Recall that multiplication represents repeated addition. The statements \( 3 + 3 + 3 + 3 = 12 \) and \( 3 \times 4 = 12 \) have the same meaning. When you multiply whole numbers, the product will always be larger than the two whole numbers being multiplied. For example, the product of \( 3 \times 4 \) is \( 12 \), and \( 12 \) is larger than both \( 3 \) and \( 4 \).
When multiplying fractions, the product is often smaller than factors being multiplied.
For Example
Chef Fractiona wants to make some pumpkin pancakes. The pancake recipe makes 30 pancakes and requires \( \frac{4}{5} \) of a cup of pumpkin. Chef Fractiona needs to make only \( \frac{1}{3} \) of the recipe.
How much pumpkin should she use?
To determine how much pumpkin is needed, you have to answer this question: "What is \( \frac{1}{3} \) of \( \frac{4}{5} \)?" Remember that the key word "of" usually means multiplication. Let's multiply \( \frac{1}{3} \) and \( \frac{4}{5} \) using a grid model. Click each row of the accordion below to see how to carry out this process.
Represent the fraction \( \frac{4}{5} \) using a grid model.
The grid is divided into five equal sections. Each section represents \( \frac{1}{5} \).
Since each section represents \( \frac{1}{5} \), shading four sections represents \( \frac{4}{5} \).
A large rectangle that is vertically divided into 5 equal columns. Four of the columns are shaded blue.
Represent \( \frac{1}{3} \) using a grid model.
It is important that the second grid model be the same overall size as the first. It is okay to divide the second model differently. Notice on this model that the divisions go across the large rectangle.
The grid is divided into three equal sections. Each section represents \( \frac{1}{3} \).
Since each section represents \( \frac{1}{3} \), shading one section represents \( \frac{1}{3} \).
A large rectangle that is horizontally divided into 3 equal rows. One row is shaded yellow.
Overlay the first grid model onto the second. The grid model is now complete.
Notice that the grid is now divided into 15 equal sections. This represents the denominator of the product \( \frac{4}{5} \times \frac{1}{3} \).
Four shaded sections on the grid model overlap. These overlapping sections are shaded green. This represents the numerator of the product \( \frac{4}{5} \times \frac{1}{3} \).
The product of \( \frac{4}{5} \times \frac{1}{3} \) is \( \frac{4}{15}. \)
A large rectangle that is vertically divided into 5 equal columns and horizontally divided into 3 equal rows. There are 15 total sections. Of the 15 total sections. 3 are shaded green, 1 is shaded yellow, and 8 are shaded blue. The green shading represents where the blue and yellow shading overlap. The label ⅘ is above the rectangle and the label ⅓ is to the left of the rectangle.
Chef Fractiona needs \( \frac{4}{15} \) of a cup of pumpkin to make her pancakes.
Multiplying fractions using a grid model is helpful for visualizing what the factors and the product look like. Grid models are also helpful for understanding the relative size of each fraction in the multiplication, along with the size of the product, in relation to each other.
Question
In the example on this page you saw that \( \frac{4}{5} \times \frac{1}{3} = \frac{4}{15} \). Is \( \frac{4}{15} \) larger or smaller than \( \frac{4}{5} \)? Explain.
The fraction \( \frac{4}{15} \) is smaller than \( \frac{4}{5} \). You can compare the grid model to see that this is true. The grid model for \( \frac{4}{15} \) is divided into 15 smaller rectangles, and of those, 4 are shaded. The grid model for \( \frac{4}{5} \) has 5 smaller rectangles, and 4 of them are shaded. Although each grid model has 4 shaded rectangles, because one grid model is divided into 15 and the other is divided into 5, the model \( \frac{4}{15} \) is smaller than \( \frac{4}{5}. \)
