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Questions and Answers
In converting recurring decimals to fractions, the power of 10 by which 'x' is multiplied depends on the number of ______ digits.
In converting recurring decimals to fractions, the power of 10 by which 'x' is multiplied depends on the number of ______ digits.
repeating
When converting a mixed recurring decimal, the first step involves multiplying 'x' by a power of 10 to move the ______ digits to the left of the decimal.
When converting a mixed recurring decimal, the first step involves multiplying 'x' by a power of 10 to move the ______ digits to the left of the decimal.
non-repeating
In the conversion of recurring decimals to fractions, the purpose of subtracting the original equation from the new equation is to ______ the repeating part of the decimal.
In the conversion of recurring decimals to fractions, the purpose of subtracting the original equation from the new equation is to ______ the repeating part of the decimal.
eliminate
Before simplifying the fraction after converting a recurring decimal, one must ______ for 'x' to find the fraction.
Before simplifying the fraction after converting a recurring decimal, one must ______ for 'x' to find the fraction.
A decimal like 0.456456456... is considered a ______ recurring decimal because all the digits after the decimal point repeat.
A decimal like 0.456456456... is considered a ______ recurring decimal because all the digits after the decimal point repeat.
A decimal like 0.1666... is considered a ______ recurring decimal because it has non-repeating digits followed by repeating digits after the decimal point.
A decimal like 0.1666... is considered a ______ recurring decimal because it has non-repeating digits followed by repeating digits after the decimal point.
To convert 0.333... to a fraction, first let x = 0.333..., then multiply by 10 to get 10x = 3.333.... Subtracting the original equation results in 9x = 3, which simplifies to x = ______.
To convert 0.333... to a fraction, first let x = 0.333..., then multiply by 10 to get 10x = 3.333.... Subtracting the original equation results in 9x = 3, which simplifies to x = ______.
To convert 0.121212... to a fraction, you would multiply by 100 because there are ______ repeating digits.
To convert 0.121212... to a fraction, you would multiply by 100 because there are ______ repeating digits.
When converting 0.1666... to a fraction, you initially multiply by 10 to get 1.666... and then by 100 to get 16.666.... Subtracting the first from the second gives 90x = 15, which simplifies to x = ______.
When converting 0.1666... to a fraction, you initially multiply by 10 to get 1.666... and then by 100 to get 16.666.... Subtracting the first from the second gives 90x = 15, which simplifies to x = ______.
When converting 0.21333... to a fraction, one multiplies by 100 and 1000, eventually leading to 900x = 192. This simplifies to x = ______.
When converting 0.21333... to a fraction, one multiplies by 100 and 1000, eventually leading to 900x = 192. This simplifies to x = ______.
The final step in converting recurring decimals to fractions is to ______ the fraction to its lowest terms.
The final step in converting recurring decimals to fractions is to ______ the fraction to its lowest terms.
A common mistake when converting recurring decimals is ______ the final fraction, which leads to an unsimplified answer.
A common mistake when converting recurring decimals is ______ the final fraction, which leads to an unsimplified answer.
When dealing with mixed recurring decimals, it is important to ensure that the correct ______ of 10 are used for multiplication to accurately eliminate the repeating decimals.
When dealing with mixed recurring decimals, it is important to ensure that the correct ______ of 10 are used for multiplication to accurately eliminate the repeating decimals.
The first step in solving recurring decimals is assigning the variable ‘x’ to the ______ decimal.
The first step in solving recurring decimals is assigning the variable ‘x’ to the ______ decimal.
Recurring decimals can be expressed as ______, through algebraic manipulation.
Recurring decimals can be expressed as ______, through algebraic manipulation.
In converting recurring decimals, multiplying ‘x’ by powers of ten helps to shift digits to the ______ side of the decimal
In converting recurring decimals, multiplying ‘x’ by powers of ten helps to shift digits to the ______ side of the decimal
A key step in converting recurring decimals involves ______ the equations to eliminate the repeating decimal part.
A key step in converting recurring decimals involves ______ the equations to eliminate the repeating decimal part.
To convert a pure recurring decimal to a fraction, if there are two digits repeating, multiply by ______.
To convert a pure recurring decimal to a fraction, if there are two digits repeating, multiply by ______.
The aim of multiplying by power of 10 and subtracting is to get the value of ______.
The aim of multiplying by power of 10 and subtracting is to get the value of ______.
Conversion of recurring decimal into fractions involves algebraic ______.
Conversion of recurring decimal into fractions involves algebraic ______.
