Lesson 6 Homework Practice Divide Whole Numbers By Fractions Answers
Lesson 6 Homework Practice Divide Whole Numbers By Fractions Answers
In this lesson, we will learn how to divide whole numbers by fractions using two methods: multiplying by the reciprocal and using visual models. We will also practice solving some word problems involving this operation.
Multiplying by the reciprocal
One way to divide a whole number by a fraction is to multiply the whole number by the reciprocal of the fraction. The reciprocal of a fraction is obtained by swapping the numerator and the denominator. For example, the reciprocal of 1/2 is 2/1, or simply 2.
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To multiply a whole number by a fraction, we multiply the numerators and the denominators separately. For example, to multiply 3 by 2/1, we do:
3 x 2/1 = (3 x 2) / (1 x 1) = 6 / 1 = 6
This is equivalent to dividing 3 by 1/2, as we can see by using the inverse operation:
6 2/1 = (6 x 1) / (2 x 1) = 6 / 2 = 3
Therefore, we can use this method to divide any whole number by any fraction. For example, to divide 12 by 3/4, we first find the reciprocal of 3/4, which is 4/3. Then, we multiply 12 by 4/3:
12 3/4 = 12 x 4/3 = (12 x 4) / (1 x 3) = 48 / 3 = 16
Using visual models
Another way to divide a whole number by a fraction is to use visual models, such as fraction bars or area models. These models help us understand what it means to divide a whole number by a fraction in terms of sharing or partitioning.
For example, to divide 6 by 1/2, we can use fraction bars to represent the whole number and the fraction. We can think of dividing 6 by 1/2 as finding how many halves are in six wholes. To do this, we can split each whole into two equal parts and count how many parts we have in total:
+---+---+ +---+---+ +---+---+ +---+---+ +---+---+ +---+---+ +---+---+ +---+---+ +---+---+ +---+---+ +---+---+ +---+---+ 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2
We can see that there are 12 halves in six wholes, so 6 1/2 = 12.
We can also use an area model to divide a whole number by a fraction. For example, to divide 8 by 1/4, we can draw a rectangle with an area of 8 square units. We can think of dividing 8 by 1/4 as finding how many quarters are in eight units. To do this, we can split the rectangle into four equal rows and count how many rows we have in total:
+---+---+---+---+---+---+---+---+ +---+---+---+---+---+---+---+---+ +---+---+---+---+---+---+---+---+ 8
We can see that there are 4 quarters in eight units, so 8 1/4 = 4.
Word problems
Now that we have learned how to divide whole numbers by fractions using two methods, let's practice solving some word problems involving this operation. We will use the following steps to solve these problems:
Read the problem carefully and identify the given information and the question.
Write an equation using a variable to represent the unknown quantity.
Solve the equation using one of the methods we learned.
Check the solution by plugging it back into the equation.
Write the answer in a complete sentence.
Here are some examples of word problems and how to solve them:
Example 1
A cake recipe calls for 3/4 cup of sugar. How many cakes can you make with 9 cups of sugar?
Solution:
We are given the amount of sugar needed for one cake (3/4 cup) and the total amount of sugar available (9 cups). We want to find how many cakes we can make with this amount of sugar. We can write an equation using a variable x to represent the number of cakes:
(3/4) x = 9
To solve this equation, we can divide both sides by 3/4, which is equivalent to multiplying by the reciprocal of 3/4, which is 4/3:
(3/4) x = 9
(4/3) (3/4) x = (4/3) 9
x = (4 x 9) / (3 x 1)
x = 36 / 3
x = 12
We can check the solution by plugging it back into the equation:
(3/4) x = 9
(3/4) 12 = 9
9 = 9
The solution is correct. Therefore, we can make 12 cakes with 9 cups of sugar. We can write the answer in a complete sentence:
You can make 12 cakes with 9 cups of sugar.
Example 2
A pizza is cut into 8 slices. Each slice is 1/8 of the pizza. How many pizzas are there if there are 32 slices?
Solution:
We are given the number of slices per pizza (8 slices) and the fraction of each slice (1/8). We want to find how many pizzas there are if there are 32 slices. We can write an equation using a variable y to represent the number of pizzas:
(1/8) y = 32
To solve this equation, we can use an area model to divide both sides by (1/8), which is equivalent to multiplying by the reciprocal of (1/8), which is (8/1):
+---------------------------------+
+---------------------------------+ 8
We can see that there are 8 eighths in eight units, so (1/8) y = 32 implies y = 32 x 8. We can multiply 32 by 8 to get the value of y:
(1/8) y = 32
y = 32 x 8
y = 256
We can check the solution by plugging it back into the equation:
(1/8) y = 32
(1/8) 256 = 32
32 = 32
The solution is correct. Therefore, there are 256 pizzas if there are 32 slices. We can write the answer in a complete sentence:
There are 256 pizzas if there are 32 slices.
Example 3
A group of 24 students went on a field trip to a museum. They were divided into equal groups of 3/4 of a class. How many groups were there?
Solution:
We are given the total number of students (24 students) and the size of each group (3/4 of a class). We want to find how many groups there were. We can write an equation using a variable z to represent the number of groups:
(3/4) z = 24
To solve this equation, we can use fraction bars to divide both sides by (3/4), which is equivalent to multiplying by the reciprocal of (3/4), which is (4/3):
+---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ 3/4 3/4 3/4 3/4
We can see that there are 4 groups of 3/4 in twenty-four units, so (3/4) z = 24 implies z = 24 x (4/3). We can multiply 24 by (4/3) to get the value of z:
(3/4) z = 24
z = 24 x (4/3)
z = (24 x 4) / (1 x 3)
z = 96 / 3
z = 32
We can check the solution by plugging it back into the equation:
(3/4) z = 24
(3/4) 32 = 24
24 = 24
The solution is correct. Therefore, there were 32 groups of students. We can write the answer in a complete sentence:
There were 32 groups of students.
Summary
In this lesson, we learned how to divide whole numbers by fractions using two methods: multiplying by the reciprocal and using visual models. We also practiced solving some word problems involving this operation. Here are some key points to remember:
To divide a whole number by a fraction, we can multiply the whole number by the reciprocal of the fraction.
The reciprocal of a fraction is obtained by swapping the numerator and the denominator.
To multiply a whole number by a fraction, we multiply the numerators and the denominators separately.
We can also use visual models, such as fraction bars or area models, to divide a whole number by a fraction.
These models help us understand what it means to divide a whole number by a fraction in terms of sharing or partitioning.
To solve word problems involving this operation, we can follow these steps:
Read the problem carefully and identify the given information and the question.
Write an equation using a variable to represent the unknown quantity.
Solve the equation using one of the methods we learned.
Check the solution by plugging it back into the equation.
Write the answer in a complete sentence.
I hope this article was helpful for you. Thank you for reading!
