Recurring Decimals: Comprehensive Notes
Welcome to our detailed guide on Recurring Decimals. Whether you're a student mastering basic math concepts or someone revisiting these essential skills, this guide offers thorough explanations, properties, and a wide range of examples to help you understand and work with recurring decimals effectively.
Introduction
Recurring decimals, also known as repeating decimals, are decimal numbers that have digits or a sequence of digits repeating infinitely. Understanding recurring decimals is fundamental in mathematics, particularly in number theory, algebra, and real-world applications where precise numerical representations are essential.
Basic Concepts of Recurring Decimals
Before delving into the methods of working with recurring decimals, it's important to grasp the basic definitions and properties associated with them.
What is a Recurring Decimal?
A recurring decimal is a decimal number in which a digit or a group of digits repeats infinitely. This repetition can be of a single digit or a sequence of digits.
- Single Recurrence: A single digit repeats, e.g., 0.3 = 0.3333...
- Multiple Recurrences: A sequence of digits repeats, e.g., 0.142857 = 0.142857142857...
Terminating vs. Recurring Decimals
Decimals can be categorized into two types:
- Terminating Decimals: Decimals that end after a finite number of digits, e.g., 0.5, 2.75.
- Recurring Decimals: Decimals that have one or more repeating digits indefinitely, e.g., 0.6, 1.3
Properties of Recurring Decimals
Understanding the properties of recurring decimals is crucial for performing accurate conversions and calculations.
Relationship with Fractions
Every recurring decimal can be expressed as a fraction of two integers (a rational number). Conversely, every rational number can be represented as either a terminating or recurring decimal.
Period of Recurrence
The period of a recurring decimal is the number of digits in the repeating sequence.
- Example: In 0.142857, the period is 6.
- Example: In 0.3, the period is 1.
Representation Notation
Recurring decimals are often represented using an overline (vinculum) to indicate the repeating part.
- Single Digit: 0.7 = 0.7777...
- Multiple Digits: 0.123 = 0.123123123...
Methods of Working with Recurring Decimals
There are several methods to convert recurring decimals to fractions and vice versa. Below are the primary methods used for each type of conversion.
1. Converting Recurring Decimals to Fractions
There are systematic steps to convert a recurring decimal to a fraction. The method involves setting up an equation where the recurring decimal is represented as a variable and then solving for that variable.
Example: Convert 0.3 to a fraction.
Solution:
Let x = 0.3333...
10x = 3.3333...
Subtract the first equation from the second:
10x - x = 3.3333... - 0.3333...
9x = 3
x = 3/9 = 1/3
Therefore, 0.3 = 1/3
2. Converting Fractions to Recurring Decimals
To convert a fraction to a recurring decimal, perform the division of the numerator by the denominator using long division. If the decimal starts repeating, identify the repeating sequence.
Example: Convert 2/7 to a decimal.
Solution:
2 ÷ 7 = 0.285714285714...
Therefore, 2/7 = 0.285714
Calculations with Recurring Decimals
Performing calculations with recurring decimals requires a clear understanding of their properties and conversion methods. Below are key formulas and examples for each type of calculation.
Converting a Single Recurring Decimal to a Fraction
Formula: If x = 0.a, then x = a/9
Example: Convert 0.6 to a fraction.
Solution: 0.6 = 6/9 = 2/3
Converting a Multi-Digit Recurring Decimal to a Fraction
Formula: If x = 0.ab, then 100x = ab.ab and subtract to solve for x.
Example: Convert 0.12 to a fraction.
Solution:
Let x = 0.121212...
100x = 12.121212...
Subtract:
100x - x = 12.121212... - 0.121212...
99x = 12
x = 12/99 = 4/33
Therefore, 0.12 = 4/33
Adding and Subtracting Recurring Decimals
To add or subtract recurring decimals, it's often easier to convert them to fractions first, perform the operation, and then convert back to decimals if needed.
Example: Add 0.3 and 0.6.
