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| 12 Theory slides |
| 10 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Fraction & Mixed Number & Decimal 14/5 & 2 45 & 2.8 Decimal numbers can be classified based on the digits after the decimal point. As an example, consider the decimal number corresponding to 13. Is there a repeating digit? How about the decimal equivalent of 17? This lesson explores decimal number types and develops methods for converting them into fractions.
Here are a few recommended readings before getting started with this lesson.
A rational number can be represented as a decimal number with the help of the long division method. Rewrite the fractions shown in the table as decimals.
Fraction | Decimal |
---|---|
2/9 | |
1/3 | |
3/4 | |
4/11 | |
3/8 |
Consider the following questions as a guide to classify decimals.
A repeating decimal number, or recurring decimal number, is a number in decimal form in which some digits after the decimal point repeat infinitely. The digits repeat their values at regular intervals and the infinitely repeated part is not zero. When writing the decimal, a line is drawn over the repeating portion to express such a number.
Repeating Decimal Numbers | ||
---|---|---|
Number | Notation | Fraction |
0.666666... | 0.6 | 2/3 |
1.533333... | 1.53 | 23/15 |
5.373737... | 5.37 | 532/99 |
A terminating decimal number is a number in decimal form with a finite number of digits. Terminating numbers can be written as fractions, which means that they are rational numbers.
Terminating Decimal Numbers | ||
---|---|---|
Number | Fraction | |
0.5 | 1/2 | |
1.53 | 153/100 | |
52.372 | 13 093/250 |
Determine whether the given number is a repeating decimal, a terminating decimal, or neither.
The fraction was found to be a repeating decimal number. 7/15 = 0.4666... The digit that repeats is 6, so a bar is drawn over that digit. 7/15 = 0.46
Vincenzo conducts a survey to find out what kind of movies his classmates like. The table shows the information he gathered from the survey.
Genre | Number of Students |
---|---|
Action | 6 |
Animation | 12 |
Comedy | 4 |
Science Fiction | 5 |
Genre | Number of Students |
---|---|
Action | 6 |
Animation | 12 |
Comedy | 4 |
Science Fiction | 5 |
Total | 27 |
For the given decimal, the repeating digit is 3. x = 0.083 There is only one digit with a bar above it. This means n= 1.
Now subtract the original equation from the equation obtained in Step 3. cl & 10x = 0.83 - & x = 0.083 Writing the decimals without using bar notation can make subtraction easier to understand. cl & 10x = 0.83333 ... - & x = 0.08333 ... & 9x = 0.75000 ... The zeros can be ignored. It is found that 9x is equal to 0.75.
.LHS /9.=.RHS /9.
Cancel out common factors
Simplify quotient
Vincenzo is organizing the information he gathered from his survey.
Genre | Number of Students |
---|---|
Action | 6 |
Animation | 12 |
Comedy | 4 |
Science Fiction | 5 |
He selected two genres at random. He used a calculator to divide the number of students who prefer the two genres by the total number of students.
His calculator showed the result as 0.629629629....
0. 629 629 629... ⇔ 0.629 There are five steps to write this repeating decimal number as a fraction.
Let x be a variable that represents the given repeating decimal number. Write an equation by setting x equal to 0.629. x = 0.629
The repeated sequence of digits in the decimal is 629. There are three repeating digits, so n= 3.
Now subtract the original equation from the equation obtained in Step 3. cr & 1000x = 629.629 [0.6em] - & x = 0.629 Consider the expanded forms of the numbers when performing this subtraction. cr & 1000x = 629.629629 ... - & x = 0.629629 ... & 999x = 629.000000 ... Since the zeros can be ignored, 999x is equal to 629.
.LHS /999.=.RHS /999.
Cancel out common factors
Simplify quotient
Now take a look at the table and determine which of the two genres gives a sum equal to 17.
Genre | Number of Students |
---|---|
Action | 6 |
Animation | 12 |
Comedy | 4 |
Science Fiction | 5 |
The number of students who prefer animation and science fiction is 17. Therefore, Vincenzo selected the animation and science fiction genres.
If the answer is a repeating decimal, do the following to submit the answer.
Here are some examples. Repeating Decimal & & How to Submit 0.409 & ⇒ & 0.4 09 [0.6em] 0.9 & ⇒ & 0. 9
Write the mixed numbers as decimals.
