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.040040040.040040040

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.040040040.040040040
  1. Identify Repeating Pattern: Identify the repeating pattern in the decimal.\newlineThe repeating pattern is ＂040＂＂040＂, which repeats after the decimal point.
  2. Express as Infinite Sum: Express the repeating decimal as an infinite sum of its repeating parts.\newline0.040040040…=0.040+0.00040+0.00000040+…0.040040040\ldots = 0.040 + 0.00040 + 0.00000040 + \ldots
  3. Convert to Fractions: Convert each term of the sum into a fraction.\newline0.040040040…=401000+401000000+40100000000+…0.040040040\ldots = \frac{40}{1000} + \frac{40}{1000000} + \frac{40}{100000000} + \ldots
  4. Simplify Fractions: Simplify each fraction to its lowest terms. 0.040040040…=125+125000+125000000+…0.040040040\ldots = \frac{1}{25} + \frac{1}{25000} + \frac{1}{25000000} + \ldots
  5. Recognize Geometric Series: Recognize that the sum forms a geometric series with the first term a1=125a_1 = \frac{1}{25} and common ratio r=11000r = \frac{1}{1000}.
  6. Use Geometric Series Formula: Use the formula for the sum of an infinite geometric series, S=a11−rS = \frac{a_1}{1 - r}, to write the repeating decimal as a fraction.\newlineSubstitute a1=125a_1 = \frac{1}{25} and r=11000r = \frac{1}{1000} into the formula.\newlineS=1251−11000S = \frac{\frac{1}{25}}{1 - \frac{1}{1000}}
  7. Calculate Denominator: Calculate the denominator of the fraction.\newline1−11000=10001000−11000=99910001 - \frac{1}{1000} = \frac{1000}{1000} - \frac{1}{1000} = \frac{999}{1000}
  8. Calculate Fraction: Calculate the fraction by dividing the first term by the denominator.\newlineS=125/9991000S = \frac{1}{25} / \frac{999}{1000}\newlineS=125⋅1000999S = \frac{1}{25} \cdot \frac{1000}{999}
  9. Multiply Fractions: Multiply the fractions.\newlineS=100025×999S = \frac{1000}{25 \times 999}\newlineS=100024975S = \frac{1000}{24975}
  10. Simplify Fraction: Simplify the fraction to its lowest terms. S=40999S = \frac{40}{999}

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