Introduction
A bread recipe calls for a cup of flour, but this recipe will make enough food for 4 people.
If we wanted to make only enough for 2 people then we would only need half of the amount of flour.
How do we figure out the exact amount of flour needed?
We will have to multiply the fractions and in order to solve for the amount of flour.
This lesson will take you through the process of multiplying mixed numbers and fractions together.
Representing the Multiplication of Fractions Using Models
We can represent this multiplication of fractions using images.
Multiplying Fractions Together Using Models
We want to multiply by .
Lets create a model to represent the fraction .
First, draw a rectangle and divide it into 4 even rectangles.
Next, shade in 3 of the 4 rectangles.
This model now represents our fraction.
Create a model to represent the fraction .
This time, be sure to divide the rectangle vertically rather than horizontally as we did with .
First, draw a rectangle and divide it into 2 even rectangles.
Next, shade in 1 of the 2 rectangles.
This model now represents our fraction.
To multiply the two fractions together, draw both rectangles over each other.
The overlapping region tells us the value of the numerator.
In this case, we have exactly 3 rectangles in the overlapping region.
The total number of rectangles present tells us the value of the denominator.
There are a total of 8 rectangles present in our model.
This means that the result of multiplying our two fractions together is .
Multiplying Fractions Together Using Models
Let's take a look at an example where our numerators are both greater than 1.
We want to multiply by .
Lets create a model to represent the fraction .
First, draw a rectangle and divide it into 4 even rectangles.
Next, shade in 3 of the 4 rectangles.
This model now represents our fraction.
Create a model to represent the fraction .
This time, be sure to divide the rectangle vertically rather than horizontally as we did with .
First, draw a rectangle and divide it into 5 even rectangles.
Next, shade in 3 of the 5 rectangles.
This model now represents our fraction.
To multiply the two fractions together, draw both rectangles over each other.
The overlapping region tells us the value of the numerator.
In this case, we have exactly 9 rectangles in the overlapping region.
The total number of rectangles present tells us the value of the denominator.
There are a total of 20 rectangles present in our model.
This means that the result of multiplying our two fractions together is .
Multiplying Fractions
Let's take a look at the problem from earlier.
Multiplying Fractions Together
We want to multiply by .
When multiplying fractions, we need to create a new fraction where we multiply the two numerators together and the two denominators together.
Simplify.
Our final answer is that multiplied by is .
Multiplying Mixed Numbers by Fractions Using Models
A lemonade recipe calls for cups of sugar. We are throwing a huge party so we want to triple the amount of lemonade.
To find the right amount of sugar, we will need to multiply the mixed number by 3.
Multiplying Mixed Numbers Using Models
Let's multiply by 3.
We can use these rectangles to represent the cups of sugar.
We have one and onethird cups of sugar so we can represent them using the shaded rectangles.
Let's use our rectangles to represent . Since the denominator of our fraction is 3 we will split our rectangle into 3 equal portions.
For the mixed number , we will have a whole shaded rectangle, and one that is shaded by one third.
We can turn this mixed number into an improper fracion by counting the total shaded portions and placing that number over 3.
Since we have 4 shaded portions, the equivalent improper fraction is .
Since we are multiplying by 3, we can redraw the same image 3 times.
We can combine the partially shaded rectangles into one, since each one is only filled.
We can count the individual shaded rectangles. We have 12 total shaded rectangles and we know our denominator will be 3 since we split our rectangles each into 3 pieces.
We can conclude that times 3 equals .
We can simplify into 4, or visually you can see that we will have 4 completely shaded rectangles.
Multiplying Mixed Numbers
We are building a treehouse that has the dimensions feet width by feet length.
If we wanted to expand each side by times, how do we solve for our new dimensions?
When multiplying mixed numbers, the biggest difference to multiplying fractions is we first have to convert the mixed numbers into improper fractions.
Improper fractions are those where the numerator is greater than the denominator.
Let's take a look at solving the problem above.
Multiplying Mixed Numbers Together
We will need to multiply each of our dimensions ( by ft) by the number since we want to make the dimensions larger.
Before we can multiply these mixed numbers however, we need to convert them into improper fractions.
To convert into an improper fraction, we need to multiply 6 by and add that to .
To convert into an improper fraction, we need to multiply 8 by and add that to .
To convert into an improper fraction, we need to multiply 1 by and add that to .
We can now multiply each of our dimensions to the amount we want to increase it by.
We will multiply our first dimension by .
We will multiply our second dimension by .
We successfully scaled up our dimensions by times.
ft by ft is equivalent to ft by ft.

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