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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.

## A Different Way of Writing Improper Fractions

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

## Mixed Numbers

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

The following are examples of mixed numbers.
Consider the graphic representation of different mixed numbers. Mixed numbers represent the rational numbers between any two integers. ## Converting a Mixed Number Into an Improper Fraction

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.
This mixed number can be rewritten as an improper fraction in three steps.
1
Multiply the Integer Part by the Denominator
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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
2
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Add the numerator of the fraction to the number from the previous step. In the given mixed number, the numerator of the fraction is
This is the numerator of the improper fraction.
3
Write the Denominator
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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 Therefore, this is also the denominator of the improper fraction.

### Extra

Interact With an Applet that Converts Different Mixed Numbers Into Improper Fractions
Submit the values of the integer part, numerator and denominator between and inclusively. Then, the process of converting the mixed number into an improper fraction will be illustrated. ## Looking for the First Clue

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. 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
b
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.

### Hint

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 and by comparing the fractions.

### Solution

a The first mixed number that should be rewritten is Start by multiplying the part by the of the fraction. Then, add the of the fraction. The result is the numerator of the improper fraction.
The numerator of the improper fraction is
The denominator of this fraction is the same as the denominator of the fraction in the mixed number. This means that it is
b The second mixed number, can be rewritten as an improper fraction by following the same method. First, multiply its integer part by the denominator and add the numerator.
The numerator of the improper fraction is
The denominator is equal to the denominator of the fraction in the mixed number.
c Each question will be answered one at a time.

### First Question

In the note, the numerator of the first improper fraction is and the denominator is
This means that and The first instruction that Izabella received is the following.
Add the known values of and to find how many steps in the direction of Tiffaniqua's room Izabella should take.

### Second Question

The numerator of the second improper fraction is and the denominator is
This means that and Now, consider the second instruction.
Substitute the values of and and evaluate the expression.
Izabella should take steps after turning left.

## Converting an Improper Fraction Into a Mixed Number

Converting an improper fraction into a mixed number can help to estimate the actual value of the fraction. Consider the following improper fraction.
This improper fraction can be rewritten as a mixed number in three steps.
1
Divide the Numerator by the Denominator
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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 by is the quotient of with a reminder of
2
Write the Integer Part
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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 3
Write the Fraction Part
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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 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 The numerator is less than the denominator, so the fraction is indeed a proper fraction. Finally, finding the mixed number corresponding to is complete.

## A Clue Hidden in the Office Room

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. External credits: @gstudioimagen
To find the numbers of steps she needs to take, Izabella needs to rewrite the improper fractions as mixed numbers.
a
b

### Hint

a Divide the numerator by the denominator by using the long division.
b The quotient of the division of and represents the integer part of the corresponding mixed number.

### Solution

a Izabella needs to rewrite as a mixed number. This requires dividing the numerator by the denominator. Long division can be applied here. The quotient is with a reminder of 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.
The calculations show that Izabella needs to take steps to the right after leaving the office room.
b This time, Izabella needs to rewrite as an improper fraction. Again, divide by using long division. The quotient is with a reminder of 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.
According to the second instruction, Izabella should turn right and take steps. Where will she end up?

## Converting Between Mixed Numbers and Improper Fractions

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. ## Introducing Decimal Numbers

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.
This observation leads to the definition of decimal numbers.

## Decimal Numbers

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 as an example. The integer part of this number is Since there is a decimal part, the number is greater than but less than Therefore, when plotting on a number line, the point will lie between and ## Converting a Decimal Number Into a Fraction

It is possible to convert a decimal number into a fraction and the other way around. Consider the following decimal number.
A decimal number can be rewritten as a fraction in three steps.
1
Count the Number of Decimal Places
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The number can be read as hundredths. There are two decimal places. 2
Write as a Fraction With the Denominator of
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The number has decimal places. This means that can be written it as a fraction with a numerator of and with a denominator of
3
Simplify the Fraction
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Next, check whether can be simplified. Start by splittting the numerator and denominator into prime factors.
The numbers share two common factors. Their product is the GCF of and
Finally, divide both the numerator and denominator by
The fractions and are equivalent and they both correspond to the decimal

### Extra

Interact With an Applet That Converts Decimals to Fractions
Submit a decimal number between and with no more than digits after the decimal point. Then, the process of converting that decimal into a fraction will be shown. ## A Clue Among the Kitchen Supplies

After successfully finding the second clue and following the instructions, Izabella ended up in the kitchen. She started looking for the third clue. External credits: @upklyak
She found the clue over the oven vent. The instructions said to rewrite a decimal number as a fraction in the simplest form.
a
b
c How many steps in total should Izabella take?

