Convert the mixed repeating (recurring) decimal number 2.0212121215. Turn it into a reduced (simplified) improper fraction, into a mixed number and write it as a percentage. Calculate other equivalent fractions to the decimal number, by expanding
Convert 2.0212121215
1. Write the mixed repeating (recurring) decimal number as a percentage.
Approximate to the desired number of decimal places (14):
2.0212121215 ≈ 2.02121212152121
Multiply the number by 100/100.
The value of the number does not change when multiplying by 100/100.
Note: 100/100 = 1
2.02121212152121 =
2.02121212152121 × 100/100 =
(2.02121212152121 × 100)/100 =
202.121212152121/100 =
202.121212152121% ≈
202.12%
(rounded off to a maximum of 2 decimal places)
In other words:
Approximate to the desired number of decimal places...
Multiply the number by 100...
... And then add the percent sign, %
2.0212121215 ≈ 202.12%
2. Write the mixed repeating (recurring) decimal number as an improper fraction.
2.0212121215 can be written as an improper fraction. The numerator is larger than or equal to the denominator.
Set up the first equation.
Let y equal the decimal number:
y = 2.0212121215
Set up the second equation.
Number of decimal places repeating: 9
Multiply both sides of the first equation by 109 = 1,000,000,000
y = 2.0212121215
1,000,000,000 × y = 1,000,000,000 × 2.0212121215
1,000,000,000 × y = 2,021,212,121.5212121215
Subtract the first equation from the second equation.
Having the same number of decimal places ...
The repeating pattern drops off by subtracting the two equations.
1,000,000,000 × y - y = 2,021,212,121.5212121215 - 2.0212121215 ⇒
(1,000,000,000 - 1) × y = 2,021,212,121.5212121215 - 2.0212121215 ⇒
We now have a new equation:
999,999,999 × y = 2,021,212,119
Solve for y in the new equation.
999,999,999 × y = 2,021,212,119 ⇒
y = 2,021,212,119/999,999,999
Let the result written as a fraction.
Write the number as a fraction.
According to our first equation:
y = 2.0212121215
According to our calculations:
y = 2,021,212,119/999,999,999
⇒ 2.0212121215 = 2,021,212,119/999,999,999
3. Reduce (simplify) the fraction above: 2,021,212,119/999,999,999
(to the lowest terms, to its simplest equivalent form, irreducible).
To reduce a fraction divide the numerator and the denominator by their greatest (highest) common factor (divisor), GCF.
Factor both the numerator and the denominator (prime factorization).
In exponential notation (an):
2,021,212,119 = 3 × 43 × 15,668,311
999,999,999 = 34 × 37 × 333,667
Calculate the greatest (highest) common factor (divisor), GCF.
Multiply all the common prime factors by the lowest exponents.
GCF (3 × 43 × 15,668,311; 34 × 37 × 333,667) = 3
Divide both the numerator and the denominator by their greatest common factor, GCF.
2,021,212,119/999,999,999 =
(3 × 43 × 15,668,311)/(34 × 37 × 333,667) =
((3 × 43 × 15,668,311) ÷ 3) / ((34 × 37 × 333,667) ÷ 3) =
(43 × 15,668,311)/(33 × 37 × 333,667) =
673,737,373/333,333,333
4. The fraction is an improper one, rewrite it as a mixed number (mixed fraction):
A mixed number = an integer number and a proper fraction, of the same sign.
Example 1: 2 1/5; Example 2: - 1 3/7.
A proper fraction = the numerator is smaller than the denominator.
673,737,373 ÷ 333,333,333 = 2, remainder = 7,070,707 ⇒
673,737,373 = 2 × 333,333,333 + 7,070,707 ⇒
673,737,373/333,333,333 =
(2 × 333,333,333 + 7,070,707) / 333,333,333 =
(2 × 333,333,333) / 333,333,333 + 7,070,707/333,333,333 =
2 + 7,070,707/333,333,333 =
2 7,070,707/333,333,333
673,737,373/333,333,333: Equivalent fractions.
The above fraction cannot be reduced.
That is, it has the smallest possible numerator and denominator.
By expanding it we can build up equivalent fractions.
Multiply the numerator & the denominator by the same number.
Example 1. By expanding the fraction by 4:
673,737,373/333,333,333 = (673,737,373 × 4)/(333,333,333 × 4) = 2,694,949,492/1,333,333,332
Example 2. By expanding the fraction by 6:
673,737,373/333,333,333 = (673,737,373 × 6)/(333,333,333 × 6) = 4,042,424,238/1,999,999,998
Of course, the above fractions are reducing...
... to the initial fraction: 673,737,373/333,333,333
:: Final answer ::
Written in 4 different ways
As a reduced (simplified) positive improper fraction:
2.0212121215 = 673,737,373/333,333,333
As a mixed number:
2.0212121215 = 2 7,070,707/333,333,333
As a percentage:
2.0212121215 ≈ 202.12%
As equivalent fractions:
2.0212121215 = 673,737,373/333,333,333 = 2,694,949,492/1,333,333,332 = 4,042,424,238/1,999,999,998
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