Solution:
0.3 = 1/3
0.6 = 2/3
Sum = 1/3 + 2/3 = 3/3 = 1
Therefore, 0.3 + 0.6 = 1
Multiplying Recurring Decimals
Multiply the fractions corresponding to the recurring decimals and simplify.
Example: Multiply 0.2 by 0.5.
Solution:
0.2 = 2/9
0.5 = 5/9
Product = (2/9) × (5/9) = 10/81 ≈ 0.123456790
Dividing Recurring Decimals
Convert the recurring decimals to fractions, perform the division, and simplify if necessary.
Example: Divide 0.4 by 0.2.
Solution:
0.4 = 4/9
0.2 = 2/9
Division = (4/9) ÷ (2/9) = (4/9) × (9/2) = 4/2 = 2
Therefore, 0.4 ÷ 0.2 = 2
Examples of Recurring Decimals
Understanding through examples is key to mastering recurring decimals. Below are a variety of problems ranging from easy to hard, each with detailed solutions.
Example 1: Converting a Single Recurring Decimal to a Fraction
Problem: Convert 0.7 to a fraction.
Solution:
Let x = 0.7777...
10x = 7.7777...
Subtract:
10x - x = 7.7777... - 0.7777...
9x = 7
x = 7/9
Therefore, 0.7 = 7/9
Therefore, 0.7 is equal to 7/9.
Example 2: Converting a Multi-Digit Recurring Decimal to a Fraction
Problem: Convert 0.18 to a fraction.
Solution:
Let x = 0.181818...
100x = 18.181818...
Subtract:
100x - x = 18.181818... - 0.181818...
99x = 18
x = 18/99 = 2/11
Therefore, 0.18 = 2/11
Therefore, 0.18 is equal to 2/11.
Example 3: Converting a Recurring Decimal to a Fraction with Non-Repeating Part
Problem: Convert 0.53 to a fraction.
Solution:
Let x = 0.5333...
10x = 5.3333...
Subtract:
10x - x = 5.3333... - 0.5333...
9x = 4.8
x = 4.8/9 = 48/90 = 8/15
Therefore, 0.53 = 8/15
Therefore, 0.53 is equal to 8/15.
Example 4: Converting a Fraction to a Recurring Decimal
Problem: Convert 4/11 to a decimal.
Solution:
4 ÷ 11 = 0.363636...
Therefore, 4/11 = 0.36
Therefore, 4/11 is equal to 0.36.
Example 5: Adding Two Recurring Decimals
Problem: Add 0.1 and 0.2.
Solution:
Convert to fractions:
0.1 = 1/9
0.2 = 2/9
Sum = 1/9 + 2/9 = 3/9 = 1/3
Therefore, 0.1 + 0.2 = 1/3 = 0.3
Therefore, 0.1 + 0.2 equals 0.3.
Word Problems: Application of Recurring Decimals
Applying recurring decimals to real-life scenarios enhances understanding and demonstrates their practical utility. Here are several word problems that incorporate these concepts, along with their solutions.
Example 1: Dividing Money
Problem: You have $10 and want to divide it equally among 3 friends. How much does each friend get?
Solution:
$10 ÷ 3 = 3.3333...
Therefore, each friend gets $3.3
Therefore, each friend receives $3.3.
Example 2: Converting Measurements
Problem: A ribbon is 2.5 meters long. Convert this length to a fraction.
Solution:
2.5 = 2 + 0.5 = 2 + 1/2 = 5/2
Therefore, 2.5 meters is equal to 5/2 meters.
Therefore, the ribbon is 5/2 meters long.
Example 3: Calculating Interest
Problem: An investment of $1,000 earns an interest rate of 4.5% per annum. Calculate the interest earned after one year.
Solution:
Interest = 4.5% of $1,000 = 0.045 × 1000 = $45
Therefore, the interest earned is $45.
Example 4: Population Percentage
Problem: In a survey, 66.666...% of respondents prefer online learning. Express this percentage as a fraction.
Solution:
66.666...% = 0.6% × 100 = 2/3
Therefore, 66.6% = 2/3
Therefore, 66.6% is equal to 2/3.
