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

Total

27

Genre

Number of Students

Action

6

Animation

12

Comedy

4

Science Fiction

5

a The number shown on the calculator is a repeating decimal number because some digits after the decimal point repeat themselves.

0.629629629 \dots \Leftrightarrow 0 . \overline{629}

There are five steps to write this repeating decimal number 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 $1 0^{n} .$
Subtract the equation in Step $1$ from the equation in Step $3 .$
Solve for the variable and express it as a fraction in its simplest form.

Step $1$

Let

x

be a variable that represents the given repeating decimal number. Write an equation by setting

x

equal to

0 . \overline{629} .

x = 0 . \overline{629}

Step $2$

The repeated sequence of digits in the decimal is $629 .$ There are three repeating digits, so $n = 3 .$

Step $3$

Since

n = 3,

multiply each side of the equation by

1 0^{3},

1000 .

x = 0 . \overline{629}

MultEqn

$LHS \cdot 1000 = RHS \cdot 1000$

x \cdot 1000 = 0 . \overline{629} \cdot 1000

Multiply

1000 x = 629 . \overline{629}

Step $4$

Now subtract the original equation from the equation obtained in Step

3 .

- 1000 x = 629 . \overline{629} x = 0 . \overline{629}

Consider the expanded forms of the numbers when performing this subtraction.

- 1000 x = 629.629629 \dots x = 0.629629 \dots 999 x = 629.000000 \dots

Since the zeros can be ignored,

999 x

is equal to

629 .

Step $5$

Lastly, solve the obtained equation for

x .

999 x = 629

DivEqn

$LHS / 999 = RHS / 999$

\frac{9 9 9 x}{9 9 9} = \frac{6 2 9}{9 9 9}

CancelCommonFac

Cancel out common factors

\frac{9 9 9 x}{9 9 9} = \frac{6 2 9}{9 9 9}

SimpQuot

Simplify quotient

x = \frac{6 2 9}{9 9 9}

Check whether this fraction can be simplified by finding the greatest common factor (GCF) of the numerator and denominator.

629 = 17 \cdot 37 999 = 3^{3} \cdot 37

Then,

GCF (629, 999)

is equal to

37 .

Finally, simplify the fraction by dividing both numerator and denominator by

37 .

x = \frac{6 2 9}{9 9 9}

ReduceFrac

$\frac{a}{b} = \frac{a / 3 7}{b / 3 7}$

x = \frac{6 2 9 / 3 7}{9 9 9 / 3 7}

CalcQuot

Calculate quotient

x = \frac{1 7}{2 7}

As a result,

x

is equal to

\frac{1 7}{2 7},

which is equal to the given repeating number.

0 . \overline{629} = \frac{1 7}{2 7}

b In Part A, it was found that

0 . \overline{629}

is equivalent to

\frac{1 7}{2 7} .

Notice that the denominator is the same as the total number of students, so

\frac{1 7}{2 7}

should be the fraction that Vincenzo entered into the calculator. The numerator of this fraction is the sum of the students who prefer two specific genres.

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.

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

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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