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There are different ways to write rational numbers. Each is useful in different situations. It is also possible to convert between these forms of numbers. This lesson will focus on exploring such number forms and converting them into each other.
### Catch-Up and Review

**Here are a few recommended readings before getting started with this lesson.**

Explore

Consider two bars split into into an equal number of parts. Try to determine the fraction that the bars represent.

Notice that the first bar is always fully shaded. This indicates that an improper fraction represents the shown diagram each time. Also, the first bar can be represented by a fraction, whose numerator and denominator are the same. Its value is always $1.$

The fraction on the right-hand side can be represented by a proper fraction. Together, $1$ and a proper fraction form a new way of writing the value of an improper fraction.

Discussion

A mixed number consists of a non-zero integer number and a proper fraction.

$acb whereais an integer,b<c,andc =0 $

$-397 ,-261 ,-1125 ,143 ,272 ,383 $

Consider the graphic representation of different mixed numbers.
Mixed numbers represent the rational numbers between any two integers.

Discussion

Improper fractions and mixed numbers are two different ways of writing numbers that can have the same value. Sometimes, it is useful to convert between them. Consider the following mixed number.
*expand_more*
*expand_more*
*expand_more*
### Extra

Interact With an Applet that Converts Different Mixed Numbers Into Improper Fractions

$592 $

This mixed number can be rewritten as an improper fraction in three steps.
1

Multiply the Integer Part by the Denominator

First, identify the integer part of the mixed number. This is the integer number written before the fraction.

Next, multiply the integer part by the denominator of the fraction. In this case, the denominator of the fraction is $9.$$5×9=45 $

2

Add the Numerator

Add the numerator of the fraction to the number from the previous step. In the given mixed number, the numerator of the fraction is $2.$

$45+2=47 $

This is the numerator of the improper fraction.
$592 =?47 $

3

Write the Denominator

Write the numerator of the fraction part as the denominator of the improper fraction. The denominator in the fraction of the given mixed number is $9.$ Therefore, this is also the denominator of the improper fraction.

$592 =947 $

Submit the values of the integer part, numerator and denominator between $1$ and $15,$ inclusively. Then, the process of converting the mixed number into an improper fraction will be illustrated.

Example

Izabella's older sister Tiffaniqua wanted to encourage her to practice converting between mixed numbers and fractions. She came up with an idea to leave some clues for Tiffaniqua all over their house leading to a hidden present. She told Izabella to start at the door of Izabella's room.
### Hint

### Solution

The numerator of the improper fraction is $13.$
The numerator of the improper fraction is $51.$
### First Question

In the note, the numerator of the first improper fraction is $x$ and the denominator is $y.$
### Second Question

The numerator of the second improper fraction is $z$ and the denominator is $w.$

Izabella sprang up, running to her door she opened it quickly. She found the first clue. It asks her to rewrite two mixed numbers as fractions. Doing that will help her decode the directions to the hidden present.

a

$431 $

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b

$683 $

{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["\\dfrac{51}{8}"]}}

b How many steps should Izabella take in the direction of Tiffaniqua's room? How many steps should Izabella take after turning left? Write each answer in list form.

{"type":"text","form":{"type":"list","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"ordermatters":true,"numinput":2,"listEditable":false,"hideNoSolution":true},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["16","35"]}}

a Multiply the integer part by the denominator of the fractional part and add the numerator to find the numerator of the improper fraction.

b The denominator of the improper fraction is the same as the denominator of the fraction in the mixed number.

c Identify the values of $x,$ $y,$ $z,$ and $w$ by comparing the fractions.

a The first mixed number that should be rewritten is $431 .$ Start by multiplying the $integer$ part by the $denominator$ of the fraction. Then, add the $numerator$ of the fraction. The result is the numerator of the improper fraction.

$?13 $

The denominator of this fraction is the same as the denominator of the fraction in the mixed number. This means that it is $3.$
$431 =313 $

b The second mixed number, $683 ,$ can be rewritten as an improper fraction by following the same method. First, multiply its integer part by the denominator and add the numerator.

$?51 $

The denominator is equal to the denominator of the fraction in the mixed number.
$683 =851 $

c Each question will be answered one at a time.