Example 5: Discount Calculation
Problem: A store offers a 33.333...% discount on a jacket priced at $150. Calculate the discount amount and the final price.
Solution:
Discount Amount = 33.333...% of $150 = 0.3 × 150 = 50
Final Price = $150 - $50 = $100
Therefore, the discount amount is $50, and the final price is $100.
Strategies and Tips for Working with Recurring Decimals
Enhancing your skills in working with recurring decimals involves employing effective strategies and consistent practice. Here are some tips to help you improve:
1. Recognize Repeating Patterns
Identify the repeating sequence in the decimal to determine the period. This recognition is crucial for accurate conversions.
Example: In 0.142857, the repeating sequence is "142857" with a period of 6.
2. Use Algebraic Methods for Conversion
Set up equations to solve for the recurring decimal as a fraction. This systematic approach ensures precision.
Example: For 0.4, let x = 0.4444..., then 10x = 4.4444..., subtract to get 9x = 4, so x = 4/9.
3. Memorize Common Recurring Decimals and Their Fraction Equivalents
Having a mental repository of common recurring decimals and their corresponding fractions speeds up problem-solving.
- 0.3 = 1/3
- 0.6 = 2/3
- 0.142857 = 1/7
4. Practice Long Division
Develop proficiency in long division to accurately convert fractions to recurring decimals.
Example: 5 ÷ 6 = 0.8333...
5. Understand the Concept of Period
The period of a recurring decimal is the length of the repeating sequence. Knowing the period helps in setting up the correct equations for conversion.
Example: In 0.0588235294117647, the period is 16.
6. Use Visual Aids
Visual representations like repeating bars or colored patterns can help in identifying and understanding recurring decimals.
Example: Highlighting the repeating part in different colors.
7. Double-Check Your Work
After performing a conversion, verify by reversing the process to ensure accuracy.
Example: If you convert 0.3 to 1/3, multiply 1/3 by 3 to check if it equals 1.
8. Break Down Complex Decimals
For decimals with both non-repeating and repeating parts, handle each part separately before combining.
Example: Convert 0.53 to a fraction by separating into 0.5 and 0.3.
9. Utilize Technology and Tools
Use calculators, online converters, or educational apps to assist in learning and verifying your conversions.
Example: Use Wolfram Alpha to verify that 0.6 = 2/3.
10. Practice Regularly with Diverse Problems
Consistent practice with various types of recurring decimals enhances proficiency and builds confidence.
Example: Work on converting different recurring decimals to fractions and vice versa daily.
Common Mistakes in Working with Recurring Decimals and How to Avoid Them
Being aware of common errors can help you avoid them and improve your calculation accuracy.
1. Misplacing the Decimal Point
Mistake: Incorrectly moving the decimal point when setting up equations for conversion.
Solution: Carefully count the number of decimal places and align the recurring sequences correctly when multiplying.
Example:
Incorrect: For x = 0.3, setting 10x = 3.3... is correct.
Correct: Ensure that the recurring part aligns correctly during subtraction to eliminate it.
2. Incorrectly Identifying the Repeating Sequence
Mistake: Misidentifying the repeating sequence, leading to incorrect equations and fractions.
Solution: Carefully observe the decimal to determine the exact repeating sequence and its length.
Example:
Incorrect: For 0.12, assuming the repeating sequence is "1".
Correct: Recognize that the repeating sequence is "12".
3. Overlooking the Non-Repeating Part
Mistake: Ignoring the non-repeating part of a decimal with both non-repeating and repeating sections.
Solution: Handle the non-repeating part separately before addressing the repeating sequence.
Example:
Incorrect: Converting 0.53 by only focusing on the recurring part.
Correct: Split into 0.5 and 0.3, then convert each part and combine.
4. Forgetting to Simplify the Fraction
Mistake: Converting a recurring decimal to a fraction but leaving the fraction unsimplified.
Solution: Always simplify the resulting fraction to its lowest terms.
