1. Write the pure repeating (recurring) decimal number as a percentage.
Approximate to the desired number of decimal places (14):
0.555555556 ≈ 0.55555555655556
Multiply the number by 100/100.
The value of the number does not change when multiplying by 100/100.
Note: 100/100 = 1
0.55555555655556 =
0.55555555655556 × 100/100 =
(0.55555555655556 × 100)/100 =
55.555555655556/100 =
55.555555655556% ≈
55.56%
(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, %
0.555555556 ≈ 55.56%
2. Write the pure repeating (recurring) decimal number as a proper fraction.
0.555555556 can be written as a proper fraction. The numerator is smaller than the denominator.
Set up the first equation.
Let y equal the decimal number:
y = 0.555555556
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 = 0.555555556
1,000,000,000 × y = 1,000,000,000 × 0.555555556
1,000,000,000 × y = 555,555,556.555555556
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 = 555,555,556.555555556 - 0.555555556 =>
(1,000,000,000 - 1) × y = 555,555,556.555555556 - 0.555555556 =>
We now have a new equation:
999,999,999 × y = 555,555,556
Solve for y in the new equation.
999,999,999 × y = 555,555,556 =>
y = 555,555,556/999,999,999
Let the result written as a fraction.
Write the number as a fraction.
According to our first equation:
y = 0.555555556
According to our calculations:
y = 555,555,556/999,999,999
=> 0.555555556 = 555,555,556/999,999,999
3. Reduce (simplify) the fraction above: 555,555,556/999,999,999
(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):
555,555,556 = 22 × 107 × 1,298,027
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.
But, the numerator and the denominator have no common factors.
GCF (22 × 107 × 1,298,027; 34 × 37 × 333,667) = 1
The numerator and the denominator are coprime numbers (no common prime factors, GCF = 1). So, the fraction cannot be reduced (simplified): irreducible fraction.
555,555,556/999,999,999: 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:
555,555,556/999,999,999 = (555,555,556 × 4)/(999,999,999 × 4) = 2,222,222,224/3,999,999,996
Example 2. By expanding the fraction by 6:
555,555,556/999,999,999 = (555,555,556 × 6)/(999,999,999 × 6) = 3,333,333,336/5,999,999,994
Of course, the above fractions are reducing...
... to the initial fraction: 555,555,556/999,999,999