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

Tearrik's mother makes a pizza with a diameter of $14 \frac{1}{3}$ inches. Tearrik wants to put it in a square box with a side length of $14.39$ inches.

External credits: @macrovector

Is the box big enough?

We want to determine whether the given square box is big enough for the pizza. We will start by rewriting the given mixed number as a fraction, then we will convert it to a decimal and, finally, compare both numbers. First, let's take another look at the given picture.

As we can see from the diagram, the box is 14.39 inches wide. Keep in mind that it is a square, which means each side has the same length. From the exercise, we also know that the diameter of the pizza is 14 13 inches. To compare these numbers, we will first rewrite 14 13 as a fraction. Let's do it!

Now that we have written the number as an improper fraction, we can write it in decimal form. We will divide the numerator 43 by the denominator 3. Let's do it!

Notice that the products and remainders repeat, which means that the remainder will never be 0. Because of this, we can say that 14 13 written in decimal form is the repeating decimal 14.3, or 14.33.... 14 13 = 14.33... The diameter of the pizza is smaller than the width of the box because the digit in the hundredths place in 14.3 3... is less than the digit in the hundredths place in 14.3 9. ccc Diameter of Pizza & & Side Length of Square 14.3 3... & < & 14.3 9 Because of this, we can say that the pizza will fit in the box.

According to a recent survey,

72 . \overline{72} %

of moviegoers in the United States prefer animated movies. The percent here is a repeating decimal. Write it as a fraction in simplest form.

We are told that according to a survey, 72.72 % of moviegoers prefer animated movie. This percent looks like a repeating decimal, so let's start by writing it as one. We can do this by dividing the number by 100 and removing the percent sign. This will move the decimal point two places to the left. 72.72 % = 0.7272 Since 7 and 2 are repeating digits, it is the same thing as 0.72. Now we can convert this number into a fraction by following five steps.

Assign a variable to represent the repeating decimal.
Count the number of repeating digits and call it n.
Multiply the repeating decimal by 10^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!

Steps 1 and 2

We will use the variable x to represent the given repeating decimal number. We can write an equation by setting this variable equal to 0.72. x = 0.72 This number has two repeating digits, so the value of n is 2 in this case.

Step 3

Now we multiply both sides of the equation by 10^2, or 100, because n= 2.

Step 4

Now we will subtract the original equation from the equation we found in Step 3. cr & 100x = 72.72 - & x = 0.72 & 99x = 72 0.46cm We found that 99x = 72.

Step 5

Finally, we can now solve the last equation for x by dividing both sides of the equation by 99 and simplifying.

We know that both 99 and 72 are divisible by 9. Let's simplify the fraction!

We found that x is equal to 811. We can conclude that the given repeating decimal is equivalent to 811. 0.72=8/11 Since 0.72 is also equal to 72.72 %, we can also express this percent as 811. This fraction can be interpreted in the context of the question as 8 out of every 11 people surveyed prefer animated movies.

The following table shows some decimal numbers and their fractional equivalents.

Decimal	Fraction
$0 . \overline{2}$	$\frac{2}{9}$
$0 . \overline{3}$	$\frac{3}{9}$
$0 . \overline{4}$	$\frac{4}{9}$
$0 . \overline{5}$	$\frac{5}{9}$

Use the table to find the fraction equivalent of

0 . \overline{6} .

We are asked to determine which of the given fractions represents 0.6. Let's first analyze the given table that shows decimal numbers and their fraction equivalents.

Decimal	Fraction
0.2	2/9
0.3	3/9
0.4	4/9
0.5	5/9

We can see that each decimal from the table is equal to a fraction with 9 in the denominator and the repeating digit in the numerator. Let's use this pattern to rewrite 0.6.

Decimal	Fraction
0.2	2/9
0.3	3/9
0.4	4/9
0.5	5/9
0.6	6/9

According to the pattern, 0.6 should be equal to 69. The answer is B.

We can check our answer by writing 69 as a decimal. To do so, we will divide the numerator by the denominator.

We can see that the pattern starts to repeat itself after a few steps. This means that the decimal is a repeating decimal. 6/9 = 0. 6 We found that 0.6 is equivalent to 69, so the correct option is B.

Tiffaniqua learned that long division can be used to convert a fraction into a decimal number. She observed that for some fractions, like

\frac{1}{4},

the division terminates after a few steps. For other fractions, like

\frac{1}{3},

the decimal does not terminate.