Flashcards
Recurring Decimals
Recurring Decimals
Decimals with repeating digits that can be expressed as fractions.
Pure Recurring Decimal
Pure Recurring Decimal
Recurring decimals where all digits after the decimal point repeat.
Converting Pure Recurring Decimals
Converting Pure Recurring Decimals
Multiply 'x' by 10, 100, 1000... based on repeating digits.
Mixed Recurring Decimal
Mixed Recurring Decimal
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Converting Mixed Recurring Decimals
Converting Mixed Recurring Decimals
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Simplify Fractions
Simplify Fractions
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Incorrect Multiplication
Incorrect Multiplication
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Forgetting to Simplify
Forgetting to Simplify
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Subtraction Errors
Subtraction Errors
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Decimal Type Confusion
Decimal Type Confusion
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Study Notes
- Recurring decimals, also known as repeating decimals, can be expressed as fractions.
- The process involves algebraic manipulation and understanding place value.
- Let 'x' equal the recurring decimal.
Pure Recurring Decimals
- Pure recurring decimals have all digits after the decimal point repeating.
- E.g., 0.333..., 0.121212..., 0.456456456... are pure recurring decimals.
- To convert them into fractions, multiply 'x' by a power of 10 (10, 100, 1000, etc.).
- The power of 10 depends on the number of repeating digits.
- If one digit repeats, multiply by 10 (e.g., for 0.333...).
- If two digits repeat, multiply by 100 (e.g., for 0.121212...).
- If three digits repeat, multiply by 1000 (e.g., for 0.456456456...).
- Subtract the original equation (x = recurring decimal) from the new equation (10x, 100x, 1000x, etc.).
- This eliminates the repeating part of the decimal.
- Solve the resulting equation for 'x,' which will be in fractional form.
- Simplify the fraction to its lowest terms if possible.
Example 1: Converting 0.333... to a Fraction
- Let x = 0.333...
- Multiply by 10: 10x = 3.333...
- Subtract the original equation: 10x - x = 3.333... - 0.333...
- Simplify: 9x = 3
- Solve for x: x = 3/9
- Simplify the fraction: x = 1/3
Example 2: Converting 0.121212... to a Fraction
- Let x = 0.121212...
- Multiply by 100: 100x = 12.121212...
- Subtract the original equation: 100x - x = 12.121212... - 0.121212...
- Simplify: 99x = 12
- Solve for x: x = 12/99
- Simplify the fraction: x = 4/33
Mixed Recurring Decimals
- Mixed recurring decimals have non-repeating digits after the decimal point, followed by repeating digits.
- E.g., 0.1666..., 0.21333..., 1.2545454... are mixed recurring decimals.
- To convert into fractions, first, multiply 'x' by a power of 10 to move the non-repeating digits to the left of the decimal.
- Then, multiply by another power of 10 to move one repeating block of digits to the left of the decimal.
- Subtract the first equation from the second equation to eliminate the repeating decimals.
- Solve the resulting equation for 'x,' and simplify the fraction.
Example 3: Converting 0.1666... to a Fraction
- Let x = 0.1666...
- Multiply by 10: 10x = 1.666...
- Multiply by 100: 100x = 16.666...
- Subtract the equations: 100x - 10x = 16.666... - 1.666...
- Simplify: 90x = 15
- Solve for x: x = 15/90
- Simplify the fraction: x = 1/6
Example 4: Converting 0.21333... to a Fraction
- Let x = 0.21333...
- Multiply by 100: 100x = 21.333...
- Multiply by 1000: 1000x = 213.333...
- Subtract the equations: 1000x - 100x = 213.333... - 21.333...
- Simplify: 900x = 192
- Solve for x: x = 192/900
- Simplify the fraction: x = 16/75
General Steps Summary
- Identify if the decimal is pure recurring or mixed recurring.
- Let x = the recurring decimal.
- Multiply x by appropriate powers of 10 to shift repeating digits to the left of the decimal.
- Subtract the equations to eliminate the repeating part.
- Solve for x.
- Simplify the resulting fraction.
Tips and Tricks
- Always simplify the fraction to its lowest terms.
- Be careful with mixed recurring decimals, ensuring correct powers of 10 are used for multiplication.
- Practice with various examples to become proficient in converting recurring decimals to fractions.
Common Mistakes
- Incorrectly multiplying by powers of 10.
- Forgetting to simplify the final fraction.
- Making errors during subtraction.
- Not recognizing the difference between pure and mixed recurring decimals.
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