Resourcesdown chevron
Testimonials
Plans
Sign in
Sign up
Resourcesright arrow
Testimonials
Plans
Bytelearn - cat image with glassesAI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Write the repeating decimal as a fraction.\newline.189189189.189189189

View step-by-step helpView step-by-step help
Homeright chevron icon
Math Problemsright chevron icon
Algebra 2right chevron icon
Write a repeating decimal as a fractionright chevron icon

Full solution

Q. Write the repeating decimal as a fraction.\newline.189189189.189189189
  1. Denote Decimal as xx: Let's denote the repeating decimal 0.189189189…0.189189189\ldots as xx.x=0.189189189…x = 0.189189189\ldots
  2. Express as Infinite Sum: To convert this repeating decimal into a fraction, we first express it as an infinite sum of its repeating parts.\newlinex=0.189+0.000189+0.000000189+…x = 0.189 + 0.000189 + 0.000000189 + \ldots
  3. Identify Geometric Series: Notice that each term is 10001000 times smaller than the previous term. This is a geometric series with the first term a=0.189a = 0.189 and the common ratio r=11000.r = \frac{1}{1000}.
  4. Use Sum Formula: We can use the formula for the sum of an infinite geometric series, which is S=a1−rS = \frac{a}{1 - r}, where SS is the sum, aa is the first term, and rr is the common ratio.
  5. Substitute Values: Substitute the values of aa and rr into the formula to find the sum SS which is equal to xx.x=0.189(1−11000)x = \frac{0.189}{(1 - \frac{1}{1000})}
  6. Simplify Denominator: Simplify the denominator. x=0.1899991000x = \frac{0.189}{\frac{999}{1000}}
  7. Clear Fraction: Multiply both the numerator and the denominator by 10001000 to clear the fraction in the denominator.\newlinex=(0.189×1000)/(9991000×1000)x = (0.189 \times 1000) / (\frac{999}{1000} \times 1000)
  8. Find GCD: Perform the multiplication. x=189999x = \frac{189}{999}
  9. Simplify Fraction: Now we simplify the fraction by finding the greatest common divisor (GCD) of 189189 and 999999. The GCD of 189189 and 999999 is 99.
  10. Find GCD Again: Divide both the numerator and the denominator by the GCD to simplify the fraction. x=(1899)/(9999)x = (\frac{189}{9}) / (\frac{999}{9})
  11. Final Simplified Fraction: Perform the division. x=21111x = \frac{21}{111}
  12. Final Simplified Fraction: Perform the division. x=21111x = \frac{21}{111} We can simplify the fraction further by finding the GCD of 2121 and 111111, which is 33.
  13. Final Simplified Fraction: Perform the division.\newlinex=21111x = \frac{21}{111}We can simplify the fraction further by finding the GCD of 2121 and 111111, which is 33.Divide both the numerator and the denominator by the GCD to get the final simplified fraction.\newlinex=(21/3)(111/3)x = \frac{(21 / 3)}{(111 / 3)}
  14. Final Simplified Fraction: Perform the division.\newlinex=21111x = \frac{21}{111}We can simplify the fraction further by finding the GCD of 2121 and 111111, which is 33.Divide both the numerator and the denominator by the GCD to get the final simplified fraction.\newlinex=(21/3)(111/3)x = \frac{(21 / 3)}{(111 / 3)}Perform the division.\newlinex=737x = \frac{7}{37}

More problems from Write a repeating decimal as a fraction

Question
Find the sum of the finite arithmetic series. ∑n=110(7n+4)\sum_{n=1}^{10} (7n+4)\newline______
Get tutor helpright-arrow

Posted 9 months ago

Question
Type the missing number in this sequence: `55,\59,\63,\text{_____},\71,\75,\79`
Get tutor helpright-arrow

Posted 5 months ago

Question
Type the missing number in this sequence: `2, 4 8`, ______,`32`
Get tutor helpright-arrow

Posted 5 months ago

Question
What kind of sequence is this? 2,10,50,250,…2, 10, 50, 250, \ldots Choices:Choices:\newline[A]arithmetic\text{[A]arithmetic}\newline[B]geometric\text{[B]geometric}\newline[C]both\text{[C]both}\newline[D]neither\text{[D]neither}
Get tutor helpright-arrow

Posted 5 months ago

Question
What is the missing number in this pattern? 1,4,9,16,25,36,49,64,81,____1, 4, 9, 16, 25, 36, 49, 64, 81, \_\_\_\_
Get tutor helpright-arrow

Posted 7 months ago

Question
Classify the series. ∑n=012(n+2)3 \sum_{n = 0}^{12} (n + 2)^3 \newlineChoices:\newline[A]arithmetic\text{[A]arithmetic}\newline[B]geometric\text{[B]geometric}\newline[C]both\text{[C]both}\newline[D]neither\text{[D]neither}
Get tutor helpright-arrow

Posted 5 months ago

Question
Find the first three partial sums of the series.\newline1+6+11+16+21+26+⋯1 + 6 + 11 + 16 + 21 + 26 + \cdots\newlineWrite your answers as integers or fractions in simplest form.\newlineS1=S_1 = ____\newlineS2=S_2 = ____\newlineS3=S_3 = ____
Get tutor helpright-arrow

Posted 7 months ago

Question
Find the third partial sum of the series. \newline3+9+15+21+27+33+⋯3 + 9 + 15 + 21 + 27 + 33 + \cdots\newlineWrite your answer as an integer or a fraction in simplest form. \newlineS3=S_3 = ____
Get tutor helpright-arrow

Posted 7 months ago

Question
Find the first three partial sums of the series. \newline1+7+13+19+25+31+⋯1 + 7 + 13 + 19 + 25 + 31 + \cdots\newlineWrite your answers as integers or fractions in simplest form. \newlineS1=S_1 = ____\newlineS2=S_2 = ____\newlineS3=S_3 = ____
Get tutor helpright-arrow

Posted 9 months ago

Question
Does the infinite geometric series converge or diverge?\newline1+34+916+2764+⋯1 + \frac{3}{4} + \frac{9}{16} + \frac{27}{64} + \cdots\newlineChoices:\newline[A]converge\text{[A]converge}\newline[B]diverge\text{[B]diverge}
Get tutor helpright-arrow

Posted 5 months ago

View all (78)

Related topics

Algebra - Order of OperationsAlgebra - Distributive Property`X` and `Y` AxesGeometry - Scalene TriangleCommon MultipleGeometry - Quadrant