\(3\frac{4}{7} - 1\frac{2}{7}\)
Mixed numbers: Numbers that contain a whole number and a fraction
Like denominators: Fractions with the same bottom number (denominator)
- Separate whole numbers and fractions
- Subtract whole numbers first
- Subtract fractions with same denominator
- Combine results
- Simplify if necessary
\(3\frac{4}{7} = 3 + \frac{4}{7}\) and \(1\frac{2}{7} = 1 + \frac{2}{7}\)
\(3 - 1 = 2\)
\(\frac{4}{7} - \frac{2}{7} = \frac{4-2}{7} = \frac{2}{7}\)
\(2 + \frac{2}{7} = 2\frac{2}{7}\)
\(3\frac{4}{7} - 1\frac{2}{7} = 2\frac{2}{7}\)
• Separation Rule: \(a\frac{b}{c} = a + \frac{b}{c}\)
• Like Denominator Subtraction: \(\frac{a}{c} - \frac{b}{c} = \frac{a-b}{c}\)
• Combination: Add whole number and fraction back together
\(4\frac{1}{5} - 2\frac{3}{5}\)
Borrowing: Converting 1 whole into fractional parts when the fraction being subtracted is larger than the fraction being subtracted from
\(\frac{1}{5} < \frac{3}{5}\), so we need to borrow from the whole number
Take 1 from 4, so 4 becomes 3
Add 5 (the denominator) to the numerator: \(1 + 5 = 6\)
\(4\frac{1}{5} = 3\frac{6}{5}\)
Whole numbers: \(3 - 2 = 1\)
Fractions: \(\frac{6}{5} - \frac{3}{5} = \frac{3}{5}\)
\(1 + \frac{3}{5} = 1\frac{3}{5}\)
\(4\frac{1}{5} - 2\frac{3}{5} = 1\frac{3}{5}\)
• Borrowing Rule: When \(\frac{a}{c} < \frac{b}{c}\), borrow 1 from whole number and add denominator to numerator
• Conversion: \(n\frac{a}{c} = (n-1)\frac{a+c}{c}\)
• Subtraction: Then subtract normally
\(5\frac{3}{8} - 2\frac{3}{8}\)
Whole number result: When fraction subtraction equals zero, the result is just the whole number difference
\(5\frac{3}{8} = 5 + \frac{3}{8}\) and \(2\frac{3}{8} = 2 + \frac{3}{8}\)
\(5 - 2 = 3\)
\(\frac{3}{8} - \frac{3}{8} = 0\)
\(3 + 0 = 3\)
\(5\frac{3}{8} - 2\frac{3}{8} = 3\)
• Zero Fraction: Any number minus itself equals zero
• Identity: Adding zero to a number doesn't change it
• Whole Number Result: When fractional part equals zero
Mixed Number: A number that combines a whole number and a proper fraction
Like Denominators: Fractions that have the same denominator
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
- Check denominators: Ensure all fractions have the same denominator
- Compare fractions: Check if borrowing is needed
- Subtract whole numbers: Subtract the whole number parts
- Subtract fractions: Subtract the fractional parts
- Combine results: Put whole number and fraction together
- Simplify: Reduce to lowest terms if possible
• Basic subtraction: \(a\frac{b}{c} - d\frac{e}{c} = (a-d) + \frac{b-e}{c}\)
• Borrowing: \(n\frac{a}{c} = (n-1)\frac{a+c}{c}\) when \(a < e\)
• Zero result: \(\frac{a}{c} - \frac{a}{c} = 0\)
• Conversion: \(a\frac{b}{c} = \frac{ac+b}{c}\)
• Reverse: \(\frac{n}{c} = q\frac{r}{c}\) where n = qc + r
Word problems: Real-life scenarios requiring mathematical operations
Left over: The remaining amount after subtraction
Start with: \(6\frac{2}{3}\) cups, Used: \(4\frac{1}{3}\) cups
Both have denominator 3, so they are like denominators
\(6 - 4 = 2\)
\(\frac{2}{3} - \frac{1}{3} = \frac{2-1}{3} = \frac{1}{3}\)
Maria has \(2\frac{1}{3}\) cups of flour left
• Like Denominator Rule: Subtract numerators, keep denominator
• Real-world context: Interpret mathematical result in problem context
• Combination: Put whole number and fraction together
\(7\frac{2}{9} - 3\frac{5}{9}\)
Complex borrowing: When subtracting mixed numbers with like denominators where the first fraction is smaller than the second
\(\frac{2}{9} < \frac{5}{9}\), so we need to borrow from the whole number
Take 1 from 7, so 7 becomes 6
Add 9 (the denominator) to the numerator: \(2 + 9 = 11\)
\(7\frac{2}{9} = 6\frac{11}{9}\)
Whole numbers: \(6 - 3 = 3\)
Fractions: \(\frac{11}{9} - \frac{5}{9} = \frac{6}{9}\)
\(\frac{6}{9} = \frac{2}{3}\) (divide by GCF of 3)
\(3 + \frac{2}{3} = 3\frac{2}{3}\)
\(7\frac{2}{9} - 3\frac{5}{9} = 3\frac{2}{3}\)
• Borrowing Rule: When \(\frac{a}{c} < \frac{b}{c}\), borrow 1 from whole number
• Simplification: Reduce fractions to lowest terms using GCF
• Like Denominator Subtraction: Subtract numerators, keep denominator
Mixed Number: A number that combines a whole number and a proper fraction (e.g., \(2\frac{3}{4}\))
Like Denominators: Fractions that have the same denominator (bottom number)
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
Borrowing: Taking 1 from the whole number and converting it to fractional parts
- Identify like denominators: Check that both mixed numbers have the same denominator
- Compare fractions: Check if the first fraction is smaller than the second (requires borrowing)
- Perform borrowing if needed: Convert 1 whole to fractional parts
- Subtract whole numbers: Subtract the whole number parts
- Subtract fractions: Subtract the fractional parts
- Combine results: Put whole number and fraction together
- Simplify if needed: Reduce to lowest terms
• Basic subtraction: \(a\frac{b}{c} - d\frac{e}{c} = (a-d) + \frac{b-e}{c}\)
• Borrowing: \(n\frac{a}{c} = (n-1)\frac{a+c}{c}\) when \(a < e\)
• Simplification: \(\frac{na}{nb} = \frac{a}{b}\)
• Zero result: \(\frac{a}{c} - \frac{a}{c} = 0\)
• Conversion: \(a\frac{b}{c} = \frac{ac+b}{c}\)
Questions & Answers
Question: When do I need to borrow when subtracting mixed numbers? It seems confusing.
