Decoding Terminating Decimals and Repeating Decimals through Long Division and Rational Numbers

Fraction	Decimal
$\frac{2}{9}$
$\frac{1}{3}$
$\frac{3}{4}$
$\frac{4}{1 1}$
$\frac{3}{8}$

Fraction

Decimal

\frac{2}{9}

\frac{1}{3}

\frac{3}{4}

\frac{4}{1 1}

\frac{3}{8}

Repeating Decimal Numbers
Number	Notation	Fraction
$0.666666 \dots$	$0 . \overline{6}$	$\frac{2}{3}$
$1.533333 \dots$	$1.5 \overline{3}$	$\frac{2 3}{1 5}$
$5.373737 \dots$	$5 . \overline{37}$	$\frac{5 3 2}{9 9}$

Repeating Decimal Numbers

Number

Notation

Fraction

0.666666 \dots

0 . \overline{6}

\frac{2}{3}

1.533333 \dots

1.5 \overline{3}

\frac{2 3}{1 5}

5.373737 \dots

5 . \overline{37}

\frac{5 3 2}{9 9}

Terminating Decimal Numbers
Number	Fraction
$0.5$	$\frac{1}{2}$
$1.53$	$\frac{1 5 3}{1 0 0}$
$52.372$	$\frac{1 3 0 9 3}{2 5 0}$

Terminating Decimal Numbers

Number

Fraction

0.5

\frac{1}{2}

1.53

\frac{1 5 3}{1 0 0}

52.372

\frac{1 3 0 9 3}{2 5 0}

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

Genre	Number of Students
Action	$6$
Animation	$12$
Comedy	$4$
Science Fiction	$5$
Total	$27$

Genre

Number of Students

Action

6

Animation

12

Comedy

4

Science Fiction

5

Total

27

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

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

Some Irrational Numbers
Euler's number, $e$	$2.718281 \dots$
Golden ratio, $ϕ$	$1.618033 \dots$
$2$	$1.414213 \dots$

Some Irrational Numbers

Euler's number,

e

2.718281 \dots

Golden ratio,

ϕ

1.618033 \dots

2

1.414213 \dots

Is the given number a terminating decimal, a repeating decimal, or neither?

0.14 \overline{15}

0.141516 \dots

0.141

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 $1.2 \overline{34},$ enter $1.234$ into the first box and $2$ into the second box because the number has two repeating digits.

\frac{9}{1 1}

1 \frac{9}{1 6}

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.

0.35 \overline{6}

- 4.0 \overline{23}

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.

Assign a variable to represent the repeating decimal.
Count the number of repeating digits and call it n.
Multiply the repeating decimal by 10 to the power of n.
Subtract the equation in Step 1 from the equation in Step 3.
Solve for the variable and express it as a fraction in simplest form.

Let's do it!

Step 1

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

Step 2

For the given decimal, the digit that repeats is 6. The number has one repeating digit, so n= 1.

Step 3

Now we multiply each side of the equation by 10^1, or 10, since we found that n=1.

Step 4

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.

Step 5

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.

Steps 1 and 2

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.

Step 3

Now we multiply each side of the equation by 10 to the power of 2, or 100, because n=2.

Step 4

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.

Step 5

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.

A rectangle is divided into 9 equal parts and 5 of the parts are shaded.

Which of the following decimal numbers is equal to the fraction modeled by Ignacio's diagram?

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.

\frac{7}{3}, - \frac{9}{1 0}, 1 \frac{1 3}{1 8}, - 2, - 3 . \overline{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.

Rewriting 73

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

Rewriting 910

The next number is - 910. Since the denominator is a multiple of 10, this is a terminating decimal. - 9/10 =- 0.9

Rewriting 1 1318

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

Comparing Numbers

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.

	12 Theory slides
	10 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

Terminating and Repeating Decimals

Catch-Up and Review

Extra

Hint

Solution

Hint

Solution

Step 1

Step 2

Step 3

Step 4

Step 5

Extra

Hint

Solution

Rewriting 371​

Rewriting 37110​

Comparing the Numbers

Extra

Step 1

Step 2

Step 3

Step 4

Step 5

Steps 1 and 2

Step 3

Step 4

Step 5

Rewriting 73

Rewriting 910

Rewriting 1 1318

Comparing Numbers

Terminating and Repeating Decimals

Recommended exercises

Step $1$

Step $2$

Step $3$

Step $4$

Step $5$

Rewriting $3 \frac{1}{7}$

Rewriting $3 \frac{1 0}{7 1}$