### Hint

a The number has two decimal places. That means it can be rewritten as a fraction with the numerator of and the denominator of
b Simplify the fraction by dividing the numerator and denominator by their greatest common factor (GCF).
c Evaluate the expressions and and then add the values.

### Solution

a The number has two decimal places. That means it can be rewritten as a fraction with the numerator of and the denominator of
Next, the fraction needs to be simplified. Start by splitting the numerator and denominator into prime factors.
The numbers share only one common factor of This is their GCF. Divide both the numerator and denominator by to simplify the fraction.
The calculations show that the decimal number corresponds to the improper fraction
b The decimal of can be converted into a fraction by following the same method. Since the number is read as thousandths, it can be written as over
Next, split the numerator and the denominator into prime factors to simplify the fraction.
The numbers share two common factors. The products of these factors is the GCF of and
Finally, divide the numerator and denominator by and simplify the fraction.
This means that is equal to
c First, find the values of and by setting the fractions corresponding to equal.
Now, substitute for and for into the expression
After going downstairs Izabella needs to take steps. Next, find and by setting the fractions corresponding to equal.
Now that the values of and are known, substitute them and evaluate the second expression.
Izabella needs to take more steps. Finally, calculate the sum of the steps that Izabella should take.

## Converting a Fraction Into a Decimal Number

It is possible to convert a fraction into a decimal number and the other way around. Consider the following fraction.
Divide the numerator of by the denominator of by using the long division to rewrite the fraction as a decimal. The result is This is the decimal number that corresponds to the fraction

### Extra

Rewriting Fractions With the Denominators That Are Powers of

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

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.

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 there is one zero. This indicates that the decimal point of will be moved one place to the left. The rest of the fractions can be rewritten into decimal numbers in a similar manner.

## Finding the Final Clue

The previous clue led Izabella to the living room. Tiffaniqua told her that there is one last clue remaining. Izabella is jumping off the walls in excitement. External credits: @upklyak
Izabella searches eagerly. There is the final clue. It is on the lamp! If she solves the last instructions and rewrites the fractions as decimals, she will find the present. The decimals should be written up to two decimal places.
a
b

### Hint

a Multiply the numerator and denominator by for the fraction to have the denominator of
b Divide the numerator by the denominator by using the long division.

### Solution

a The fraction needs to be converted into a decimal. Start by multiplying the numerator and the denominator by This will result in a fraction with the denominator of
Now, the denominator is a power of and has two zeros. This means that the fraction can be rewritten as a decimal by moving the decimal point of the numerator two places to the left. The calculations show that the fraction of or corresponds to the decimal of
Izabella needs to take steps.
b The fraction needs to be rewritten as a decimal number. Divide the numerator of by the denominator of by using long division. Write the decimal up to two decimal places. The decimal is This means that the fraction corresponds to about Then, Izabella needs to take steps. After taking those steps, Izabella found a huge present wrapped beautifully. External credits: @pikisuperstar
A badminton racket and birdie! Izabella has wanted this for years. Now she can finally play badminton at the park with her neighborhood buddies!

## Converting Between Decimal Numbers and Fractions

Convert the given decimal number into the corresponding fraction or the other way around — the given fraction into a decimal number. Round the decimal number to two decimal places, if needed. ## Advantages of Each Type of Numbers

In this lesson, three forms of numbers were discussed: fractions, mixed numbers, and decimal numbers. Each number form has their own advantages and disadvantages. Consider what those may be.

Fractions Mixed Numbers Decimals
Pros Precise and accurate Show the actual value of a number Easy to use in calculations
Cons More difficult to use in calculations Very inconvenient in calculations Sometimes, decimals are approximations of the exact value.

Depending on the situation, some forms of numbers might be more useful than other. Here are some real-world examples.

• Two farmers want to sell a portion of their harvest to the other. One farmer says "Let's each sell of our harvests to each other!" The other farmer says "Hold up! That is such an inconvenient number. How about of our harvests?" Now they agree.
• Imagine being told that a tree grew feet last year. Now, imagine being told that same tree grew feet last year. The second number form is more commonly used because it gives a clearer image of height.
• A local market writes the price of one kilogram of apples on one box as and on another box as Imagine trying to figure out how much cash to give at the register! It would be much easier to understand how to pay
Converting between different forms of numbers is so great! Look around and try to find more situations where some forms of numbers are more useful than others.

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