1. Write the pure repeating (recurring) decimal number as a percentage.
Approximate to the desired number of decimal places (14):
0.037 ≈ 0.03703703703704
Multiply the number by 100/100.
The value of the number does not change when multiplying by 100/100.
Note: 100/100 = 1
0.03703703703704 =
0.03703703703704 × 100/100 =
(0.03703703703704 × 100)/100 =
3.703703703704/100 =
3.703703703704% ≈
3.7%
(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, %
0.037 ≈ 3.7%
2. Write the pure repeating (recurring) decimal number as a proper fraction.
0.037 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 = 0.037
Set up the second equation.
Number of decimal places repeating: 3
Multiply both sides of the first equation by 103 = 1,000
y = 0.037
1,000 × y = 1,000 × 0.037
1,000 × y = 37.037
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 = 37.037 - 0.037 ⇒
(1,000 - 1) × y = 37.037 - 0.037 ⇒
We now have a new equation:
999 × y = 37
Solve for y in the new equation.
999 × y = 37 ⇒
y = 37/999
Let the result written as a fraction.
Write the number as a fraction.
According to our first equation:
y = 0.037
According to our calculations:
y = 37/999
⇒ 0.037 = 37/999
3. Reduce (simplify) the fraction above: 37/999
(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):
37 is a prime number, it cannot be factored into other prime factors
999 = 33 × 37
Calculate the greatest (highest) common factor (divisor), GCF.
Multiply all the common prime factors by the lowest exponents.
GCF (37; 33 × 37) = 37
Divide both the numerator and the denominator by their greatest common factor, GCF.
37/999 =
37/(33 × 37) =
(37 ÷ 37) / ((33 × 37) ÷ 37) =
1/33 =
1/27
1/27: 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 3:
1/27 = (1 × 3)/(27 × 3) = 3/81
Example 2. By expanding the fraction by 6:
1/27 = (1 × 6)/(27 × 6) = 6/162
Of course, the above fractions are reducing...
... to the initial fraction: 1/27