1. Write the pure repeating (recurring) decimal number as a percentage.
Approximate to the desired number of decimal places (14):
0.966 ≈ 0.96696696696697
Multiply the number by 100/100.
The value of the number does not change when multiplying by 100/100.
Note: 100/100 = 1
0.96696696696697 =
0.96696696696697 × 100/100 =
(0.96696696696697 × 100)/100 =
96.696696696697/100 =
96.696696696697% ≈
96.7%
(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, %
0.966 ≈ 96.7%
2. Write the pure repeating (recurring) decimal number as a proper fraction.
0.966 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 = 0.966
Set up the second equation.
Number of decimal places repeating: 3
Multiply both sides of the first equation by 103 = 1,000
y = 0.966
1,000 × y = 1,000 × 0.966
1,000 × y = 966.966
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 × y - y = 966.966 - 0.966 ⇒
(1,000 - 1) × y = 966.966 - 0.966 ⇒
We now have a new equation:
999 × y = 966
Solve for y in the new equation.
999 × y = 966 ⇒
y = 966/999
Let the result written as a fraction.
Write the number as a fraction.
According to our first equation:
y = 0.966
According to our calculations:
y = 966/999
⇒ 0.966 = 966/999
3. Reduce (simplify) the fraction above: 966/999
(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):
966 = 2 × 3 × 7 × 23
999 = 33 × 37
Calculate the greatest (highest) common factor (divisor), GCF.
Multiply all the common prime factors by the lowest exponents.
GCF (2 × 3 × 7 × 23; 33 × 37) = 3
Divide both the numerator and the denominator by their greatest common factor, GCF.
966/999 =
(2 × 3 × 7 × 23)/(33 × 37) =
((2 × 3 × 7 × 23) ÷ 3) / ((33 × 37) ÷ 3) =
(2 × 7 × 23)/(32 × 37) =
322/333
322/333: 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 5:
322/333 = (322 × 5)/(333 × 5) = 1,610/1,665
Example 2. By expanding the fraction by 9:
322/333 = (322 × 9)/(333 × 9) = 2,898/2,997
Of course, the above fractions are reducing...
... to the initial fraction: 322/333