Example:
Incorrect: 0.6 = 6/9
Correct: 0.6 = 6/9 = 2/3
5. Misapplying Algebraic Methods
Mistake: Incorrectly setting up equations for conversion, leading to wrong fractions.
Solution: Follow the algebraic steps methodically, ensuring each step aligns with the properties of recurring decimals.
Example:
Incorrect: For x = 0.3, assuming 10x = 0.3
Correct: For x = 0.3, 10x = 3.3
6. Rounding Intermediate Steps Prematurely
Mistake: Rounding decimals during intermediate steps can lead to inaccuracies in the final result.
Solution: Maintain precision throughout the calculation and round only the final answer if necessary.
Example:
Incorrect: 0.3 = 0.3
Correct: 0.3 = 0.3333...
7. Confusing Recurring Decimals with Non-Recurring Ones
Mistake: Mistaking terminating decimals for recurring decimals or vice versa.
Solution: Carefully examine the decimal to determine if it terminates or repeats indefinitely.
Example:
Incorrect: Treating 0.5 as recurring.
Correct: Recognize that 0.5 is a terminating decimal, not recurring.
8. Not Practicing Enough
Mistake: Lack of practice leads to confusion and difficulty in performing conversions accurately.
Solution: Engage in regular practice with a variety of recurring decimals to build familiarity and confidence.
Example:
Practice converting 0.7, 0.58, and 0.142857 to fractions.
Practice Questions: Test Your Recurring Decimals Skills
Practicing with a variety of problems is key to mastering recurring decimals. Below are practice questions categorized by difficulty level, along with their solutions.
Level 1: Easy
- Convert 0.5 to a fraction.
- Convert 1/3 to a decimal.
- Convert 0.2 to a fraction.
- Convert 4/9 to a decimal.
- Convert 0.9 to a fraction.
Solutions:
-
Solution:
Let x = 0.5
10x = 5.5
10x - x = 5.5 - 0.5
9x = 5
x = 5/9 -
Solution:
1 ÷ 3 = 0.3 -
Solution:
Let x = 0.2
10x = 2.2
10x - x = 2.2 - 0.2
9x = 2
x = 2/9 -
Solution:
4 ÷ 9 = 0.4 -
Solution:
Let x = 0.9
10x = 9.9
10x - x = 9.9 - 0.9
9x = 9
x = 1
Level 2: Medium
- Convert 0.12 to a fraction.
- Convert 5/6 to a decimal.
- Convert 0.46 to a fraction.
- Convert 7/11 to a decimal.
- Convert 0.58 to a fraction.
Solutions:
-
Solution:
Let x = 0.12
100x = 12.12
100x - x = 12.12 - 0.12
99x = 12
x = 12/99 = 4/33 -
Solution:
5 ÷ 6 = 0.8333...
Therefore, 5/6 = 0.8333 -
Solution:
Let x = 0.46
10x = 4.66
100x = 46.66
100x - 10x = 46.66 - 4.66
90x = 42
x = 42/90 = 7/15 -
Solution:
7 ÷ 11 ≈ 0.6363...
Therefore, 7/11 = 0.63 -
Solution:
Let x = 0.58
100x = 58.58
100x - x = 58.58 - 0.58
99x = 58
x = 58/99
Therefore, 0.58 = 58/99
Level 3: Hard
- Convert 0.142857 to a fraction.
- Convert 7/9 to a decimal.
- Convert 0.273 to a fraction.
- Convert 22/7 to a decimal.
- Convert 0.0588235294117647 to a fraction.
Solutions:
-
Solution:
Let x = 0.142857
1000000x = 142857.142857
1000000x - x = 142857.142857 - 0.142857
999999x = 142857
x = 142857/999999 = 1/7 -
Solution:
7 ÷ 9 ≈ 0.7 -
Solution:
Let x = 0.273
100x = 27.33
10x = 2.7333...
100x - 10x = 27.333... - 2.7333...
90x = 24.6
x = 24.6/90 = 246/900 = 41/150 -
Solution:
22 ÷ 7 ≈ 3.142857142857...
