Solved Exercises on Multiplying Rational Numbers in Grade 7

Master multiplying rational numbers: fractions, decimals, mixed numbers through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Fraction × Fraction
Exercise 1
Calculate: \( \frac{2}{3} \times \frac{4}{5} \)
Definition:

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

Fraction multiplication method:
  1. Multiply numerators together
  2. Multiply denominators together
  3. Simplify if possible
Original Expression
\(\frac{2}{3} \times \frac{4}{5}\)
Multiply Numerators
\(2 \times 4 = 8\)
Multiply Denominators
\(3 \times 5 = 15\)
Result
\(\frac{8}{15}\)
Step 1: Identify the fractions

We have \(\frac{2}{3}\) and \(\frac{4}{5}\)

Step 2: Multiply numerators

\(2 \times 4 = 8\)

Step 3: Multiply denominators

\(3 \times 5 = 15\)

Step 4: Write the result

\(\frac{8}{15}\)

Step 5: Check for simplification

GCD(8, 15) = 1, so \(\frac{8}{15}\) is already in simplest form

\( \frac{2}{3} \times \frac{4}{5} = \frac{8}{15} \)
Final answer:

\( \frac{2}{3} \times \frac{4}{5} = \frac{8}{15} \)

Applied rules:

• Fraction multiplication: \(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\)

• Simplification: Reduce to lowest terms if possible

• Sign rule: Same as integer multiplication

2 Negative Fraction × Positive Fraction
Exercise 2
Calculate: \( -\frac{3}{4} \times \frac{2}{7} \)
Definition:

Sign rule: When multiplying rational numbers with different signs, the result is negative

Original Expression
\(-\frac{3}{4} \times \frac{2}{7}\)
Multiply Numerators
\((-3) \times 2 = -6\)
Multiply Denominators
\(4 \times 7 = 28\)
Apply Sign Rule
Negative
Simplify
\(\frac{-6}{28} = -\frac{3}{14}\)
Step 1: Identify the signs

One negative \(-\frac{3}{4}\) and one positive \(\frac{2}{7}\)

Step 2: Multiply numerators

\((-3) \times 2 = -6\)

Step 3: Multiply denominators

\(4 \times 7 = 28\)

Step 4: Apply sign rule

Negative × Positive = Negative

Step 5: Simplify the fraction

\(\frac{-6}{28} = \frac{-6 \div 2}{28 \div 2} = -\frac{3}{14}\)

\( -\frac{3}{4} \times \frac{2}{7} = -\frac{3}{14} \)
Final answer:

\( -\frac{3}{4} \times \frac{2}{7} = -\frac{3}{14} \)

Applied rules:

• Sign rule: Negative × Positive = Negative

• Fraction multiplication: \(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\)

• Simplification: Divide numerator and denominator by GCD

3 Mixed Number × Fraction
Exercise 3
Calculate: \( 2\frac{1}{3} \times \frac{3}{4} \)
Definition:

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

Convert Mixed Number
\(2\frac{1}{3} = \frac{7}{3}\)
Multiply Fractions
\(\frac{7}{3} \times \frac{3}{4}\)
Multiply Numerators
\(7 \times 3 = 21\)
Multiply Denominators
\(3 \times 4 = 12\)
Simplify
\(\frac{21}{12} = \frac{7}{4} = 1\frac{3}{4}\)
Step 1: Convert mixed number to improper fraction

\(2\frac{1}{3} = \frac{(2 \times 3) + 1}{3} = \frac{6 + 1}{3} = \frac{7}{3}\)

Step 2: Multiply the fractions

\(\frac{7}{3} \times \frac{3}{4} = \frac{7 \times 3}{3 \times 4} = \frac{21}{12}\)

Step 3: Simplify the fraction

\(\frac{21}{12} = \frac{21 \div 3}{12 \div 3} = \frac{7}{4}\)

Step 4: Convert to mixed number (if needed)

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

\( 2\frac{1}{3} \times \frac{3}{4} = 1\frac{3}{4} \)
Final answer:

\( 2\frac{1}{3} \times \frac{3}{4} = 1\frac{3}{4} \)

Applied rules:

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

• Fraction multiplication: \(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\)

• Simplification: Reduce to lowest terms

Rules and methods, laws,...
\( \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd} \)
Fraction Multiplication
Fraction × Fraction
\( \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd} \)
Multiply numerators and denominators
Mixed Number
\( a\frac{b}{c} = \frac{ac + b}{c} \)
Convert to improper fraction first
Sign Rules
\( (+) \times (+) = + \) and \( (-) \times (-) = + \)
\( (+) \times (-) = - \) and \( (-) \times (+) = - \)
Key definitions:

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

Fraction: A number representing a part of a whole: \(\frac{\text{numerator}}{\text{denominator}}\)

