Solved Exercises on Dividing Rational Numbers in Grade 7

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

Solution: Exercises 1 to 3
1 Fraction ÷ Fraction
Exercise 1
Calculate: \( \frac{3}{4} \div \frac{2}{5} \)
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 division method:
  1. Keep the first fraction
  2. Change division to multiplication
  3. Flip (take reciprocal of) the second fraction
  4. Multiply the fractions
Original Expression
\(\frac{3}{4} \div \frac{2}{5}\)
Keep-Change-Flip
\(\frac{3}{4} \times \frac{5}{2}\)
Multiply Numerators
\(3 \times 5 = 15\)
Multiply Denominators
\(4 \times 2 = 8\)
Result
\(\frac{15}{8}\)
Step 1: Apply "Keep-Change-Flip" method

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

Step 2: Multiply numerators

\(3 \times 5 = 15\)

Step 3: Multiply denominators

\(4 \times 2 = 8\)

Step 4: Write the result

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

Step 5: Convert to mixed number (if needed)

\(\frac{15}{8} = 1\frac{7}{8}\)

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

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

Applied rules:

• Fraction division: \(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\)

• Keep-Change-Flip: Keep first fraction, change division to multiplication, flip second fraction

• Sign rule: Same as integer division

2 Negative Fraction ÷ Positive Fraction
Exercise 2
Calculate: \( -\frac{5}{6} \div \frac{3}{4} \)
Definition:

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

Original Expression
\(-\frac{5}{6} \div \frac{3}{4}\)
Keep-Change-Flip
\(-\frac{5}{6} \times \frac{4}{3}\)
Apply Sign Rule
Negative
Multiply Numerators
\((-5) \times 4 = -20\)
Multiply Denominators
\(6 \times 3 = 18\)
Simplify
\(\frac{-20}{18} = -\frac{10}{9}\)
Step 1: Apply "Keep-Change-Flip" method

\(-\frac{5}{6} \div \frac{3}{4} = -\frac{5}{6} \times \frac{4}{3}\)

Step 2: Apply sign rule

Negative ÷ Positive = Negative

Step 3: Multiply numerators

\((-5) \times 4 = -20\)

Step 4: Multiply denominators

\(6 \times 3 = 18\)

Step 5: Simplify the fraction

\(\frac{-20}{18} = \frac{-20 \div 2}{18 \div 2} = -\frac{10}{9}\)

\( -\frac{5}{6} \div \frac{3}{4} = -\frac{10}{9} \)
Final answer:

\( -\frac{5}{6} \div \frac{3}{4} = -\frac{10}{9} \)

Applied rules:

• Sign rule: Negative ÷ Positive = Negative

• Fraction division: \(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\)

• Simplification: Divide numerator and denominator by GCD

3 Mixed Number ÷ Fraction
Exercise 3
Calculate: \( 2\frac{1}{2} \div \frac{5}{8} \)
Definition:

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

Convert Mixed Number
\(2\frac{1}{2} = \frac{5}{2}\)
Apply Division Method
\(\frac{5}{2} \div \frac{5}{8}\)
Keep-Change-Flip
\(\frac{5}{2} \times \frac{8}{5}\)
Multiply
\(\frac{40}{10} = 4\)
Step 1: Convert mixed number to improper fraction

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

Step 2: Apply "Keep-Change-Flip" method

\(\frac{5}{2} \div \frac{5}{8} = \frac{5}{2} \times \frac{8}{5}\)

Step 3: Multiply the fractions

\(\frac{5}{2} \times \frac{8}{5} = \frac{5 \times 8}{2 \times 5} = \frac{40}{10}\)

Step 4: Simplify the fraction

\(\frac{40}{10} = 4\)

\( 2\frac{1}{2} \div \frac{5}{8} = 4 \)
Final answer:

\( 2\frac{1}{2} \div \frac{5}{8} = 4 \)

Applied rules:

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

• Fraction division: \(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\)

• Simplification: Reduce to lowest terms

Rules and methods, laws,...
\( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} \)
Fraction Division Rule
Fraction ÷ Fraction
\( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} \)
Keep-Change-Flip method
Mixed Number
\( a\frac{b}{c} = \frac{ac + b}{c} \)
Convert to improper fraction first
Sign Rules
\( (+) \div (+) = + \) and \( (-) \div (-) = + \)
\( (+) \div (-) = - \) and \( (-) \div (+) = - \)
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

Reciprocal: The multiplicative inverse of a number

Equivalent fractions: Fractions that represent the same value

Rational number division methods:
  1. Fractions: Use "Keep-Change-Flip" method
  2. Mixed numbers: Convert to improper fractions first
  3. Decimals: Convert to fractions or use long division
  4. Apply sign rules: Determine sign of result
Tip 1: Always use "Keep-Change-Flip" for fraction division
Tip 2: Convert mixed numbers to improper fractions before dividing
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 flip the second fraction, not applying sign rules, not simplifying fractions.
Exam preparation: Practice all forms of rational numbers, memorize sign rules, learn to simplify efficiently.
Formulas to know by heart:

• Fraction division: \(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\)

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

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

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

Solution: Exercises 4 to 5
4 Decimal ÷ Fraction
Exercise 4
Calculate: \( 0.75 \div \frac{3}{8} \)
Definition:

Decimal to fraction: Convert decimal to fraction to divide with fraction

Convert Decimal
\(0.75 = \frac{75}{100} = \frac{3}{4}\)
Apply Division Method
\(\frac{3}{4} \div \frac{3}{8}\)
Keep-Change-Flip
\(\frac{3}{4} \times \frac{8}{3}\)
Multiply
\(\frac{24}{12} = 2\)
Step 1: Convert decimal to fraction

