Lesson 48: Subtracting Mixed Numbers with Regrouping, Part 1

New Concept

To subtract mixed numbers when the top fraction is smaller than the bottom one, we must regroup by borrowing 1 from the whole number.

What’s next

This introduces the core idea of regrouping with fractions. Soon, you'll work through examples and practice problems to build speed and accuracy.

Property

When subtracting mixed numbers, if the top fraction is smaller than the bottom fraction, you must regroup. Borrow 1 from the whole number and add it to the fraction. For example: $5\frac{1}{3} = 4 + 1 + \frac{1}{3} = 4 + \frac{3}{3} + \frac{1}{3} = 4\frac{4}{3}$.

Examples

$4\frac{1}{3} - 1\frac{2}{3} \rightarrow 3\frac{4}{3} - 1\frac{2}{3} = 2\frac{1}{3}$
$6\frac{1}{4} - 2\frac{3}{4} \rightarrow 5\frac{5}{4} - 2\frac{3}{4} = 3\frac{2}{4} = 3\frac{1}{2}$
$7\frac{3}{12} - 4\frac{10}{12} \rightarrow 6\frac{15}{12} - 4\frac{10}{12} = 2\frac{5}{12}$

Explanation

Think of it like trading money. If you have five dollars and one dime but need to pay for something that costs two dollars and two dimes, you can't! So, you trade one of your dollars for ten dimes. Similarly, when you can't subtract fractions, you 'trade' one whole for its fractional parts, giving you enough pieces to subtract.

Property

To subtract a mixed number from a whole number, you must first regroup. Borrow 1 from the whole number and rewrite it as a fraction with the same denominator as the number you are subtracting. For example, to solve $12 - 5\frac{2}{3}$, you first rewrite 12 as $11\frac{3}{3}$.

Examples

$12 - 5\frac{2}{3} \rightarrow 11\frac{3}{3} - 5\frac{2}{3} = 6\frac{1}{3}$
$8 - 3\frac{1}{4} \rightarrow 7\frac{4}{4} - 3\frac{1}{4} = 4\frac{3}{4}$
$10 - 2\frac{4}{5} \rightarrow 9\frac{5}{5} - 2\frac{4}{5} = 7\frac{1}{5}$

Explanation

Imagine you have 12 whole pizzas and need to give away $5\frac{2}{3}$ of them. You can't just subtract the fractional part from nothing! So, you open one of the 12 pizza boxes and slice it into thirds. Now you have 11 whole pizzas and three fresh slices, making it super easy to hand over the required amount.

Property

Visual models, like pies or circles, can illustrate the process of regrouping in mixed number subtraction. A whole unit is 'sliced' into fractional parts to enable subtraction. For instance, the number $4\frac{1}{6}$ can be shown as $\bigcirc \bigcirc \bigcirc \bigcirc \frac{1}{6}$, which is then regrouped to $\bigcirc \bigcirc \bigcirc \frac{7}{6}$.

Examples

To solve $3\frac{1}{4} - 1\frac{3}{4}$, picture 3 circles and a quarter slice. Convert one circle to four quarters, making it $2\frac{5}{4}$.
$4\frac{1}{6}$ is shown as $\bigcirc \bigcirc \bigcirc \bigcirc \frac{1}{6}$. Regroup by 'slicing' a whole to get $\bigcirc \bigcirc \bigcirc \frac{7}{6}$ before subtracting.
For $2 - \frac{1}{3}$, draw two rectangles. Then, divide one rectangle into three parts to show $1\frac{3}{3}$, making it easy to remove $\frac{1}{3}$.

Explanation

Don't just trust the numbers, see it for yourself! When math looks tricky, drawing it out can make everything click. Turning a whole pie into slices is the same as borrowing from a whole number in an equation. This method helps you understand what's really happening when you regroup, turning an abstract problem into a delicious, solvable puzzle.