Proper fraction: A fraction where numerator < denominator

Improper fraction: A fraction where numerator ≥ denominator

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

Equivalent fractions: Fractions that represent the same value

Rational number multiplication methods:
  1. Fractions: Multiply numerators together and denominators together
  2. Mixed numbers: Convert to improper fractions first
  3. Decimals: Convert to fractions or multiply as decimals
  4. Apply sign rules: Determine sign of result
Tip 1: Always simplify fractions to lowest terms after multiplication
Tip 2: Convert mixed numbers to improper fractions before multiplying
Tip 3: Remember sign rules: same signs → positive, different signs → negative
Tip 4: Look for opportunities to cancel common factors before multiplying
Common errors: Forgetting to apply sign rules, not simplifying fractions, multiplying denominators incorrectly.
Exam preparation: Practice all forms of rational numbers, memorize sign rules, learn to simplify efficiently.
Formulas to know by heart:

• Fraction multiplication: \(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\)

• Mixed number conversion: \(a\frac{b}{c} = \frac{ac + b}{c}\)

• Sign rules: \((+)\times(+)=(+)\), \((-)\times(-)=(+)\), \((+)\times(-)=(-)\), \((-)\times(+)=(−)\)

• Identity: \(\frac{a}{b} \times 1 = \frac{a}{b}\)

Solution: Exercises 4 to 5
4 Decimal × Fraction
Exercise 4
Calculate: \( 0.6 \times \frac{3}{8} \)
Definition:

Decimal to fraction: Convert decimal to fraction to multiply with fraction

Convert Decimal
\(0.6 = \frac{6}{10} = \frac{3}{5}\)
Multiply Fractions
\(\frac{3}{5} \times \frac{3}{8}\)
Multiply Numerators
\(3 \times 3 = 9\)
Multiply Denominators
\(5 \times 8 = 40\)
Result
\(\frac{9}{40}\)
Step 1: Convert decimal to fraction

\(0.6 = \frac{6}{10} = \frac{3}{5}\) (divide numerator and denominator by 2)

Step 2: Multiply the fractions

\(\frac{3}{5} \times \frac{3}{8} = \frac{3 \times 3}{5 \times 8} = \frac{9}{40}\)

Step 3: Check for simplification

GCD(9, 40) = 1, so \(\frac{9}{40}\) is already in simplest form

Step 4: Write the final answer

\(\frac{9}{40}\)

\( 0.6 \times \frac{3}{8} = \frac{9}{40} \)
Final answer:

\( 0.6 \times \frac{3}{8} = \frac{9}{40} \)

Applied rules:

• Decimal to fraction: Write decimal over appropriate power of 10

• Fraction multiplication: \(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\)

• Simplification: Reduce to lowest terms if possible

5 Negative Mixed Number × Negative Fraction
Exercise 5
Calculate: \( -2\frac{1}{4} \times \left(-\frac{2}{3}\right) \)
Definition:

Negative × Negative: This always results in a positive number

Convert Mixed Number
\(-2\frac{1}{4} = -\frac{9}{4}\)
Apply Sign Rule
Negative × Negative = Positive
Multiply Absolutes
\(\frac{9}{4} \times \frac{2}{3} = \frac{18}{12}\)
Simplify
\(\frac{18}{12} = \frac{3}{2} = 1\frac{1}{2}\)
Step 1: Convert mixed number to improper fraction

\(-2\frac{1}{4} = -\frac{(2 \times 4) + 1}{4} = -\frac{9}{4}\)

Step 2: Apply sign rule

Since both numbers are negative: \((-) \times (-) = (+)\)

Step 3: Multiply the absolute values

\(\frac{9}{4} \times \frac{2}{3} = \frac{9 \times 2}{4 \times 3} = \frac{18}{12}\)

Step 4: Simplify the fraction

\(\frac{18}{12} = \frac{18 \div 6}{12 \div 6} = \frac{3}{2}\)

Step 5: Convert to mixed number

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

\( -2\frac{1}{4} \times \left(-\frac{2}{3}\right) = 1\frac{1}{2} \)
Final answer:

\( -2\frac{1}{4} \times \left(-\frac{2}{3}\right) = 1\frac{1}{2} \)

Applied rules:

• Sign rule: Negative × Negative = Positive

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

• Fraction multiplication: \(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\)

Key Concepts: Laws, Methods, Rules, Definitions
\( \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd} \)
Fundamental Fraction Multiplication Rule
Key definitions:

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

Fraction: A number representing a part of a whole: \(\frac{\text{numerator}}{\text{denominator}}\)

