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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
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 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
Mixed Numbers
A mixed number consists of a non-zero integer number and a proper fraction.
acbwhere a is an integer, b<c, and c=0
-397, -261, -1125, 143, 272, 383
Consider the graphic representation of different mixed numbers.
Discussion
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.
Next, multiply the integer part by the denominator of the fraction. In this case, the denominator of the fraction is 9.
592
This mixed number can be rewritten as an improper fraction in three steps.
1
Multiply the Integer Part by the Denominator
expand_more First, identify the integer part of the mixed number. This is the integer number written before the fraction.
5×9=45
2
Add the Numerator
expand_more 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
expand_more 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
Extra
Interact With an Applet that Converts Different Mixed Numbers Into Improper FractionsSubmit 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
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
431
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b
683
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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.
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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 x, y, z, and w by comparing the fractions.
Solution
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.
First Question
In the note, the numerator of the first improper fraction is x and the denominator is y.yx=313
This means that x=13 and y=3. The first instruction that Izabella received is the following.
1.Take x+y steps 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=16 steps
Second Question
The numerator of the second improper fraction is z and the denominator is w.wz=851
This means that z=51 and w=8. Now, consider the second instruction.
2.Turn left and take z−2w steps.
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
Converting an improper fraction into a mixed number can help to estimate the actual value of the fraction. Consider the following improper fraction.
421
This improper fraction can be rewritten as a mixed number in three steps.
1
Divide the Numerator by the Denominator
expand_more 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
expand_more 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
expand_more 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
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.
To find the numbers of steps she needs to take, Izabella needs to rewrite the improper fractions as mixed numbers.
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.
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.
External credits: @gstudioimagen
a
587
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b
9115
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Hint
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.
Solution
a Izabella needs to rewrite 587 as a mixed number. This requires dividing the numerator by the denominator. Long division can be applied here.
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.
9115=1297
According to the second instruction, Izabella should turn right and take 7 steps. Where will she end up?
Pop Quiz
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.
Discussion
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.
354→3.la
This observation leads to the definition of decimal numbers. Concept
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 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
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.
0.56
A decimal number can be rewritten as a fraction in three steps.
1
Count the Number of Decimal Places n
expand_more The number 0.56 can be read as 56 hundredths.
There are two decimal places.
2
Write as a Fraction With the Denominator of 10n
expand_more 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 102.
0.56=10256⇓0.56=10056
3
Simplify the Fraction
expand_more 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. Extra
Interact With an Applet That Converts Decimals to FractionsSubmit 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
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. 
She found the clue over the oven vent. The instructions said to rewrite a decimal number as a fraction in the simplest form.
External credits: @upklyak
a
1.98
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b
0.564
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c How many steps in total should Izabella take?
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Hint
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.
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.
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=81 steps
Discussion
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.
The result is 0.64. This is the decimal number that corresponds to the fraction 2516.
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.
The rest of the fractions can be rewritten into decimal numbers in a similar manner.
2516
Divide the numerator of 16 by the denominator of 25 by using the long division to rewrite the fraction as a decimal.
Extra
Rewriting Fractions With the Denominators That Are Powers of 10A 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 |
| Fraction | 107 | 10026 | 1000782 |
|---|---|---|---|
| Number of Zeros | 1 | 2 | 3 |
| Decimal | 0.7 | 0.26 | 0.782 |
Example
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. 
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.
The calculations show that the fraction of 254 or 10016 corresponds to the decimal of 0.16.
The decimal is 0.42. This means that the fraction 16871 corresponds to about 0.42. Then, Izabella needs to take 0.42⋅100=42 steps. After taking those steps, Izabella found a huge present wrapped beautifully. 
A badminton racket and birdie! Izabella has wanted this for years. Now she can finally play badminton at the park with her neighborhood buddies!
External credits: @upklyak
a
254
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b
16871
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Hint
a Multiply the numerator and denominator by 4 for the fraction to have the denominator of 100.
b Divide the numerator by the denominator by using the long division.
Solution
a The fraction 254 needs to be converted into a decimal. Start by multiplying the numerator and the denominator by 4. This will result in a fraction with the denominator of 100.
10016=0.16
Izabella needs to take 0.16⋅100=16 steps.
b The fraction 16871 needs to be rewritten as a decimal number. Divide the numerator of 71 by the denominator of 168 by using long division. Write the decimal up to two decimal places.
External credits: @pikisuperstar
Pop Quiz
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.
Closure
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 0.166666… of our harvests to each other!" The other farmer says "Hold up! That is such an inconvenient number. How about 61 of our harvests?" Now they agree.
- Imagine being told that a tree grew 413 feet last year. Now, imagine being told that same tree grew 341 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 $42519 and on another box as $25119. Imagine trying to figure out how much cash to give at the register! It would be much easier to understand how to pay $4.76.
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