Repeating Decimal \frac{1}{3} Terminating Decimal \frac{1}{4}

Tiffaniqua wonders which fractions have terminating decimals and which fractions have repeating decimals. Help her draw a conclusion by answering the following questions.

Which fractions have terminating decimals? Select all that apply.

Which of the statements about repeating and terminating decimals are correct? Select all that apply.

I . II . III . IV . If 4 is a factor of the denominator, then the result will always be a terminating decimal . If 3 is a factor of the denominator, then the result will always be a repeating decimal . If the denominator of a fraction in simplest form is a factor of some power of 10, then the result will be a terminating decimal . If the denominator of a fraction in simplest form has a prime factor other than 2 or 5, then the result will be a repeating decimal .

Let's list the given fractions. 1/4, 1/5, 1/6, 1/8, 1/12, 1/15, 1/16, and 1/18 We want to write these fractions as decimals to see which ones are terminating. Recall that if the denominator of a fraction is a power of 10, then the fraction is a terminating number. Let's check if we can find a number that, when multiplied by the denominator, is equal to some power of 10. 10 & = 5 * 2 100 & = 4 * 25 1000 & = 8 * 125 10 000 & = 16 * 625 We were able to find factors for four of the denominators that made their product equal a power of 10. This means that the fractions 1 4, 1 5, 1 8, and 1 16 are terminating decimals.

Fraction	Expand	Decimal Equivalent
1/5	1 * 2/5 * 2 = 2/10	0.2
1/4	1 * 25/4 * 25 = 25/100	0.25
1/8	1 * 125/8 * 125 = 125/1000	0.125
1/16	1 * 625/16 * 625 = 625/10 000	0.0625

The denominators of the other fractions — 6, 12, 15, and 16 — are not factors of any power of 10. This suggests that they are not terminating decimals. We can use long division confirm our suspicions. Let's find 118.

As we can see, it is a repeating decimal. We can find the decimal form of the other fractions in the same way.

Fraction	Decimal
1/6	0.16
1/12	0.83
1/15	0.06
1/18	0.05

As a result, the fractions with terminating decimals are as follows. 1/4, 1/5, 1/8, and 1/16

Let's make a table to organize what we found in the previous part.

Denominators 4, 5, 8, and 16		Denominators 6, 12, 15, and 18
Fraction	Terminating Decimal	Fraction	Repeating Decimal
1/4	0.25	1/6	0.16
1/5	0.2	1/12	0.83
1/8	0.125	1/15	0.06
1/16	0.0625	1/18	0.05

It makes sense to say that a fraction's denominator determines whether it has a terminating or repeating decimal. If the denominator of a fraction in simplest form is a factor of some power of 10, then we can be sure that the result will be a terminating decimal. We can see that the numbers 4, 5, 8, and 16 are factors of some power of 10. 10 & = 5 * 2 100 & = 4 * 25 1000 & = 8 * 125 10 000 & = 16 * 625 Therefore, Statement III is true. On the other hand, the denominators 6, 12, 15, and 18 are not factors of any power of 10. Notice that they are all multiples of 3, and no power of 10 is divisible by 3. 6 & = 3 * 2 12 & = 3 * 4 15 & = 3 * 5 18 & = 3 * 6 As we can see in the table, simplified fractions with a prime factor other than 2 or 5 in the denominator produce repeating decimals. This means that Statement IV is also true. The first two statements are not correct. The following examples disprove the first two expressions. 4/12 = 0.3 and 3/15 = 0.2

Terminating and Repeating Decimals

Catch-Up and Review

Decimal Forms of Fractions

Extra

Repeating Decimals

Terminating Decimals

Identifying Types of Decimals

Converting a Fraction Into a Repeating Decimal Number

Favorite Movie Genres

Hint

Solution

Converting a Repeating Decimal Number Into a Fraction

Which Two Genres Did Vincenzo Choose?

Hint

Solution

Step $1$

Step $2$

Step $3$

Step $4$

Step $5$

Practice Converting Between Fractions and Repeating Decimals

Extra

Was Archimedes Correct?

Hint

Solution

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

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

Comparing the Numbers

Extra

Why Is $π$ So Special?

Steps 1 and 2

Step 3

Step 4

Step 5

Terminating and Repeating Decimals

Recommended exercises

	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

Steps 1 and 2

Step 3

Step 4

Step 5

Terminating and Repeating Decimals

Recommended exercises

Step $1$

Step $2$

Step $3$

Step $4$

Step $5$

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

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