Proper fraction: A fraction where numerator < denominator

Improper fraction: A fraction where numerator ≥ denominator

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

Equivalent fractions: Fractions that represent the same value

Decimal: A number expressed in base 10 using a decimal point

Complete rational number multiplication methodology:
  1. Identify the forms: Determine if numbers are fractions, decimals, or mixed numbers
  2. Convert if necessary: Convert mixed numbers to improper fractions, decimals to fractions
  3. Apply sign rules: Determine the sign of the product based on signs of factors
  4. Multiply fractions: Multiply numerators together and denominators together
  5. Simplify: Reduce to lowest terms if possible
  6. Convert back: If needed, convert improper fractions to mixed numbers
  7. Verify: Check with estimation or alternative method
Tip 1: Always convert mixed numbers to improper fractions before multiplying
Tip 2: Look for opportunities to cancel common factors before multiplying to simplify calculations
Tip 3: Remember that sign rules for rational numbers are the same as for integers
Tip 4: Always simplify fractions to lowest terms for the final answer
Common errors: Forgetting to convert mixed numbers, not applying sign rules correctly, failing to simplify fractions.
Exam preparation: Practice converting between forms, master sign rules, and practice simplification techniques.
Formulas to know by heart:

• Fraction multiplication: \(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\)

• Mixed number conversion: \(a\frac{b}{c} = \frac{ac + b}{c}\)

• Sign rules: \((+)\times(+)=(+)\), \((-)\times(-)=(+)\), \((+)\times(-)=(-)\), \((-)\times(+)=(−)\)

• Decimal to fraction: \(0.d_1d_2...d_n = \frac{d_1d_2...d_n}{10^n}\)

• Identity: \(\frac{a}{b} \times 1 = \frac{a}{b}\)

• Commutative: \(\frac{a}{b} \times \frac{c}{d} = \frac{c}{d} \times \frac{a}{b}\)

Exercise with Visualization: Rational Number Multiplication
Exercise 6: Multiplication of Different Forms
Compare these multiplication forms:
\( \frac{1}{2} \times \frac{3}{4} \)
\( 0.5 \times 0.75 \)
\( \frac{1}{2} \times 0.75 \)
\( 0.5 \times \frac{3}{4} \)

Analysis: The chart shows how different forms of rational numbers multiply to the same result.

  • \( \frac{1}{2} \times \frac{3}{4} = \frac{3}{8} = 0.375 \)
  • \( 0.5 \times 0.75 = 0.375 \)
  • \( \frac{1}{2} \times 0.75 = \frac{3}{8} = 0.375 \)
  • \( 0.5 \times \frac{3}{4} = \frac{3}{8} = 0.375 \)

Questions & Answers

Question: Why do I need to convert mixed numbers to improper fractions before multiplying? Can't I just multiply the whole number parts separately?

Answer: You must convert mixed numbers to improper fractions because multiplication distributes across addition. Here's why:

  • A mixed number like \(2\frac{1}{3}\) actually means \(2 + \frac{1}{3}\), not \(2 \times \frac{1}{3}\)
  • When you multiply \(2\frac{1}{3} \times \frac{3}{4}\), you're really doing \((2 + \frac{1}{3}) \times \frac{3}{4}\)
  • This requires the distributive property: \(2 \times \frac{3}{4} + \frac{1}{3} \times \frac{3}{4}\)
  • Converting to improper fractions first is the correct way to handle this: \(2\frac{1}{3} = \frac{7}{3}\)

If you tried to multiply the whole number and fraction parts separately, you'd get an incorrect result!

Question: What's the easiest way to remember the sign rules for multiplying rational numbers?

Answer: The sign rules for rational numbers are exactly the same as for integers! Here are some memory aids:

  • Same signs → Positive: "Friends of friends are friends" and "Enemies of enemies are friends"
  • Different signs → Negative: "Friends of enemies are enemies" and "Enemies of friends are enemies"
  • Simple rhyme: "Same signs positive, different signs negative"
  • Remember: The sign rules are consistent across all number types!

Sign Rules Summary:

  • Positive × Positive = Positive
  • Negative × Negative = Positive
  • Positive × Negative = Negative
  • Negative × Positive = Negative

Question: How do I know if my multiplication of rational numbers is correct?

Answer: Here are several ways to verify your rational number multiplication:

  1. Sign check: Verify you applied the correct sign rule
  2. Estimation: Round numbers to check if answer is reasonable
  3. Alternative form: Convert to decimals and verify the result
  4. Reverse operation: Divide your answer by one factor to get the other
  5. Simplification: Ensure fraction is in lowest terms

Example verification: For \(\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}\)

  • Sign check: Positive × Positive = Positive ✓
  • Estimation: \(\frac{2}{3} \approx 0.67\), \(\frac{4}{5} = 0.8\), \(0.67 \times 0.8 \approx 0.53\), \(\frac{8}{15} \approx 0.53\) ✓
  • Reverse: \(\frac{8}{15} \div \frac{2}{3} = \frac{8}{15} \times \frac{3}{2} = \frac{24}{30} = \frac{4}{5}\) ✓