Learn Dividing Fractions with Mixed Numbers and Reciprocal Concepts

Type	Reciprocal	Example
Natural number $a$	$\frac{1}{a}$	The reciprocal of $2$ is $\frac{1}{2} .$
Integer numbers $a,$ $a \neq = 0$	$\frac{1}{a}$	The reciprocal of $- 6$ is $- \frac{1}{6} .$
Fraction $\frac{a}{b},$ $b \neq = 0$	$\frac{b}{a}$	The reciprocal of $\frac{3}{2}$ is $\frac{2}{3} .$
Decimal $a$	$\frac{1}{a}$	The reciprocal of $0.2$ is $\frac{1}{0 . 2} .$

Type

Reciprocal

Example

a

\frac{1}{a}

The reciprocal of

2

\frac{1}{2} .

Integer numbers

a,

a \neq = 0

\frac{1}{a}

The reciprocal of

- 6

- \frac{1}{6} .

Fraction

\frac{a}{b},

b \neq = 0

\frac{b}{a}

The reciprocal of

\frac{3}{2}

\frac{2}{3} .

Decimal

a

\frac{1}{a}

The reciprocal of

0.2

\frac{1}{0 . 2} .

Dividing by $0$ then becomes impossible.

Complete the sentence with the correct phrase.

The reciprocal of a fraction is a whole number when $.$

We want to complete the given sentence.

The reciprocal of a fraction is a whole number when .

Remember that we switch the numerator and denominator of the number to find its reciprocal. We also know the product of a number and its reciprocal is 1.

Fraction	Reciprocal	Product
a/b	b/a	a/b*b/a=1

We want to know in which cases b a= b. Now, recall that whole numbers are all fractions with a denominator of 1. This means that a must equal 1. \begin{gathered} \dfrac{{\color{#FF0000}{b}}}{{\color{#0000FF}{1}}} & = \quad {\color{#FF0000}{b}} \\ \downarrow & \\ \ ^\text{ whole}_\text{number} & \end{gathered} If we want the reciprocal of the fraction to be a whole number, the fraction must be a fraction with a numerator of 1. This type of fraction is called a unit fraction. \begin{gathered} \dfrac{{\color{#0000FF}{1}}}{{\color{#FF0000}{b}}} & \times& \dfrac{{\color{#FF0000}{b}}}{{\color{#0000FF}{1}}} & = \quad 1 \\ \downarrow & & \downarrow \\ ^{\ \ \text{ unit}}_\text{fraction} & & ^{\, \text{ whole}}_\text{number} \end{gathered} With this in mind, we can now complete the sentence.

The reciprocal of a fraction is a whole number when the numerator of the fraction is 1.

Maya and her father make a table that shows the portions of their budget. The table includes expenses for food, housing, cleaning, and education.

Expense	Portion of Budget
Food	$\frac{3}{7}$
Housing	$\frac{3}{8}$
Cleaning	$\frac{1}{3 5}$
Education	$\frac{1}{1 6}$

How many times more is the expense for food than for cleaning?

How many times more is the expense for food than for housing?

We are given a table of the portions of the budget of Maya's family.

Expense	Portion of Budget
Food	3/7
Housing	3/8
Cleaning	1/35
Education	1/16

We want to find how many times more the expense for food is when compared to the expense for cleaning. We can find it by dividing 37 by 135. 3/7 ÷ 1/35 Recall that dividing fractions is the same as multiplying the first fraction by the reciprocal of the second fraction. 3/7 ÷ 1/35=2/5 * 35/1 The product of two fractions is equal to the product of the numerators over the product of the denominators.

Therefore, the expense for food is 15 times more than the expense for cleaning.

The expense for food is 37 of the budget and the expense for housing is 38 of the budget.

Expense	Portion of Budget
Food	3/7
Housing	3/8

We need to divide 37 by 38 to find how many times more the expense for food is than the expense for housing. Remember that dividing fractions is the same as multiplying by the reciprocal of the second fraction. 3/7 ÷ 3/8=3/7 * 8/3 Let's multiply!

We will rewrite it as a mixed number.