Archimedes believed that π was between two rational numbers, 3 17 and 3 1071. π = 3.141592... To check whether Archimedes was correct, both of the numbers will be written as a decimal. Since the first six decimal places of π are given, the numbers will be calculated up to 6 decimal places. In this way, the numbers can be compared.
Now that the decimal forms of both numbers have been found, the numbers can be compared. 3 17 &= 3.142857... 3 1071 &= 3.140845... π &= 3.141592... Notice that the first three digits of the numbers are all the same. 3 17 &= 3.142857... 3 1071 &= 3.140845... π &= 3.141592... This means that the digits in the thousandth place must be compared. 3 17 &= 3.14 2857... 3 1071 &= 3.14 0845... π &= 3.14 1592... The number with the greatest digit in the thousandth place is the greatest, and the number with the least digit is the least. 3.140845... < 3.141592... < 3.142857... ⇓ 3 1071 < π < 3 17 Therefore, π is between 3 1071 and 3 17. Archimedes was correct. The answer is B.
This lesson presented two types of decimal numbers, terminating and repeating decimals. The lesson also introduced methods for converting both fractions. Now comes a tough question! Are there decimal numbers that have an infinite number of places but do not repeat? The answer is yes and they are called irrational numbers. Take, for example, π. π = 3.141592653 ... No repeating patterns are seen in this type of decimal number, so their decimal representations have no end. Here are a few other commonly used irrational numbers.
Some Irrational Numbers | |
---|---|
Euler's number, e | 2.718281 ... |
Golden ratio, ϕ | 1.618033 ... |
sqrt(2) | 1.414213 ... |
Irrational numbers cannot be converted into rational numbers. All that can be done is to get better and better approximations. The diagram shows how decimal numbers are classified.
Is the given number a terminating decimal, a repeating decimal, or neither?
We want to determine whether the given number is a repeating decimal. Let's begin by recalling the definitions.
Definition | |
---|---|
Terminating Decimal | A decimal number whose digits after the decimal point are finite. When written as a fraction and dividing the numerator by the denominator, the remainder is 0. |
Repeating Decimal | A decimal number whose some digits after the decimal point repeat infinitely. When written as a fraction and dividing the numerator by the denominator, the remainder will never be 0 because the products and differences repeat. |
With this in mind, let's take a look at the given number. 0.1415 Notice that a bar is drawn over two digits after the decimal point. This means that the digits 1 and 5 repeat infinitely. Because of this, we can say that 0.1415 is a repeating decimal.
We know that the dots at the end of a decimal number mean that the number continues indefinitely.
0.141516 ...
This number has infinitely many digits after the decimal point. Additionally, the digits do not appear to repeat. Therefore, the decimal number is neither terminating nor repeating. Additionally, the number cannot be written as the ratio of two integers, which means that it is an irrational number.
Let's take a look at our last number.
0.141
Notice that there are only three digits after the decimal point, 141, with no dots to indicate that the decimal continues. This means the number of digits is finite. Because of this, we can say that 0.141 is a terminating decimal.
Write the fraction or mixed number as decimal. If the answer is a repeating decimal such as 1.234, enter 1.234 into the first box and 2 into the second box because the number has two repeating digits.
We want to write the given fraction as a decimal. 9/11 We will divide the numerator by the denominator. Let's do it by using long division!
We can stop dividing because we get the same remainders as we continue — the remainder will never become 0. Because of this, we can conclude that the quotient is a repeating decimal. 9/11 = 0.818181... We see that the number has two repeating digits, 8 and 1, so we can also write the decimal with a bar above those digits. 9/11 = 0.81
We are given a mixed number.
1 916
Before we convert this number into a decimal, let's rewrite it as an improper fraction. This will make it easier for us to convert it into decimal form.
The mixed number 1 916 is equal to 2516. Next we will divide the numerator by the denominator to write this number in decimal form.
We eventually got 0 as the remainder, so we can say that 1 916 written in decimal form is a terminating decimal and is equal to 1.5625. 1 916 = 1.5625
Write the given decimal as a fraction.
We want to write the given repeating decimal number as a fraction. 0.356 We will follow five steps to rewrite it as a fraction.
Let's do it!
Let's use the variable x to represent the given repeating decimal number. We can write an equation by setting this variable equal to 0.356. x = 0.356
For the given decimal, the digit that repeats is 6. The number has one repeating digit, so n= 1.
Now we multiply each side of the equation by 10^1, or 10, since we found that n=1.
