5.11: Fundamentals 11A - Fractions–Multiplying and Dividing
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By the end of this lesson, you should understand
- multiplying and dividing fractions does not require a common denominator
- when you multiply a number by a fraction that is less than 1, the result is smaller than the original value
By the end of this lesson, you should be able to
- multiply fractions
- cancel common factors to simplify fractions before multiplying.
- divide fractions by multiplying by the reciprocal.
FUNDAMENTALS OF THE LESSON
A fraction is a quantity that is made up of two parts called the numerator and denominator. The denominator is the bottom number of the fraction and it represents how many equal parts your item was split into. The numerator is the top number of the fraction and it represents how many of those equal parts you are concerned with (for example, how many of the pieces someone ate, or how many of the pieces are colored in).
Sharing a Sweet Treat
Imagine that you are holding a chocolate bar in your hand. A friend walks up to you, so you decide to share the chocolate bar by splitting it into two equal parts. She walks away with her portion of the chocolate bar. Then, before you have eaten any of your portion, a second friend walks up to you. You split what you have left into two equal parts and give him one.
1. Draw a picture to show how the chocolate bar has been split among you and your two friends.
2. How can you describe how much of the original chocolate bar each of you ends up with? Can you use fractions in your descriptions? Hint: Use fractions to describe the amount of the bar that you have left and the amounts given to your friends.
3. When we multiply by a fraction (or a decimal or a percent), we can think of the word “of” as where the multiplication sign goes. For example, we can say that your second friend received \(\dfrac{1}{2}\) of \(\dfrac{1}{2}\) of the chocolate bar, since you gave him one half of your half. Now let’s replace the word “of” with a multiplication sign. We will use the dot symbol for multiplication. Write \(\dfrac{1}{2}\) of \(\dfrac{1}{2}\) as a mathematical expression.
The first friend received \(\dfrac{1}{2}\) of the chocolate bar. The second friend received \(\dfrac{1}{4}\) of the chocolate bar, which is \(\dfrac{1}{2}\) of \(\dfrac{1}{2}\). To multiply \(\dfrac{1}{2}\) by \(\dfrac{1}{2}\), we simply multiply across the numerators, and multiply across the denominators. This is shown below.
4. Let’s consider a different scenario. You have a chocolate bar, and you give half of the bar to a friend. A second friend comes along. Before eating any of the remaining bar, you break it into three equal parts, and you give your second friend one of the pieces. What fraction of the original chocolate bar did you give to your second friend?
NEXT STEPS
Passing An Exam
The Pre-Algebra class at Whitmore Community College had an exam at the end of last week. \(\dfrac{2}{3}\) of the students in the class decided to get extra tutoring help before the exam. \(\dfrac{5}{6}\) of those students who got extra help passed the exam. Let’s figure out what fraction of the entire class passed the class after receiving the extra tutoring.
5. The fraction of all students in the class who received extra tutoring is a fraction of a fraction. Fill in the blanks below.
______ of the class received extra tutoring. ______ of those students pass the class. So, ______ of
______ of the class are in this group.
6. Write the multiplication that represents this situation, and find the results. Explain the meaning of the results.
You might notice that the multiplied answer is not in its simplest form. We can find an equivalent fraction by multiplying or dividing the numerator and denominator by the same number. If we can divide the numerator and denominator by a common factor other than 1, then that will help us put the fraction in simplest form. A fraction is in simplest form, or lowest terms, when the only factor that the numerator and denominator have in common is 1.
7. A. What is the greatest factor that the numerator and denominator in the fraction \(\dfrac{10}{18}\) have in common?
B. Divide the top and bottom of the fraction \(\dfrac{10}{18}\) by the greatest common factor. What fraction do you get?
We also could have simplified our fractions before multiplying. Let’s look back at the multiplication problem we were working with:
\(\dfrac{5}{6}\cdot \dfrac{2}{3}\)
Any time we are multiplying two fractions and we can find a common factor between any numerator and any denominator, we can cancel that factor from those two numbers before multiplying.