Therefore, 22/7 ≈ 3.142857 -
Solution:
Let x = 0.0588235294117647
10000000000000000x = 588235294117647.0588235294117647
10000000000000000x - x = 588235294117647.0588235294117647 - 0.0588235294117647
9999999999999999x = 588235294117647
x = 588235294117647 / 9999999999999999 = 1/17
Combined Exercises: Examples and Solutions
Many mathematical problems require the use of recurring decimals in conjunction with other operations. Below are examples that incorporate these concepts alongside logical reasoning and application to real-world scenarios.
Example 1: Mixed Conversions
Problem: Convert 0.27 to a fraction, and then multiply it by 3.
Solution:
Convert to fraction:
Let x = 0.2777...
10x = 2.777...
Subtract:
10x - x = 2.777... - 0.2777...
9x = 2.5
x = 2.5/9 = 5/18
Multiply by 3:
5/18 × 3 = 15/18 = 5/6
Therefore, 0.27 as a fraction is 5/18, and multiplying by 3 gives 5/6.
Example 2: Financial Calculations
Problem: An item is sold at a price of $45.666... (with the "6" repeating). Express this price as a fraction.
Solution:
Let x = 45.666...
Subtract the integer part:
x - 45 = 0.666...
Let y = 0.666...
10y = 6.666...
10y - y = 6.666... - 0.666...
9y = 6
y = 6/9 = 2/3
Therefore, x = 45 + 2/3 = 135/3 + 2/3 = 137/3
Therefore, $45.666... is equal to 137/3 dollars.
Example 3: Engineering Measurements
Problem: A component requires a measurement of 0.0588235294117647 meters. Express this measurement as a fraction.
Solution:
Let x = 0.0588235294117647
Recognize the repeating sequence has a period of 16.
Multiply by 10^16:
10000000000000000x = 588235294117647.0588235294117647
Subtract:
10000000000000000x - x = 588235294117647.0588235294117647 - 0.0588235294117647
9999999999999999x = 588235294117647
x = 588235294117647 / 9999999999999999 = 1/17
Therefore, 0.0588235294117647 meters is equal to 1/17 meters.
Example 4: Comparing Fractions and Decimals
Problem: Determine whether 3/7 is greater than or less than 0.428571.
Solution:
Convert 3/7 to a decimal:
3 ÷ 7 ≈ 0.428571
Therefore, 3/7 = 0.428571, so they are equal.
Therefore, 3/7 is equal to 0.428571.
Example 5: Applying Recurring Decimals in Real-World Context
Problem: A recipe requires 0.3 cups of sugar. Express this measurement as a fraction and calculate how much sugar is needed for 4 servings.
Solution:
Convert to fraction:
Let x = 0.3
10x = 3.3
Subtract:
10x - x = 3.3 - 0.3
9x = 3
x = 3/9 = 1/3
Therefore, 0.3 cups = 1/3 cups
For 4 servings:
1/3 × 4 = 4/3 = 1 1/3 cups
Therefore, 0.3 cups of sugar is equal to 1/3 cups, and for 4 servings, you need 1 1/3 cups of sugar.
Practice Questions: Test Your Recurring Decimals Skills
Practicing with a variety of problems is key to mastering recurring decimals. Below are additional practice questions categorized by difficulty level, along with their solutions.
Level 1: Easy
- Convert 0.1 to a fraction.
- Convert 1/9 to a decimal.
- Convert 0.8 to a fraction.
- Convert 2/3 to a decimal.
- Convert 0.4 to a fraction.
Solutions:
-
Solution:
Let x = 0.1
10x = 1.1
10x - x = 1.1 - 0.1
9x = 1
x = 1/9 -
Solution:
1 ÷ 9 ≈ 0.1 -
Solution:
Let x = 0.8
10x = 8.8
10x - x = 8.8 - 0.8
9x = 8
x = 8/9 -
Solution:
2 ÷ 3 ≈ 0.6 -
Solution:
Let x = 0.4
10x = 4.4
10x - x = 4.4 - 0.4
9x = 4
x = 4/9
Level 2: Medium
- Convert 0.27 to a fraction.
