately

• Decimals: Align decimal points

• Always simplify final answers

Exercise with Visualization: Fraction Addition on Number Line
Exercise 6: Visualizing Fraction Addition
Consider the addition of fractions on a number line:
\(\frac{1}{4} + \frac{1}{3}\)
Step 1: Convert to equivalent fractions with LCD
Step 2: Visualize the addition on the number line
Step 3: Verify the result

Analysis: The chart shows how \(\frac{1}{4} + \frac{1}{3} = \frac{7}{12}\) visually.

  • Step 1: Find LCD of 4 and 3 → LCD = 12
  • Step 2: Convert fractions → \(\frac{1}{4} = \frac{3}{12}\) and \(\frac{1}{3} = \frac{4}{12}\)
  • Step 3: Add numerators → \(\frac{3}{12} + \frac{4}{12} = \frac{7}{12}\)

Questions & Answers

Question: I'm confused about when to find the LCD. Do I need it for all fraction addition?

Answer: You only need to find the LCD when adding fractions with different denominators. Here's when you do and don't need it:

  • Same denominators: Just add the numerators. Example: \(\frac{2}{5} + \frac{1}{5} = \frac{3}{5}\)
  • Different denominators: Must find LCD first. Example: \(\frac{1}{2} + \frac{1}{3}\) needs LCD of 6

The LCD ensures you're adding like-sized pieces. You can't directly add thirds and fourths because they're different sized pieces of a whole. Converting to twelfths (the LCD) gives you the same sized pieces to add.

Question: How do I quickly find the LCD of two numbers? Is there a shortcut?

Answer: Here are several strategies to find LCD quickly:

  • Check if one is a multiple of the other: For 4 and 12, LCD = 12
  • Prime factorization: Break down each number into prime factors and take the highest power of each prime
  • Multiply denominators: If they share no common factors, just multiply them
  • List multiples: Write out multiples of each denominator until you find a common one

For example, to find LCD of 6 and 8:

  • 6 = 2 × 3
  • 8 = 2³
  • LCD = 2³ × 3 = 24

Question: When adding mixed numbers, can I add the whole numbers and fractions separately?

Answer: Yes, you absolutely can! Adding mixed numbers by combining whole numbers and fractions separately is often easier:

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

  • Add whole numbers: \(2 + 1 = 3\)
  • Add fractions: \(\frac{1}{3} + \frac{2}{3} = \frac{3}{3} = 1\)
  • Total: \(3 + 1 = 4\)

However, be careful when the fraction sum is greater than 1. If you get \(\frac{5}{3}\) as the fraction part, convert it to \(1\frac{2}{3}\) and add the 1 to the whole number sum.

Both methods (separate parts vs. converting to improper fractions) will give the same result.

Question: What should I do when adding fractions and decimals together? Which form should I convert to?

Answer: You can convert either way, but here are some guidelines:

  • Convert to fractions when: The decimal terminates (like 0.5, 0.25) or the fractions have simple denominators
  • Convert to decimals when: The fractions convert to terminating decimals or you're comfortable with decimal arithmetic

For example, with \(\frac{1}{4} + 0.7\):

  • Fraction approach: \(\frac{1}{4} + \frac{7}{10} = \frac{5}{20} + \frac{14}{20} = \frac{19}{20}\)
  • Decimal approach: \(0.25 + 0.7 = 0.95\)

Both approaches are valid. Choose the one that feels more comfortable and leads to simpler calculations.

Question: How can I check if my fraction addition is correct?

Answer: Here are several ways to verify your fraction addition:

  • Estimation: Round fractions to nearby benchmarks (\(\frac{1}{2}, \frac{1}{4}, \frac{3}{4}\)) and estimate the sum
  • Convert to decimals: Calculate decimal equivalents and add them to see if they match
  • Visual representation: Draw models or number lines to verify
  • Work backwards: Subtract one addend from the sum to see if you get the other addend

For example, if you calculated \(\frac{1}{3} + \frac{1}{4} = \frac{7}{12}\):

  • Estimation: \(\frac{1}{3} ≈ 0.33\) and \(\frac{1}{4} = 0.25\), so sum ≈ 0.58
  • \(\frac{7}{12} ≈ 0.583\), which matches our estimate

Always develop a habit of checking your work using at least one verification method.