Convert the pure repeating (recurring) decimal number 0.00000009. Turn it into a reduced (simplified) proper fraction and write it as a percentage. Calculate other equivalent fractions to the decimal number, by expanding
Convert 0.00000009
1. Write the pure repeating (recurring) decimal number as a percentage.
Approximate to the desired number of decimal places:
0.00000009 ≈ 0.00000009
Multiply the number by 100/100.
The value of the number does not change when multiplying by 100/100.
Note: 100/100 = 1
0.00000009 =
0.00000009 × 100/100 =
(0.00000009 × 100)/100 =
0.000009/100 =
0.000009% ≈
0%
(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, %
0.00000009 ≈ 0%
2. Write the pure repeating (recurring) decimal number as a proper fraction.
0.00000009 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 = 0.00000009
Set up the second equation.
Number of decimal places repeating: 8
Multiply both sides of the first equation by 108 = 100,000,000
y = 0.00000009
100,000,000 × y = 100,000,000 × 0.00000009
100,000,000 × y = 9.00000009
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.
100,000,000 × y - y = 9.00000009 - 0.00000009 ⇒
(100,000,000 - 1) × y = 9.00000009 - 0.00000009 ⇒
We now have a new equation:
99,999,999 × y = 9
Solve for y in the new equation.
99,999,999 × y = 9 ⇒
y = 9/99,999,999
Let the result written as a fraction.
Write the number as a fraction.
According to our first equation:
y = 0.00000009
According to our calculations:
y = 9/99,999,999
⇒ 0.00000009 = 9/99,999,999
3. Reduce (simplify) the fraction above: 9/99,999,999
(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):
9 = 32
99,999,999 = 32 × 11 × 73 × 101 × 137
Calculate the greatest (highest) common factor (divisor), GCF.
Multiply all the common prime factors by the lowest exponents.
GCF (32; 32 × 11 × 73 × 101 × 137) = 32
Divide both the numerator and the denominator by their greatest common factor, GCF.
9/99,999,999 =
32/(32 × 11 × 73 × 101 × 137) =
(32 ÷ 32) / ((32 × 11 × 73 × 101 × 137) ÷ 32) =
1/(11 × 73 × 101 × 137) =
1/11,111,111
1/11,111,111: 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 5:
1/11,111,111 = (1 × 5)/(11,111,111 × 5) = 5/55,555,555
Example 2. By expanding the fraction by 8:
1/11,111,111 = (1 × 8)/(11,111,111 × 8) = 8/88,888,888
Of course, the above fractions are reducing...
... to the initial fraction: 1/11,111,111
:: Final answer ::
Written in 3 different ways
As a reduced (simplified) positive proper fraction:
0.00000009 = 1/11,111,111
As a percentage:
0.00000009 ≈ 0%
As equivalent fractions:
0.00000009 = 1/11,111,111 = 5/55,555,555 = 8/88,888,888
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