- Convert 5/6 to a decimal.
- Convert 0.16 to a fraction.
- Convert 7/11 to a decimal.
- Convert 0.54 to a fraction.
Solutions:
-
Solution:
Let x = 0.27
100x = 27.27
100x - x = 27.27 - 0.27
99x = 27
x = 27/99 = 3/11 -
Solution:
5 ÷ 6 ≈ 0.8333 -
Solution:
Let x = 0.16
10x = 1.6
100x = 16.6
100x - 10x = 16.6 - 1.6
90x = 15
x = 15/90 = 1/6 -
Solution:
7 ÷ 11 ≈ 0.63 -
Solution:
Let x = 0.54
100x = 54.54
100x - x = 54.54 - 0.54
99x = 54
x = 54/99 = 6/11
Level 3: Hard
- Convert 0.142857 to a fraction.
- Convert 13/90 to a decimal.
- Convert 0.833 to a fraction.
- Convert 22/7 to a decimal.
- Convert 0.0588235294117647 to a fraction.
Solutions:
-
Solution:
Let x = 0.142857
1000000x = 142857.142857
1000000x - x = 142857.142857 - 0.142857
999999x = 142857
x = 142857/999999 = 1/7 -
Solution:
13 ÷ 90 ≈ 0.1444 -
Solution:
Let x = 0.833
10x = 8.33
100x = 83.33
100x - 10x = 83.333... - 8.333...
90x = 75
x = 75/90 = 5/6 -
Solution:
22 ÷ 7 ≈ 3.142857 -
Solution:
Let x = 0.0588235294117647
10000000000000000x = 588235294117647.0588235294117647
10000000000000000x - x = 588235294117647.0588235294117647 - 0.0588235294117647
9999999999999999x = 588235294117647
x = 588235294117647 / 9999999999999999 = 1/17
Combined Exercises: Examples and Solutions
Many mathematical problems require the use of recurring decimals in conjunction with other operations. Below are additional examples that incorporate these concepts alongside logical reasoning and application to real-world scenarios.
Example 1: Multi-Step Conversions
Problem: Convert 0.76 to a fraction, then subtract 1/4 from it.
Solution:
Convert to fraction:
Let x = 0.76
10x = 7.6
100x = 76.6
Subtract:
100x - 10x = 76.6 - 7.6
90x = 69
x = 69/90 = 23/30
Subtract 1/4:
23/30 - 1/4 = (23×4 - 1×30)/(30×4) = (92 - 30)/120 = 62/120 = 31/60
Therefore, 0.76 as a fraction is 23/30, and 23/30 - 1/4 = 31/60.
Example 2: Financial Calculations
Problem: An investment grows to 0.8333 times its original value. Express this growth factor as a fraction and calculate the original investment if the final amount is $1,000.
Solution:
Convert to fraction:
Let x = 0.8333
10x = 8.3333...
1000x = 833.3333...
Subtract:
1000x - 10x = 833.3333... - 8.3333...
990x = 825
x = 825/990 = 55/66 = 5/6
Therefore, growth factor = 5/6
Let original investment = y
5/6 * y = 1000
y = 1000 * 6/5 = 1200
Therefore, the growth factor is 5/6, and the original investment was $1,200.
Example 3: Engineering Measurements
Problem: A component requires a measurement of 0.142857 meters. Express this measurement as a fraction and calculate the total length required for 10 such components.
Solution:
Convert to fraction:
Let x = 0.142857
1000000x = 142857.142857
Subtract:
1000000x - x = 142857.142857 - 0.142857
999999x = 142857
x = 142857/999999 = 1/7
Total length for 10 components:
1/7 × 10 = 10/7 = 1 3/7 meters
Therefore, 0.142857 meters is equal to 1/7 meters, and 10 components require 10/7 meters or 1 3/7 meters.
Example 4: Comparing Decimals and Fractions
Problem: Determine whether 0.25 is greater than, less than, or equal to 1/4.