$yx =313 $

This means that $x=13$ and $y=3.$ The first instruction that Izabella received is the following.
$1.Takex+ysteps toward my room. $

Add the known values of $x$ and $y$ to find how many steps in the direction of Tiffaniqua's room Izabella should take.
$13+3=16steps $

$wz =851 $

This means that $z=51$ and $w=8.$ Now, consider the second instruction.
$2.Turn left and takez−2wsteps. $

Substitute the values of $z$ and $w$ and evaluate the expression.
Izabella should take $35$ steps after turning left.
Discussion

Converting an improper fraction into a mixed number can help to estimate the actual value of the fraction. Consider the following improper fraction.
*expand_more*
*expand_more*
*expand_more*

$421 $

This improper fraction can be rewritten as a mixed number in three steps.
1

Divide the Numerator by the Denominator

Start by dividing the numerator of the improper fraction by the denominator. Note that the quotient must be an integer number.

Here, the result of the division of $21$ by $4$ is the quotient of $5$ with a reminder of $1.$

2

Write the Integer Part

Recall that a mixed number consists of an integer part and a proper fraction. The integer part is equal to the quotient of the improper fraction. In this case, it is $5.$

3

Write the Fraction Part

Now the numerator and denominator of the fraction part will be identified. The numerator is equal to the remainder of the division from the first step. In this case, the reminder is $1$ and becomes the numerator of the fraction part.

Note that the numerator must be less than the denominator since the fraction part of a mixed number is a proper fraction. The denominator is the same as the denominator of the improper fraction. Therefore, its value is $4.$

The numerator is less than the denominator, so the fraction is indeed a proper fraction. Finally, finding the mixed number corresponding to $421 $ is complete.

Example

Izabella completed the instructions from the first clue and she ended up in her dad's home office. Looking through piles of papers, she finally found the second clue in the book shelf.
To find the numbers of steps she needs to take, Izabella needs to rewrite the improper fractions as mixed numbers.
### Hint

### Solution

External credits: @gstudioimagen

a

$587 $

{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["17\\frac{2}{5}"]}}

b

$9115 $

{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["12\\frac{7}{9}"]}}

a Divide the numerator by the denominator by using the long division.

b The quotient of the division of $115$ and $9$ represents the integer part of the corresponding mixed number.

a Izabella needs to rewrite $587 $ as a mixed number. This requires dividing the numerator by the denominator. Long division can be applied here.

The quotient is $17$ with a reminder of $2.$ Recall that the quotient represents the integer part of the corresponding mixed number. The remainder represents the numerator of the fraction in the mixed number. The denominator is the same as the denominator of improper fraction.

$587 =1752 $

The calculations show that Izabella needs to take $17$ steps to the right after leaving the office room.
b This time, Izabella needs to rewrite $9115 $ as an improper fraction. Again, divide $115$ by $9$ using long division.

The quotient is $12$ with a reminder of $7.$ The quotient represents the integer part and the reminder represents the numerator of the fraction in the mixed number. The denominator is equal to the denominator of improper fraction.

$9115 =1297 $

According to the second instruction, Izabella should turn right and take $7$ steps. Where will she end up?
Pop Quiz

Convert the given mixed number into the corresponding improper fraction or the other way around — the given improper fraction into a mixed number. If the improper fraction corresponds to an integer number, leave the fraction fields empty. Do not simplify the fraction part in a mixed number.

Discussion

While mixed numbers help estimate the value of an improper fraction, they are not very convenient in calculations. For that reason, the equivalent number to the fraction of a mixed number can be written after the integer followed by a dot. *decimal numbers*. ## Decimal Numbers

$354 →3.la $

This observation leads to the definition of Concept

Numbers that lie between integers on the number line can be written as decimal numbers. These consist of an integer part, a decimal point as a separator, and a non-zero decimal part written to the right of the decimal point. Consider the number $12.346$ as an example.

The integer part of this number is $12.$ Since there is a decimal part, $0.346,$ the number is greater than $12$ but less than $13.$ Therefore, when plotting $12.346$ on a number line, the point will lie between $12$ and $13.$

Discussion

It is possible to convert a decimal number into a fraction and the other way around. Consider the following decimal number.
*expand_more*
*expand_more*
*expand_more*
### Extra

Interact With an Applet That Converts Decimals to Fractions

$0.56 $

A decimal number can be rewritten as a fraction in three steps.
1

Count the Number of Decimal Places $n$

The number $0.56$ can be read as $56$ hundredths.

There are two decimal places.

