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# Division of Fractions

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Division of Fractions

Have you ever been to a bakery? Have you ever had a chance to bake at a bakery?

Marcella is a student in Mr. Carroll’s seventh grade class. Her Uncle Aldo owns a bakery. Uncle Aldo has told the class that they can have some extra bread dough on his baking day. He said that he usually has some extra and if a few students wish to come into the bakery, that they can take the extra dough and make some bread to sell at the bake sale. Mr. Carroll thinks it is a great idea and assigns four students the task of baking bread on a Saturday morning.

On Saturday, Marcella, Juan, Julia and Christopher head to the bakery. Marcella’s Mom drives them over and the students plan on baking all morning. When they arrive and walk in the door, they are overcome with the smell of fresh bread baking.

“Come on in kids. Here is the extra dough that you can bake with,” He says handing a huge bowl of dough to Juan.

Uncle Aldo gives the students $8 \frac{1}{4}$ pounds of dough.

“Each loaf takes three-quarters of a pound. You can use that scale and get to work. I’ll be over here if you have any questions.”

The students look at the bowl and dough and the scale.

“How many loaves can we make from this?” Juan asks picking up a bread pan.

“I’m not sure,” Marcella answers. “I guess we need to do a little math.”

They certainly do, and you will too. Pay attention throughout this Concept and you will know how to solve this problem.

### Guidance

By now you have a pretty solid understanding of how fractions work. You can add, subtract and multiply fractions. Of course, it will also be extremely helpful to learn to divide fractions. There are plenty of real-world situations in which you can use your expertise at dividing fractions.

Dividing fractions is a lot like multiplying fractions. In fact, it’s exactly like multiplying fractions! Remember that when you divide two numbers, one number is the dividend and the other number is the divisor. For example, in the division problem $a \div b, a$ is the dividend and $b$ is the divisor. To divide two fractions, you simply multiply the dividend by the inverted divisor. How do you invert the divisor ? Just flip the fraction over! $\frac{1}{2}$ inverted becomes $\frac{2}{1}, \frac{3}{4}$ inverted becomes $\frac{4}{3}$ . This inverted fraction is also called a reciprocal.

Let’s divide.

$\frac{6}{8} \div \frac{1}{2}$

In this problem, one-half is the divisor. We need to flip the divisor so that we multiply by the reciprocal. Here is a rhyme to help you remember

When dividing fractions, never wonder why

Flip the second and multiply

Now we rewrite the problem as a multiplication problem.

$\frac{6}{8} \cdot \frac{2}{1}$

Next, we multiply across.

$\frac{12}{8}=1 \frac{4}{8}$

Our last step is to simplify the fraction part of the mixed number.

Our answer is $1 \frac{1}{2}$ .

How do we divide mixed numbers?

Just like with multiplying fractions, if you are dividing mixed numbers, you have to first convert the mixed number to an improper fraction. Remember how multiplying by fractions usually gave us a product that was smaller than one of the factors? Well, with dividing you usually get an answer that is larger than the divisor or the dividend. But either way, the rhyme is still going to apply here too.

$4 \frac{1}{3} \div 2 \frac{1}{6}$

First, we convert both of these mixed numbers to improper fractions. Let’s rewrite the problem with these numbers.

$\frac{13}{3} \div \frac{13}{6}$

Now we change this to a multiplication problem by multiplying by the reciprocal.

$\frac{13}{3} \cdot \frac{6}{13}$

Next, we can simplify on the diagonals.

$\xcancel{\frac{13}{3} \cdot \frac{6}{13}} = \frac{1}{1} \cdot \frac{2}{1} = 2$

Now it's time for you to try a few on your own.

#### Example A

$\frac{5}{10} \div \frac{1}{2}$

Solution: $1$

#### Example B

$\frac{6}{8} \div \frac{1}{4}$

Solution: $3$

#### Example C

$6 \frac{1}{4} \div 1 \frac{1}{2}$

Solution: $4 \frac{1}{6}$

Here is the original problem once again.

Marcella is a student in Mr. Carroll’s seventh grade class. Her Uncle Aldo owns a bakery. Uncle Aldo has told the class that they can have some extra bread dough on his baking day. He said that he usually has some extra and if a few students wish to come into the bakery, that they can take the extra dough and make some bread to sell at the bake sale. Mr. Carroll thinks it is a great idea and assigns four students the task of baking bread on a Saturday morning.