Solution:
Convert 1/4 to a decimal:
1 ÷ 4 = 0.25
Compare:
0.25 = 0.252525...
0.25 < 0.252525...
Therefore, 0.25 is greater than 1/4.
Therefore, 0.25 is greater than 1/4.
Example 5: Recurring Decimals in Real-World Context
Problem: A car's fuel efficiency is recorded as 0.6 miles per gallon. Convert this to a fraction and determine how many gallons of fuel are needed to drive 150 miles.
Solution:
Convert to fraction:
Let x = 0.6
10x = 6.6
10x - x = 6.6 - 0.6
9x = 6
x = 6/9 = 2/3
Fuel Efficiency = 2/3 miles per gallon
Gallons needed = 150 ÷ (2/3) = 150 × (3/2) = 225 gallons
Therefore, 0.6 miles per gallon is equal to 2/3 miles per gallon, and 225 gallons of fuel are needed to drive 150 miles.
Practice Questions: Test Your Recurring Decimals Skills
Practicing with a variety of problems is key to mastering recurring decimals. Below are additional practice questions categorized by difficulty level, along with their solutions.
Level 1: Easy
- Convert 0.2 to a fraction.
- Convert 1/5 to a decimal.
- Convert 0.7 to a fraction.
- Convert 3/4 to a decimal.
- Convert 0.1 to a fraction.
Solutions:
-
Solution:
Let x = 0.2
10x = 2.2
10x - x = 2.2 - 0.2
9x = 2
x = 2/9 -
Solution:
1 ÷ 5 = 0.2 -
Solution:
Let x = 0.7
10x = 7.7
10x - x = 7.7 - 0.7
9x = 7
x = 7/9 -
Solution:
3 ÷ 4 = 0.75 -
Solution:
Let x = 0.1
10x = 1.1
10x - x = 1.1 - 0.1
9x = 1
x = 1/9
Level 2: Medium
- Convert 0.45 to a fraction.
- Convert 7/8 to a decimal.
- Convert 0.16 to a fraction.
- Convert 5/12 to a decimal.
- Convert 0.63 to a fraction.
Solutions:
-
Solution:
Let x = 0.45
100x = 45.45
100x - x = 45.45 - 0.45
99x = 45
x = 45/99 = 5/11 -
Solution:
7 ÷ 8 = 0.875 -
Solution:
Let x = 0.16
10x = 1.66
100x = 16.66
100x - 10x = 16.666... - 1.666...
90x = 15
x = 15/90 = 1/6 -
Solution:
5 ÷ 12 ≈ 0.4166...
Therefore, 5/12 ≈ 0.4166 -
Solution:
Let x = 0.63
100x = 63.63
100x - x = 63.63 - 0.63
99x = 63
x = 63/99 = 7/11
Level 3: Hard
- Convert 0.0588235294117647 to a fraction.
- Convert 22/7 to a decimal.
- Convert 0.833 to a fraction.
- Convert 1.6 to a fraction.
- Convert 0.123456 to a fraction.
Solutions:
-
Solution:
Let x = 0.0588235294117647
Recognize the repeating sequence has a period of 16.
Multiply by 10^16:
10000000000000000x = 588235294117647.0588235294117647
Subtract:
10000000000000000x - x = 588235294117647.0588235294117647 - 0.0588235294117647
9999999999999999x = 588235294117647
x = 588235294117647 / 9999999999999999 = 1/17
Therefore, 0.0588235294117647 = 1/17 -
Solution:
22 ÷ 7 ≈ 3.142857 -
Solution:
Let x = 0.833
100x = 83.33
10x = 8.333...
100x - 10x = 83.333... - 8.333...