Therefore, the expense for food is 1 17 times more than the expense for housing.

Kevin and Jordan start a training program about javelin throwing. After a few practice sessions, Kevin and Jordan was able to throw a javelin $21$ feet $8$ inches away and $16$ feet $3$ inches away, respectively.

How many times greater is Kevin's throw than Jordan' throw?

We are told about javelin throws made by Kevin and Jordan.

Let's start by writing the given distances in feet. Since 1 inch is 112 foot, 8 inches is 812= 23 of a foot and 3 inches is 312= 14 of a foot. With this in mind, we can rewrite the distances in feet as mixed numbers. 21feet8inches = 21 23ft [0.7em] 16feet3inches = 16 14ft We want to find how many times greater the throw made by Kevin is than the throw made by Jordan. We can find this by dividing 21 23 by 16 14. 21 23 ÷ 16 14 Our first step is to rewrite the mixed numbers as improper fractions to evaluate this quotient.

We can now use the fact that dividing fractions is the same as multiplying the first fraction by the reciprocal of the second fraction. 65/3 ÷ 65/4=65/3 * 4/65 Let's multiply the fractions. Notice that we can simplify 65 when we multiply the fraction.

We found that the throw made by Kevin was 1 13 times greater than the throw made by Jordan.

The diagram shows how Davontay divides two mixed numbers.

Is Davontay correct?

We are given the steps Davontay followed to divide 5 14 and 1 45. Let's take a look at his work.

We want to determine whether these steps are correct. We see that in the first step, Davontay rewrites 1 45 as 1 54. However, 1 54 is not the reciprocal of 1 45. We should first rewrite the mixed numbers as improper fractions to find their reciprocals. So, let's start by rewriting the mixed numbers so that all of them are fractions.

Now, recall that dividing fractions is the same as multiplying the first fraction by the reciprocal of the second fraction. 21/4 ÷ 9/5=21/4 * 5/9 We know that the product of two fractions is equal to the product of the numerators over the product of the denominators.

Finally, let's rewrite the answer as a mixed number and find the correct answer.

Therefore, Davontay is not correct and the quotient is equal to 2 1112. This matches with option D.

Izabella uses a model to find the following quotient.

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

She draws the following diagram to model the quotient.

$A diagram modeling the division of the fractions$

Izabella states that the quotient is equal to

2 \frac{1}{4} .

2 \frac{3}{4} \div 1 \frac{1}{4} = 2 \frac{1}{4}

Which statement(s) is correct? Select all that apply.

Let's take a look the quotient. 2 34 ÷ 1 14 This quotient means how many one and one-fourth can fit into two and three-fourths. The blue region in the given diagram represents two and three-fourths.

We can see that at least two parts sized one and one-fourth is contained in the blue region. There is one small part left. We have to express that part as a fraction of 1 14. This is because we are looking for groups of 1 14.

We know that each smaller part represents one-fourth of a unit. Therefore, we need to find a fraction that is equal to 14 when multiplied by 1 14. / * 1 14 = 1/4 Let's rewrite the mixed number as an improper fraction. / * 54 = 1/4 The product of 15 and 54 is 14. Therefore, 14 of a unit is the same as 15 of 1 14. As a result, the given quotient is equal to 2 plus 15, or 2 15. 2 12 ÷ 1 16 = 2 15 ✓ Izabella's answer is incorrect. We cannot say the expression is equal to 2 plus 14, or 2 14. 2 34 ÷ 1 14 = 2 14 * Therefore, the statements III and IV are correct.

We can find the given quotient algebraically. 2 34÷ 1 14 We first rewrite the mixed numbers so that all of the numbers are fractions.

Let's write it as a mixed number.

Divide Fractions

Catch-Up and Review

Numbers With a Product of $1$

Reciprocals

Dividing Fractions

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Hint

Solution

Alternative Solution

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Hint

Solution

Dividing Fractions

Dividing Mixed Numbers

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Solution

Comparing the Time Spent on Completing the Box

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Solution

Dividing Mixed Numbers

Is It Possible to Divide by Zero?

Extra

Divide Fractions

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