On Saturday, Marcella, Juan, Julia and Christopher head to the bakery. Marcella’s Mom drives them over and the students plan on baking all morning. When they arrive and walk in the door, they are overcome with the smell of fresh bread baking.

“Come on in kids. Here is the extra dough that you can bake with,” He says handing a huge bowl of dough to Juan.

Uncle Aldo gives the students $8 \frac{1}{4}$ pounds of dough.

“Each loaf takes three-quarters of a pound. You can use that scale and get to work. I’ll be over here if you have any questions.”

The students look at the bowl and dough and the scale.

“How many loaves can we make from this?” Juan asks picking up a bread pan.

“I’m not sure,” Marcella answers. “I guess we need to do a little math.”

The students want to figure out how many loaves they can make from the extra dough. To do this, we can write this equation.

Pounds of dough $\div$ # of pounds needed per loaf = number of loaves

Now we can fill in the given values.

$8 \frac{1}{4} \div \frac{3}{4}=x$

Next, we convert the mixed number to an improper fraction.

$\frac{25}{4} \div \frac{3}{4}$

Now we can rewrite this as a multiplication problem, simplify and solve it.

$\frac{25}{4} \cdot \frac{4}{3}=\frac{25}{1} \cdot \frac{1}{3}=\frac{25}{3}=8 \frac{1}{3}$

The students can make 8 loaves of bread and they will have $\frac{1}{3}$ of a pound of dough left over.

### Vocabulary

Reciprocal
the flip or inverted form of a fraction
Improper Fraction
a fraction where the numerator is greater than the denominator
Quotient
the answer in a division problem.

### Guided Practice

Here is one for you to try on your own.

Trina is building a sailboat. She needs 8 planks that are $\frac{3}{4}$ foot in length. She has a piece of wood that is $6 \frac{1}{2}$ feet long. How many planks can she cut from this board and does she have enough to make her sailboat?

Let’s assess the information that the problem has given us. We know that we need planks that are $\frac{3}{4}$ foot in length and we know that we need 8 of them. But, we are not really sure, if we have 8 planks because we are working with a piece of wood that is $6 \frac{1}{2}$ feet long. How many $\frac{3}{4}$ foot planks can be obtained from a $6 \frac{1}{2}$ foot long board? That’s a simple division problem. We set the problem up like this.

Total length of wood $\div$ length of plank needed = number of planks

Let’s plug in the values: $6 \frac{1}{2} \div \frac{3}{4} =$ number of planks.

We convert $6 \frac{1}{2}$ to an improper fraction and we set up the division problem as a multiplication problem with inverted divisor.

$\frac{13}{2} \cdot \frac{4}{3}$

We can cancel out the factors of 2 to get a new multiplication problem, which looks like this.

$\frac{13}{1} \cdot \frac{2}{3}$

We get $\frac{26}{3}$ or $8 \frac{2}{3}$ .

Trina can cut $8 \frac{2}{3}$ planks from the piece of wood that she has. She has enough to make her sailboat.

### Practice

Directions: Divide.

1. $\frac{1}{10} \div \frac{3}{4}$

2. $\frac{5}{9} \div \frac{1}{6}$

3. $\frac{1}{5} \div \frac{5}{8}$

4. $\frac{2}{3} \div \frac{1}{4}$

5. $2 \div \frac{1}{2}$

6. $1 \frac{1}{3} \div 3 \frac{1}{8}$

7. $5 \frac{4}{5} \div 1 \frac{1}{5}$

8. $2 \frac{1}{4} \div \frac{1}{7}$

9. $4 \frac{5}{8} \div 2$

10. $\frac{1}{7} \div \frac{1}{6}$

11. $3 \frac{5}{6} \div 1\frac{2}{3}$

12. $6 \frac{1}{5} \div \frac{7}{12}$

13. $\frac{1}{5} \div \frac{7}{12}$

14. $9\frac{1}{5} \div \frac{3}{12}$

15. $4\frac{1}{5} \div \frac{8}{9}$

### Vocabulary Language: English

improper fraction

improper fraction

An improper fraction is a fraction in which the absolute value of the numerator is greater than the absolute value of the denominator.
Quotient

Quotient

The quotient is the result after two amounts have been divided.
reciprocal

reciprocal

The reciprocal of a number is the number you can multiply it by to get one. The reciprocal of 2 is 1/2. It is also called the multiplicative inverse, or just inverse.

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