Question

How to divide a mixed number divided by a fraction using a model?

Answer 1

Dividing a mixed number by a fraction can be visualized using a model. Here’s a step-by-step approach to help you understand the process:

### Step 1: Understand the Mixed Number and the Fraction
A mixed number has a whole part and a fractional part. For example, let's take the mixed number \( 2 \frac{1}{2} \) and the fraction \( \frac{1}{3} \).

### Step 2: Convert the Mixed Number to an Improper Fraction
First, convert the mixed number \( 2 \frac{1}{2} \) into an improper fraction. You do this by multiplying the whole number by the denominator and adding the numerator.

\[
2 \frac{1}{2} = \frac{2 \times 2 + 1}{2} = \frac{5}{2}
\]

### Step 3: Set Up the Division Problem
Now, you express the division of the mixed number (now as an improper fraction) by the fraction:

\[
\frac{5}{2} \div \frac{1}{3}
\]

### Step 4: Convert the Division to Multiplication
To divide by a fraction, you multiply by its reciprocal. The reciprocal of \( \frac{1}{3} \) is \( 3 \):

\[
\frac{5}{2} \div \frac{1}{3} = \frac{5}{2} \times 3
\]

### Step 5: Perform the Multiplication
Now, perform the multiplication:

\[
\frac{5}{2} \times 3 = \frac{5 \times 3}{2 \times 1} = \frac{15}{2}
\]

### Step 6: Convert Back to a Mixed Number
If required, convert \( \frac{15}{2} \) back to a mixed number. This means dividing 15 by 2, which gives you a quotient of 7 and a remainder of 1:

\[
\frac{15}{2} = 7 \frac{1}{2}
\]

### Visualization with a Model
You can use a area model to help visualize the division. If you wanted to divide \( 2 \frac{1}{2} \) by \( \frac{1}{3} \):

1. **Whole Area Model**: Draw a rectangle that represents \( 2 \frac{1}{2} \) (which is 2 and a half squares). Each whole square represents 1 whole, and the extra half square represents \( \frac{1}{2} \).

2. **Partitioning by the Fraction**: Next, to model dividing this by \( \frac{1}{3} \), consider partitioning the whole area into sections that each represent \( \frac{1}{3} \).

3. **Counting the Sections**: Count how many sections of \( \frac{1}{3} \) fit into the area. Each whole will give you 3 sections, and the half will give you 1. Therefore, you will find that \( 2 \frac{1}{2} \div \frac{1}{3} = 7 \frac{1}{2} \).

In summary, to divide a mixed number by a fraction using a model, convert the mixed number to an improper fraction, multiply by the reciprocal of the fraction, and visualize the division to understand the relationship between the parts.