Solved Exercises on Subtracting Rational Numbers in Grade 7

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

Solution: Exercises 1 to 3
1 Subtracting Fractions with Same Denominator
Exercise 1
Subtract: \(\frac{7}{8} - \frac{3}{8}\)
Definition:

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

Subtraction method for same denominators:
  1. Keep the common denominator
  2. Subtract the numerators (minuend - subtrahend)
  3. Simplify the fraction if possible
Expression
\(\frac{7}{8} - \frac{3}{8}\)
Same Denominator
\(\frac{7-3}{8}\)
Result
\(\frac{4}{8}\)
Simplified
\(\frac{1}{2}\)
Step 1: Identify the common denominator

Both fractions have denominator 8

Step 2: Subtract the numerators

\(7 - 3 = 4\)

Step 3: Write the difference over the common denominator

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

Step 4: Simplify the fraction

\(\frac{4}{8} = \frac{4 \div 4}{8 \div 4} = \frac{1}{2}\)

\(\frac{7}{8} - \frac{3}{8} = \frac{1}{2}\)
Final answer:

\(\frac{7}{8} - \frac{3}{8} = \frac{1}{2}\)

Applied rules:

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

• Simplification: Reduce to lowest terms when possible

2 Subtracting Fractions with Different Denominators
Exercise 2
Subtract: \(\frac{3}{4} - \frac{1}{6}\)
Definition:

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

Expression
\(\frac{3}{4} - \frac{1}{6}\)
Find LCD
\(12\)
Convert
\(\frac{9}{12} - \frac{2}{12}\)
Result
\(\frac{7}{12}\)
Step 1: Find the least common denominator (LCD)

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

Factors of 6: 6, 12, 18...

LCD = 12

Step 2: Convert each fraction to have the LCD

\(\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}\)

\(\frac{1}{6} = \frac{1 \times 2}{6 \times 2} = \frac{2}{12}\)

Step 3: Subtract the fractions with common denominators

\(\frac{9}{12} - \frac{2}{12} = \frac{7}{12}\)

Step 4: Check if simplification is needed

GCD of 7 and 12 is 1, so \(\frac{7}{12}\) is already in simplest form

\(\frac{3}{4} - \frac{1}{6} = \frac{7}{12}\)
Final answer:

\(\frac{3}{4} - \frac{1}{6} = \frac{7}{12}\)

Applied rules:

• Find LCD: Smallest common multiple of denominators

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

• Subtract Numerators: With common denominators

3 Subtracting Mixed Numbers
Exercise 3
Subtract: \(3\frac{1}{4} - 1\frac{2}{3}\)
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
\(3\frac{1}{4} - 1\frac{2}{3}\)
Convert to Improper
\(\frac{13}{4} - \frac{5}{3}\)
Find LCD
\(12\)
Convert
\(\frac{39}{12} - \frac{20}{12}\)
Result
\(\frac{19}{12}\)
Back to Mixed
\(1\frac{7}{12}\)
Step 1: Convert mixed numbers to improper fractions

\(3\frac{1}{4} = \frac{(3 \times 4) + 1}{4} = \frac{13}{4}\)

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

Step 2: Find the LCD

Denominators are 4 and 3

LCD = 12

Step 3: Convert to equivalent fractions with LCD

\(\frac{13}{4} = \frac{13 \times 3}{4 \times 3} = \frac{39}{12}\)

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

Step 4: Subtract the fractions

\(\frac{39}{12} - \frac{20}{12} = \frac{19}{12}\)

Step 5: Convert back to mixed number

\(19 \div 12 = 1\) remainder \(7\)

\(\frac{19}{12} = 1\frac{7}{12}\)

\(3\frac{1}{4} - 1\frac{2}{3} = 1\frac{7}{12}\)
Final answer:

\(3\frac{1}{4} - 1\frac{2}{3} = 1\frac{7}{12}\)

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

Subtracting Rational Numbers: Concepts and Methods
\(\frac{a}{c} - \frac{b}{c} = \frac{a-b}{c}\)
Subtracting 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

Minuend: The number being subtracted from

Subtrahend: The number being subtracted

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. Subtract numerators: Keep the common denominator
  6. Simplify: Reduce to lowest terms if possible
Tip 1: Always find the LCD before subtracting 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: Subtracting 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: Subtract numerators only

• Different denominators: Find LCD first

• Mixed numbers: Convert to improper fractions first

• Always simplify final answers

Solution: Exercises 4 to 5
4 Subtracting Decimals
Exercise 4
Subtract: \(5.72 - 2.45\)
Definition:

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

Align Decimals
\(\begin{align} & 5.72 \\ - & 2.45 \\ \hline & 3.27 \end{align}\)
Step 1: Align decimal points vertically

\(\begin{align} & 5.72 \\ - & 2.45 \end{align}\)

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

Hundredths: \(2 - 5\) (need to borrow)

Tenths: \(6 - 4 = 2\) (after borrowing)

Ones: \(5 - 2 = 3\)

Step 3: Place the decimal point in the answer

Line up with the original decimal points

\(5.72 - 2.45 = 3.27\)
Final answer:

\(5.72 - 2.45 = 3.27\)

Applied rules:

• Align Decimal Points: Essential for accurate subtraction

• Borrow When Needed: From higher place values when subtracting

• Place Decimal Point: Directly below original points

5 Subtracting Mixed Forms
Exercise 5
Subtract: \(2.5 - \frac{3}{4}\)
Definition:

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

Convert to Same Form
\(2.5 - 0.75\)
Result
\(1.75\)
Or Fraction
\(\frac{7}{4}\)
Step 1: Convert to the same form (decimals)

\(\frac{3}{4} = 0.75\)

Step 2: Perform the subtraction

\(2.5 - 0.75 = 1.75\)

Step 3: Convert to fraction if needed

\(1.75 = 1\frac{3}{4} = \frac{7}{4}\)

Step 4: Alternative method using fractions

\(2.5 = \frac{5}{2}\)

\(\frac{5}{2} - \frac{3}{4} = \frac{10}{4} - \frac{3}{4} = \frac{7}{4}\)

\(2.5 - \frac{3}{4} = 1.75 = \frac{7}{4}\)
Final answer:

\(2.5 - \frac{3}{4} = 1.75\) or \(\frac{7}{4}\)

Applied rules:

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

• Find LCD: For fraction subtraction

• Multiple Representations: Answer can be in different forms

Subtracting 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

Minuend: The number from which another number (the subtrahend) is to be subtracted

Subtrahend: The number that is to be subtracted from the minuend

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 subtraction with different denominators
  5. Perform the subtraction: Following the appropriate method
  6. Simplify the result: Reduce fractions to lowest terms
Tip 1: When subtracting mixed numbers, you can subtract whole numbers and fractions separately if the fraction part of the minuend is larger.
Tip 2: To quickly find LCD, multiply the denominators if they have no common factors.

Tip 3: When subtracting decimals, annex zeros to make columns align properly.

Tip 4: Always check if your answer is reasonable by estimating.

Common errors: Subtracting 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: Subtract numerators only

• Different denominators: Find LCD first

• Mixed numbers: Convert or subtract parts separately

• Decimals: Align decimal points

• Always simplify final answers

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

Analysis: The chart shows how \(\frac{3}{4} - \frac{1}{3} = \frac{5}{12}\) visually.

  • Step 1: Find LCD of 4 and 3 → LCD = 12
  • Step 2: Convert fractions → \(\frac{3}{4} = \frac{9}{12}\) and \(\frac{1}{3} = \frac{4}{12}\)
  • Step 3: Subtract numerators → \(\frac{9}{12} - \frac{4}{12} = \frac{5}{12}\)

Questions & Answers

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

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

  • Same denominators: Just subtract the numerators. Example: \(\frac{5}{7} - \frac{2}{7} = \frac{3}{7}\)
  • Different denominators: Must find LCD first. Example: \(\frac{3}{4} - \frac{1}{3}\) needs LCD of 12

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

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 subtracting mixed numbers, what happens if the fraction part of the minuend is smaller than the fraction part of the subtrahend?

Answer: When the fraction part of the minuend is smaller than the fraction part of the subtrahend, you need to "borrow" from the whole number part:

For example: \(3\frac{1}{4} - 1\frac{3}{4}\)

  • You can't subtract \(\frac{3}{4}\) from \(\frac{1}{4}\)
  • Borrow 1 from the 3, making it 2, and add \(\frac{4}{4}\) to \(\frac{1}{4}\)
  • This gives \(2\frac{5}{4} - 1\frac{3}{4}\)
  • Now subtract: \(2 - 1 = 1\) and \(\frac{5}{4} - \frac{3}{4} = \frac{2}{4} = \frac{1}{2}\)
  • Result: \(1\frac{1}{2}\)

Alternatively, convert both to improper fractions and subtract: \(\frac{13}{4} - \frac{7}{4} = \frac{6}{4} = \frac{3}{2} = 1\frac{1}{2}\)

Question: What should I do when subtracting 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 \(1.25 - \frac{1}{4}\):

  • Fraction approach: \(\frac{5}{4} - \frac{1}{4} = \frac{4}{4} = 1\)
  • Decimal approach: \(1.25 - 0.25 = 1.00\)

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

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

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

  • Estimation: Round fractions to nearby benchmarks (\(\frac{1}{2}, \frac{1}{4}, \frac{3}{4}\)) and estimate the difference
  • Convert to decimals: Calculate decimal equivalents and subtract them to see if they match
  • Visual representation: Draw models or number lines to verify
  • Add back: Add the subtrahend to your result to see if you get the minuend

For example, if you calculated \(\frac{3}{4} - \frac{1}{3} = \frac{5}{12}\):

  • Estimation: \(\frac{3}{4} ≈ 0.75\) and \(\frac{1}{3} ≈ 0.33\), so difference ≈ 0.42
  • \(\frac{5}{12} ≈ 0.417\), which matches our estimate
  • Add back: \(\frac{5}{12} + \frac{1}{3} = \frac{5}{12} + \frac{4}{12} = \frac{9}{12} = \frac{3}{4}\) ✓

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