C. In this example, do you see a numerator and denominator that have a common factor? What is that common factor?
We can cross cancel by crossing out the 2 and 6, as shown below. When we cross cancel, we divide a number in the numerator and a number in the denominator by the same number. Below, both 2 and 6 are divided by 2, resulting in 1 and 3 respectively. Then we multiply.
When we simplify before multiplying, we obtain the same product as when we multiply first and then simplify.
8 Multiply the following fractions. First cross cancel any common factors, then multiply.
A. \(\dfrac{3}{10}\times \dfrac{5}{4} =\)
B. \(\dfrac{4}{9}\times \dfrac{5}{12} =\)
C. \(\dfrac{12}{13}\times \dfrac{1}{18} =\)
FURTHER APPLICATIONS
Dividing fractions is very closely related to multiplying fractions! In fact, whenever we have a division problem, we will rewrite it in a certain way to turn it into a related multiplication problem.
9. Dan is scooping ice cream for the campers in his group at summer camp. He is not sure there will be enough for everyone. He knows that he has 8 cups of ice cream in the container, and each camper is receiving \(\dfrac{1}{2}\) of a cup in his or her bowl. How many campers can he feed?
Remember that when we want to split something into equal parts, that is dividing. In this case we want to take the 8 cups and divide it into portions of \(\dfrac{1}{2}\) of a cup and see how many there are.
A. Try to write this out as a division problem. (Hint: How can you write a whole number such as 8 as a fraction?)
We need to solve 8 divided by ½. To figure out how many times ½ goes into 8, we can first think about how many times ½ goes into other numbers.
- How many times does ½ go into 1? The result is 2. \(1\div \dfrac{1}{2}\) = 2
- How many times does ½ go into 2? The result is 4. \(2\div \dfrac{1}{2}\) = 4
- How many times does ½ go into 3? The result is 6. \(3\div \dfrac{1}{2}\) = 6
Usually when we divide, we obtain a smaller number. However, when we divide by a proper fraction, we obtain a larger number.
B. What is \(8\div \dfrac{1}{2}\)?
We can divide by a fraction by changing the division problem into an equivalent multiplication problem. In order to do that, we must keep three steps in mind:
1. Keep the first number (divided) as it is.
2. Change the division sign to multiplication.
3. Flip or invert the fraction we are dividing by (divisor), so that it is upside-down (This is called the reciprocal of the original fraction).
In the example that we just solved, we can convert the division to multiplication as follows. Notice that the number 8 has been written as the fraction 8/1.
C. Suppose Dan decides to scoop the ice cream into larger servings. He still has 8 cups of ice cream in the container, and now each camper will receive \(\dfrac{2}{3}\) of a cup in his or her bowl. How many campers can he feed?
Questions: Fractions
Multiplying and Dividing
1. Write two expressions that are equivalent to the division: \(7\div \dfrac{1}{8}\)
2. Write two expressions that are equivalent to the division: \(\dfrac{35}{5}\div \dfrac{8}{6}\)
3. Tanya makes \(\dfrac{2}{4}\) of the foul shots she attempts during basketball practice. If she took 36 shots, how many did she make?
4. A pizza store offers 30 different topping choices. If \(\dfrac{2}{5}\) of these choices are non-meat options, how many meat options does the store offer?
5. Janita brings \(\dfrac{3}{5}\) of her birthday cake to school with her. She and her classmates eat \(\dfrac{2}{3}\) of what she brought. What fraction of the original cake did they eat? Write the fraction in simplest form.
6. What fraction of the original cake will Janita be bringing home from school? Hint: Start with the amount that they did not eat, and then determine the total amount she will be bringing home.
7. According to https://www.catster.com/cat-food/how-much-should-i-feed-my-cat, a guideline for feeding is that an 8-lb cat should be given about \(\dfrac{4}{5}\) cups of dry cat food per day. If you have a cat of this size and have a bag of cat food that contains 52 cups of food in it, how many days will that last your cat at this feeding rate?
8. Alex is making muffins and has \(\dfrac{14}{6}\) sticks of butter. Each batch of muffins will take \(\dfrac{2}{6}\) sticks of butter. How many batches of muffins can he make?