Now we will subtract the original equation from the equation we found in Step 3. cr & 10x = 3.566 - & x = 0.356 Let's expand the decimals a bit to make it easier to see how the repeating digits cancel each other out. cr & 10x = 3.566666 ... - & x = 0.356666 ... & 9x = 3.210000 ... We can ignore the zeros because they do not change the value of the difference. We found that 9x = 3.21.
Lastly, we solve the final equation for x. We need to divide both sides of the equation by 9 to isolate x on the left-hand side.
We can eliminate the decimal in the numerator by multiplying both the numerator and the denominator by 100. This will move the decimal point in the numerator two place to the right and make it a whole number.
Finally, we will check if this fraction can be simplified. Let's find the greatest common factor (GCF) of the numerator and denominator. 321 = 3 * 107 900 = 3^2 * 10^2 The GCF of the numbers is 3. Let's simplify the fraction by dividing both the numerator and the denominator by 3.
We found that x is equal to 107300. We can conclude that the given repeating number is equal to 107300. 0.356=107/300
We are given a negative repeating decimal.
- 4.023
Let's ignore the negative sign and write the number as a fraction. We will put the negative sign back at the end.
4.023
We will follow the same steps we followed in Part A to write this number as a fraction.
Let's use x to represent our decimal. x = 4.023 We can see that the number has two repeating digits, meaning that n= 2.
Now we multiply each side of the equation by 10 to the power of 2, or 100, because n=2.
Now we will subtract the original equation from the equation we found in Step 3. cr & 100x = 402.323 - & x = 4.023 Let's use the expanded forms of the numbers to make it easier to see how the repeating digits cancel each other out. cr & 100x = 402.3232323 ... - & x = 4.0232323 ... & 99x = 398.3000000 ... Since we can ignore the zeros, we have 99x = 398.3.
Finally, we solve the new equation for x. Dividing both sides of the equation by 99 will isolate x on the left-hand side.
We can eliminate the decimal in the numerator by multiplying both the numerator and the denominator by 10.
Let's check if the numerator and denominator have any common factors. 3983 & = 7 * 569 990 & = 2 * 3^2 * 5 11 The numbers do not have any common factors, so we cannot simplify the fraction. We can conclude that the given repeating number is 3983990. 4.023=3983/990 We can also write this improper fraction as a mixed number.
We found that 4.023=4 23990. Therefore, - 4.023 is equal to - 4 23990. - 4.023 = - 4 23/990
Ignacio draws the diagram below to represent a fraction.
There are 9 equal parts in the diagram, 5 of which are shaded.
We can then say that this model represents the fraction 59. We want to write this fraction as a decimal number. We can divide the numerator by the denominator to find its decimal form. Let's use long division to do it!
We can see that a specific pattern starts to repeat itself after a few steps. This means that the decimal is a repeating decimal. 5/9 = 0.5555... We can write the decimal using bar notation. Remember, the bar goes over the decimal part that repeats itself! 5/9 = 0.5 This corresponds option B.
Ramsha arranges the following rational numbers from least to greatest along a number line. 7/3, - 9/10, 1 1318, - 2, - 3.1 Which number line did Ramsha draw?
We will write all the given numbers as decimals so we can easily compare them. 7/3, - 9/10, 1 1318, - 2, - 3.1 We see that - 3.1 is a repeating decimal and - 2 is an integer, so they are already in decimal form. Let's rewrite the other numbers.
Let's start with the improper fraction 73. We will use long division to divide 7 by 3.
We keep getting the same remainder as we continue dividing. Therefore, this is a repeating decimal. 7/3 = 2.3
The next number is - 910. Since the denominator is a multiple of 10, this is a terminating decimal. - 9/10 =- 0.9
The third number is a mixed number. Let's first write it as an improper fraction.
Now we use long division again to write 3118 as a decimal number.
We eventually start getting the same remainder 4 at each step. Let's stop here because the quotient is a repeating decimal. The mixed number 1 1313 is equivalent to 1.72. 1 1318 = 1.72
We have now found the decimal forms of all of the numbers!
Numbers | Decimal Form |
---|---|
7/3 | 2.3 |
- 9/10 | - 0.9 |
1 1318 | 1.72 |
- 2 | - 2 |
- 3.1 | - 3.1 |
As we can see, - 3.1 is the least and 2.3 is the greatest. - 3.1 < - 2 < - 0.9 < 1.72 < 2.3 We can show these number on a number line.
This number line corresponds to option C.