2

Write as a Fraction With the Denominator of $10_{n}$

The number has $2$ decimal places. This means that $0.56$ can be written it as a fraction with a numerator of $56$ and with a denominator of $10_{2}.$

$0.56=10_{2}56 ⇓0.56=10056 $

3

Simplify the Fraction

Next, check whether $10056 $ can be simplified. Start by splittting the numerator and denominator into prime factors.

$56100 =2⋅2⋅2⋅7=2⋅2⋅5⋅5 $

The numbers share two common factors. Their product is the GCF of $56$ and $100.$
$GCF(56,100)=2⋅2=4 $

Finally, divide both the numerator and denominator by $4.$
The fractions $2514 $ and $10056 $ are equivalent and they both correspond to the decimal $0.56.$ Submit a decimal number between $1$ and $0.001$ with no more than $3$ digits after the decimal point. Then, the process of converting that decimal into a fraction will be shown.

Example

After successfully finding the second clue and following the instructions, Izabella ended up in the kitchen. She started looking for the third clue.
She found the clue over the oven vent. The instructions said to rewrite a decimal number as a fraction in the simplest form.
### Hint

### Solution

a The number $1.98$ has two decimal places. That means it can be rewritten as a fraction with the numerator of $198$ and the denominator of $100.$

External credits: @upklyak

a

$1.98 $

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b

$0.564 $

{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["\\dfrac{141}{250}"]}}

c How many steps in total should Izabella take?

{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["81"]}}

a The number $1.98$ has two decimal places. That means it can be rewritten as a fraction with the numerator of $198$ and the denominator of $100.$

b Simplify the fraction by dividing the numerator and denominator by their greatest common factor (GCF).

c Evaluate the expressions $a−b$ and $2c−d$ and then add the values.

$1.98=100198 $

Next, the fraction needs to be simplified. Start by splitting the numerator and denominator into prime factors. $198100 =2⋅3⋅3⋅11=2⋅2⋅5⋅5 $

The numbers share only one common factor of $2.$ This is their GCF. Divide both the numerator and denominator by $2$ to simplify the fraction.
The calculations show that the decimal number $1.98$ corresponds to the improper fraction $5099 .$
b The decimal of $0.564$ can be converted into a fraction by following the same method. Since the number $0.564$ is read as

$564$ thousandths,it can be written as $564$ over $1000.$

$0.564=1000564 $

Next, split the numerator and the denominator into prime factors to simplify the fraction.
$5641000 =2⋅2⋅3⋅47=2⋅2⋅2⋅5⋅5⋅5 $

The numbers share two common factors. The products of these factors is the GCF of $564$ and $1000.$
$GCF(564,1000)=2⋅2=4 $

Finally, divide the numerator and denominator by $4$ and simplify the fraction.
This means that $0.564$ is equal to $250141 .$
c First, find the values of $a$ and $b$ by setting the fractions corresponding to $1.98$ equal.

$ba =5099 $

Now, substitute $99$ for $a$ and $50$ for $b$ into the expression $a−b.$
After going downstairs Izabella needs to take $49$ steps. Next, find $c$ and $d$ by setting the fractions corresponding to $0.564$ equal.
$dc =250141 $

Now that the values of $c$ and $d$ are known, substitute them and evaluate the second expression.
Izabella needs to take $32$ more steps. Finally, calculate the sum of the steps that Izabella should take.
$49+32=81steps $

Discussion

It is possible to convert a fraction into a decimal number and the other way around. Consider the following fraction.
### Extra

Rewriting Fractions With the Denominators That Are Powers of $10$

Then, move the decimal point of the numerator to the left the number of times equal to the number of zeros in the denominator. For example, in the case of $107 ,$ there is **one** zero. This indicates that the decimal point of $7$ will be moved **one** place to the left.

$2516 $

Divide the numerator of $16$ by the denominator of $25$ by using the long division to rewrite the fraction as a decimal.
The result is $0.64.$ This is the decimal number that corresponds to the fraction $2516 .$

A fraction can have a denominator that is a power of $10.$ Consider a few examples.

Fraction | $107 $ | $10026 $ | $1000782 $ |
---|

In that case, the procedure of the long division of the numerator by the denominator is not the best way to go. Instead, the fraction can be rewritten directly as a decimal. First, count how many zeros each denominator has.

Fraction | $107 $ | $10026 $ | $1000782 $ |
---|---|---|---|

Number of Zeros | $1$ | $2$ | $3$ |

The rest of the fractions can be rewritten into decimal numbers in a similar manner.

Fraction | $10$ |
---|