\(0.75 = \frac{75}{100} = \frac{3}{4}\) (divide numerator and denominator by 25)

Step 2: Apply "Keep-Change-Flip" method

\(\frac{3}{4} \div \frac{3}{8} = \frac{3}{4} \times \frac{8}{3}\)

Step 3: Multiply the fractions

\(\frac{3}{4} \times \frac{8}{3} = \frac{3 \times 8}{4 \times 3} = \frac{24}{12}\)

Step 4: Simplify the fraction

\(\frac{24}{12} = 2\)

\( 0.75 \div \frac{3}{8} = 2 \)
Final answer:

\( 0.75 \div \frac{3}{8} = 2 \)

Applied rules:

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

• Fraction division: \(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\)

• Simplification: Reduce to lowest terms if possible

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

Negative ÷ Negative: This always results in a positive number

Convert Mixed Number
\(-1\frac{2}{3} = -\frac{5}{3}\)
Apply Sign Rule
Negative ÷ Negative = Positive
Keep-Change-Flip
\(\frac{4}{5} \times \frac{3}{5}\)
Multiply
\(\frac{12}{25}\)
Step 1: Convert mixed number to improper fraction

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

Step 2: Apply sign rule

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

Step 3: Apply "Keep-Change-Flip" method

\(\frac{4}{5} \div \frac{5}{3} = \frac{4}{5} \times \frac{3}{5}\)

Step 4: Multiply the fractions

\(\frac{4}{5} \times \frac{3}{5} = \frac{4 \times 3}{5 \times 5} = \frac{12}{25}\)

Step 5: Check for simplification

GCD(12, 25) = 1, so \(\frac{12}{25}\) is already in simplest form

\( -\frac{4}{5} \div \left(-1\frac{2}{3}\right) = \frac{12}{25} \)
Final answer:

\( -\frac{4}{5} \div \left(-1\frac{2}{3}\right) = \frac{12}{25} \)

Applied rules:

• Sign rule: Negative ÷ Negative = Positive

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

• Fraction division: \(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\)

Key Concepts: Laws, Methods, Rules, Definitions
\( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} \)
Fundamental Fraction Division 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

Reciprocal: The multiplicative inverse of a number; for \(\frac{a}{b}\), the reciprocal is \(\frac{b}{a}\)

Equivalent fractions: Fractions that represent the same value

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

Complete rational number division 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 quotient based on signs of dividend and divisor
  4. Apply "Keep-Change-Flip": Keep first fraction, change division to multiplication, flip second fraction
  5. Multiply fractions: Multiply numerators together and denominators together
  6. Simplify: Reduce to lowest terms if possible
  7. Convert back: If needed, convert improper fractions to mixed numbers
  8. Verify: Check with multiplication (quotient × divisor = dividend)
Tip 1: Always convert mixed numbers to improper fractions before dividing
Tip 2: Remember "Keep-Change-Flip" for fraction division - this is crucial!
Tip 3: Look for opportunities to cancel common factors before multiplying to simplify calculations
Tip 4: Remember that sign rules for rational numbers are the same as for integers
Common errors: Forgetting to flip the second fraction, not applying sign rules correctly, failing to simplify fractions.
Exam preparation: Practice converting between forms, master sign rules, and practice the "Keep-Change-Flip" method.
Formulas to know by heart:

• Fraction division: \(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\)

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

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

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

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

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

Exercise with Visualization: Rational Number Division
Exercise 6: Division of Different Forms
Compare these division forms:
\( \frac{1}{2} \div \frac{1}{4} \)
\( 0.5 \div 0.25 \)
\( \frac{1}{2} \div 0.25 \)
\( 0.5 \div \frac{1}{4} \)

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

  • \( \frac{1}{2} \div \frac{1}{4} = \frac{1}{2} \times \frac{4}{1} = 2 \)
  • \( 0.5 \div 0.25 = 2 \)
  • \( \frac{1}{2} \div 0.25 = \frac{1}{2} \div \frac{1}{4} = 2 \)
  • \( 0.5 \div \frac{1}{4} = 0.5 \times 4 = 2 \)

Questions & Answers

Question: Why do I need to "flip" the second fraction when dividing? Why can't I just divide the numerators and denominators?

Answer: Division is the inverse operation of multiplication. When you divide by a fraction, you're essentially asking "how many times does this fraction fit into the first number?" Here's why flipping works:

  • Dividing by \(\frac{a}{b}\) is the same as multiplying by its reciprocal \(\frac{b}{a}\)
  • This is because \(\frac{1}{\frac{a}{b}} = \frac{b}{a}\)
  • For example: \(\frac{1}{2} \div \frac{1}{4}\) asks "how many \(\frac{1}{4}\)'s are in \(\frac{1}{2}\)?"
  • The answer is 2, which we get by \(\frac{1}{2} \times \frac{4}{1} = 2\)

Simply dividing numerators and denominators would give an incorrect result!

Question: What's the easiest way to remember the sign rules for dividing 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 division of rational numbers is correct?

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

  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: Multiply your answer by the divisor to get the dividend
  5. Simplification: Ensure fraction is in lowest terms

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

  • Sign check: Positive ÷ Positive = Positive ✓
  • Estimation: \(\frac{3}{4} \approx 0.75\), \(\frac{2}{5} = 0.4\), \(0.75 \div 0.4 \approx 1.875\), \(\frac{15}{8} = 1.875\) ✓
  • Reverse: \(\frac{15}{8} \times \frac{2}{5} = \frac{30}{40} = \frac{3}{4}\) ✓