90x = 75
x = 75/90 = 5/6 -
Solution:
Let x = 1.6
x = 1 + 0.6
0.6 = 6/9 = 2/3
x = 1 + 2/3 = 5/3 -
Solution:
Let x = 0.123456
1000000x = 123456.123456
Subtract:
1000000x - x = 123456.123456 - 0.123456
999999x = 123456
x = 123456/999999 = 13717/111111
Simplify:
13717 ÷ 13717 = 1
111111 ÷ 13717 ≈ 8.111 (no further simplification)
Therefore, x = 13717/111111
Combined Exercises: Examples and Solutions
Many mathematical problems require the use of recurring decimals in conjunction with other operations. Below are additional examples that incorporate these concepts alongside logical reasoning and application to real-world scenarios.
Example 1: Multi-Step Conversions
Problem: Convert 0.63 to a fraction, then add it to 1/2.
Solution:
Convert to fraction:
Let x = 0.63
10x = 6.3
100x = 63.3
Subtract:
100x - 10x = 63.3 - 6.3
90x = 57
x = 57/90 = 19/30
Add to 1/2:
19/30 + 15/30 = 34/30 = 17/15
Therefore, 0.63 + 1/2 = 17/15.
Example 2: Financial Calculations
Problem: An investment grows to 1.25 times its original value, expressed as a recurring decimal. Convert this growth factor to a fraction and calculate the original investment if the final amount is $1,250.
Solution:
Convert to fraction:
1.25 is a terminating decimal, not recurring. Therefore, 1.25 = 5/4
Let original investment = x
5/4 × x = 1250
x = 1250 × 4/5 = 1000
Therefore, the original investment was $1,000.
Example 3: Engineering Measurements
Problem: A machine produces 0.142857 units per minute. Convert this rate to a fraction and calculate how many units it produces in 10 minutes.
Solution:
Convert to fraction:
Let x = 0.142857
1000000x = 142857.142857
Subtract:
1000000x - x = 142857.142857 - 0.142857
999999x = 142857
x = 142857/999999 = 1/7
Units per minute = 1/7
Units in 10 minutes = 10 × (1/7) = 10/7 ≈ 1.42857 units
Therefore, the machine produces approximately 1.42857 units per minute, and in 10 minutes, it produces approximately 10/7 units or 1.42857 units.
Example 4: Comparing Recurring Decimals and Fractions
Problem: Compare 0.285714 and 2/7.
Solution:
Convert 2/7 to a decimal:
2 ÷ 7 ≈ 0.285714
Therefore, 0.285714 = 2/7
Therefore, 0.285714 is equal to 2/7.
Example 5: Real-World Application in Finance
Problem: A loan requires monthly payments of 0.83 dollars. Express this payment as a fraction and calculate the total payment over 12 months.
Solution:
Convert to fraction:
Let x = 0.83
100x = 83.83
100x - x = 83.83 - 0.83
99x = 83
x = 83/99
Total payment over 12 months:
0.83 × 12 = 83/99 × 12 = 996/99 = 332/33 = 10 2/33 dollars
≈ $10.06
Therefore, 0.83 dollars is equal to 83/99 dollars, and the total payment over 12 months is approximately $10.06.
Summary
Understanding and working with recurring decimals are essential mathematical skills that facilitate easier calculations and clearer communication of numerical information. By grasping the fundamental concepts, mastering the conversion methods, and practicing consistently, you can confidently handle recurring decimals in various mathematical and real-world contexts.
Remember to:
- Recognize and identify recurring patterns in decimals.
- Use algebraic methods to convert recurring decimals to fractions accurately.
- Convert fractions to recurring decimals using long division.
- Understand the concept of the period of a recurring decimal.
- Memorize common recurring decimals and their corresponding fractions.
- Practice long division to improve conversion accuracy.
- Double-check your work by reversing conversions to ensure correctness.
- Handle decimals with both non-repeating and repeating parts by breaking them down.
- Utilize visual aids and step-by-step guides to track the conversion process.
- Leverage technology, such as calculators and online tools, to assist in complex conversions.
- Avoid common mistakes by carefully following conversion steps and verifying results.
- Teach others or explain your solutions to reinforce your understanding and identify any gaps.
With dedication and consistent practice, recurring decimals will become a fundamental skill in your mathematical toolkit, enhancing your problem-solving and analytical abilities.
Additional Resources
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