Answer: You need to borrow when the fraction in the first mixed number is smaller than the fraction in the second mixed number:
- Look at the fractions: in \(4\frac{1}{5} - 2\frac{3}{5}\), compare \(\frac{1}{5}\) and \(\frac{3}{5}\)
- Since \(\frac{1}{5} < \frac{3}{5}\), you need to borrow
- Take 1 from the whole number and add it to the fraction: \(4\frac{1}{5} = 3\frac{6}{5}\)
Think of it like this: if you have $4.20 and want to spend $2.80, you can't spend 80 cents from 20 cents, so you "borrow" $1 to make it $3.120 (which is impossible, but you get the idea!).
Remember: \(\frac{1}{5} = \frac{5}{5}\), so borrowing 1 adds 5 to the numerator!
Question: What happens if the fractional parts are the same when subtracting mixed numbers?
Answer: When the fractional parts are the same, they subtract to zero, and you only have the difference of the whole numbers:
- Example: \(5\frac{3}{8} - 2\frac{3}{8}\)
- Subtract whole numbers: \(5 - 2 = 3\)
- Subtract fractions: \(\frac{3}{8} - \frac{3}{8} = 0\)
- Combine: \(3 + 0 = 3\)
Any number minus itself equals zero, so the fractional part disappears, leaving only the whole number difference.
This is a special case where the result is a whole number instead of a mixed number!
Question: How do I know if I need to simplify my answer after subtracting mixed numbers?
Answer: You should always check if your fractional part can be simplified to lowest terms:
- Find the Greatest Common Factor (GCF) of the numerator and denominator
- If the GCF is greater than 1, divide both by it
- If the GCF is 1, the fraction is already in lowest terms
Example: If your answer is \(3\frac{6}{9}\):
- Factors of 6: 1, 2, 3, 6
- Factors of 9: 1, 3, 9
- GCF is 3, so \(\frac{6}{9} = \frac{2}{3}\)
- Final answer: \(3\frac{2}{3}\)
Always simplify your final answer unless specifically told otherwise!
Complete Summary: Subtracting Mixed Numbers with Like Denominators
Mixed Number: A number that combines a whole number and a proper fraction (e.g., \(2\frac{3}{4}\)).
Like Denominators: Fractions that have the same denominator (bottom number).
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.
Borrowing: Taking 1 from the whole number and converting it to fractional parts when the first fraction is smaller than the second.
Basic Rule: \(a\frac{b}{c} - d\frac{e}{c} = (a-d) + \frac{b-e}{c}\)
Borrowing Rule: When \(\frac{b}{c} < \frac{e}{c}\), convert \(n\frac{b}{c} = (n-1)\frac{b+c}{c}\)
Zero Rule: \(\frac{a}{c} - \frac{a}{c} = 0\)
Like Denominator Rule: Only subtract numerators, keep denominator the same.
Simplification Rule: Reduce fractions to lowest terms using GCF.
- Check denominators: Ensure both fractions have the same denominator.
- Compare fractions: Determine if borrowing is needed.
- Perform borrowing if needed: Convert 1 whole to fractional parts.
- Subtract whole numbers: Subtract the whole number parts.
- Subtract fractions: Subtract the fractional parts.
- Combine results: Put whole number and fraction together.
- Simplify if needed: Reduce to lowest terms.
Simple Case: \(3\frac{4}{7} - 1\frac{2}{7} = 2\frac{2}{7}\)
Borrowing Case: \(4\frac{1}{5} - 2\frac{3}{5} = 3\frac{6}{5} - 2\frac{3}{5} = 1\frac{3}{5}\)
Zero Fraction Case: \(5\frac{3}{8} - 2\frac{3}{8} = 3\frac{0}{8} = 3\)
Word Problem: Maria has \(6\frac{2}{3}\) cups of flour and uses \(4\frac{1}{3}\) cups, so she has \(2\frac{1}{3}\) cups left.
Complex Case: \(7\frac{2}{9} - 3\frac{5}{9} = 6\frac{11}{9} - 3\frac{5}{9} = 3\frac{6}{9} = 3\frac{2}{3}\)
Tips: Always check if borrowing is needed first; use visual models to understand; verify your answer by adding back.
Common Mistakes: Forgetting to borrow when needed; subtracting denominators; not simplifying results; confusing addition and subtraction.
Memory Aid: "Subtract wholes, subtract parts, combine together! But check if borrowing is needed first."
Verification: Add your answer to the subtracted number to get the original: \(1\frac{3}{5} + 2\frac{3}{5} = 4\frac{1}{5}\) (after borrowing).
Memorize: The basic subtraction formula and borrowing rule.
Understand: Why borrowing is necessary when the first fraction is smaller.
Practice: Both simple cases and those requiring borrowing.
Apply: To real-world problems involving measurements and quantities.
Check: Always verify your answer makes sense and is in lowest terms.