Subtracting Mixed Numbers with Regrouping
Subtracting mixed numbers involves two key steps in Grade 6 Saxon Math Course 1: first convert fractions to a common denominator, then regroup (borrow 1 from the whole number) if the top fraction is smaller. For 6¼ − 3¾, both fractions are already in fourths; since 1/4 < 3/4, regroup 6¼ to 5⁵⁄₄ and subtract: 5⁵⁄₄ − 3¾ = 2²⁄₄ = 2½. Completing both steps in sequence prevents the most common errors students make in mixed-number subtraction.
Key Concepts
Property To subtract mixed numbers, first rewrite the fractions to have common denominators. If the top fraction is smaller than the bottom fraction, you must regroup by borrowing 1 from the whole number and adding it to the fraction part.
Examples $5\frac{1}{2} 1\frac{2}{3} \rightarrow 5\frac{3}{6} 1\frac{4}{6} \rightarrow 4\frac{9}{6} 1\frac{4}{6} = 3\frac{5}{6}$ $6\frac{1}{4} 2\frac{3}{4} \rightarrow 5\frac{5}{4} 2\frac{3}{4} = 3\frac{2}{4} = 3\frac{1}{2}$ $7\frac{1}{3} 3\frac{5}{6} \rightarrow 7\frac{2}{6} 3\frac{5}{6} \rightarrow 6\frac{8}{6} 3\frac{5}{6} = 3\frac{3}{6} = 3\frac{1}{2}$.
Explanation Think of it as a pizza problem! You can't give away more slices than you have. So, you trade one whole pizza for more slices, making sure all slices are the same size (common denominators). Now you have a big pile of slices, making it easy to subtract and see what's left for a midnight snack.
Common Questions
What two steps are always needed for mixed-number subtraction?
Step 1: Find a common denominator. Step 2: Regroup if the top fraction is smaller than the bottom fraction.
Solve: 8⅓ − 4⅔
Both are thirds. Since 1/3 < 2/3, regroup: 8⅓ = 7⁴⁄₃. Then 7⁴⁄₃ − 4⅔ = 3²⁄₃.
What does borrowing 1 from the whole number give you as a fraction?
It becomes a fraction equal to 1 written with the common denominator. For thirds: 1 = 3/3; for fifths: 1 = 5/5.
Must you find the common denominator before checking if regrouping is needed?
Yes. Always establish the common denominator first; then compare the new numerators to determine whether to regroup.
What error occurs if you skip regrouping?
You subtract a larger fraction from a smaller one, producing a negative fractional part in the answer, which is wrong for this problem type.