Answer key for converting decimals to fractions worksheet.
Worksheet titled "Converting Decimals to Fractions (A) Answers" showing decimal numbers converted to fractions, with examples like 0.45 = 9/20 and 0.18 = 2/11, from Math-Drills.com.
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Step-by-step solution for: Converting Terminating and Repeating Decimals to Fractions (A)
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Show Answer Key & Explanations
Step-by-step solution for: Converting Terminating and Repeating Decimals to Fractions (A)
The task involves converting decimals to fractions. Below, I will explain the process for converting each decimal to a fraction and verify the solutions provided in the image.
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1. Identify the Place Value of the Decimal:
- For terminating decimals, write the decimal as a fraction where the numerator is the decimal number without the decimal point, and the denominator is a power of 10 corresponding to the place value.
- For repeating decimals, use algebraic methods to express them as fractions.
2. Simplify the Fraction:
- Reduce the fraction to its simplest form by dividing both the numerator and the denominator by their greatest common divisor (GCD).
3. Verify the Solution:
- Convert the fraction back to a decimal to ensure it matches the original decimal.
---
#### 1. Terminating Decimals:
- 0.45:
- Write as \( \frac{45}{100} \).
- Simplify by dividing numerator and denominator by their GCD (which is 5): \( \frac{45 \div 5}{100 \div 5} = \frac{9}{20} \).
- Answer: \( \frac{9}{20} \)
- 0.18:
- Write as \( \frac{18}{100} \).
- Simplify by dividing numerator and denominator by their GCD (which is 2): \( \frac{18 \div 2}{100 \div 2} = \frac{9}{50} \).
- However, the provided answer is \( \frac{2}{11} \), which suggests an error in the problem statement or solution key.
- 0.3:
- Write as \( \frac{3}{10} \).
- This is already in simplest form.
- Answer: \( \frac{3}{10} \)
- 0.95:
- Write as \( \frac{95}{100} \).
- Simplify by dividing numerator and denominator by their GCD (which is 5): \( \frac{95 \div 5}{100 \div 5} = \frac{19}{20} \).
- Answer: \( \frac{19}{20} \)
- 0.8:
- Write as \( \frac{8}{10} \).
- Simplify by dividing numerator and denominator by their GCD (which is 2): \( \frac{8 \div 2}{10 \div 2} = \frac{4}{5} \).
- Answer: \( \frac{4}{5} \)
- 0.63:
- Write as \( \frac{63}{100} \).
- This is already in simplest form.
- Answer: \( \frac{63}{100} \) (Note: The provided answer is \( \frac{7}{11} \), which suggests an error.)
- 0.54:
- Write as \( \frac{54}{100} \).
- Simplify by dividing numerator and denominator by their GCD (which is 2): \( \frac{54 \div 2}{100 \div 2} = \frac{27}{50} \).
- Answer: \( \frac{27}{50} \) (Note: The provided answer is \( \frac{6}{11} \), which suggests an error.)
- 0.125:
- Write as \( \frac{125}{1000} \).
- Simplify by dividing numerator and denominator by their GCD (which is 125): \( \frac{125 \div 125}{1000 \div 125} = \frac{1}{8} \).
- Answer: \( \frac{1}{8} \)
- 0.83:
- Write as \( \frac{83}{100} \).
- This is already in simplest form.
- Answer: \( \frac{83}{100} \) (Note: The provided answer is \( \frac{5}{6} \), which suggests an error.)
- 0.90:
- Write as \( \frac{90}{100} \).
- Simplify by dividing numerator and denominator by their GCD (which is 10): \( \frac{90 \div 10}{100 \div 10} = \frac{9}{10} \).
- Answer: \( \frac{9}{10} \) (Note: The provided answer is \( \frac{10}{11} \), which suggests an error.)
#### 2. Repeating Decimals:
- 0.285714:
- Let \( x = 0.\overline{285714} \).
- Multiply by \( 10^6 \) (since the repeating block has 6 digits): \( 10^6x = 285714.\overline{285714} \).
- Subtract the original \( x \): \( 10^6x - x = 285714 \).
- Simplify: \( 999999x = 285714 \).
- Solve for \( x \): \( x = \frac{285714}{999999} \).
- Simplify by dividing numerator and denominator by their GCD (which is 142857): \( \frac{285714 \div 142857}{999999 \div 142857} = \frac{2}{7} \).
- Answer: \( \frac{2}{7} \)
- 0.\(\overline{1}\):
- Let \( x = 0.\overline{1} \).
- Multiply by 10: \( 10x = 1.\overline{1} \).
- Subtract the original \( x \): \( 10x - x = 1 \).
- Simplify: \( 9x = 1 \).
- Solve for \( x \): \( x = \frac{1}{9} \).
- Answer: \( \frac{1}{9} \)
- 0.3:
- This is a terminating decimal, not a repeating decimal. See above for the solution.
- Answer: \( \frac{1}{3} \) (Note: This is incorrect for a terminating decimal.)
- 0.16:
- Write as \( \frac{16}{100} \).
- Simplify by dividing numerator and denominator by their GCD (which is 4): \( \frac{16 \div 4}{100 \div 4} = \frac{4}{25} \).
- Answer: \( \frac{4}{25} \) (Note: The provided answer is \( \frac{1}{6} \), which suggests an error.)
- 0.857142:
- Let \( x = 0.\overline{857142} \).
- Multiply by \( 10^6 \) (since the repeating block has 6 digits): \( 10^6x = 857142.\overline{857142} \).
- Subtract the original \( x \): \( 10^6x - x = 857142 \).
- Simplify: \( 999999x = 857142 \).
- Solve for \( x \): \( x = \frac{857142}{999999} \).
- Simplify by dividing numerator and denominator by their GCD (which is 142857): \( \frac{857142 \div 142857}{999999 \div 142857} = \frac{6}{7} \).
- Answer: \( \frac{6}{7} \)
- 0.\(\overline{4}\):
- Let \( x = 0.\overline{4} \).
- Multiply by 10: \( 10x = 4.\overline{4} \).
- Subtract the original \( x \): \( 10x - x = 4 \).
- Simplify: \( 9x = 4 \).
- Solve for \( x \): \( x = \frac{4}{9} \).
- Answer: \( \frac{4}{9} \)
- 0.875:
- Write as \( \frac{875}{1000} \).
- Simplify by dividing numerator and denominator by their GCD (which is 125): \( \frac{875 \div 125}{1000 \div 125} = \frac{7}{8} \).
- Answer: \( \frac{7}{8} \)
- 0.25:
- Write as \( \frac{25}{100} \).
- Simplify by dividing numerator and denominator by their GCD (which is 25): \( \frac{25 \div 25}{100 \div 25} = \frac{1}{4} \).
- Answer: \( \frac{1}{4} \)
- 0.428571:
- Let \( x = 0.\overline{428571} \).
- Multiply by \( 10^6 \) (since the repeating block has 6 digits): \( 10^6x = 428571.\overline{428571} \).
- Subtract the original \( x \): \( 10^6x - x = 428571 \).
- Simplify: \( 999999x = 428571 \).
- Solve for \( x \): \( x = \frac{428571}{999999} \).
- Simplify by dividing numerator and denominator by their GCD (which is 142857): \( \frac{428571 \div 142857}{999999 \div 142857} = \frac{3}{7} \).
- Answer: \( \frac{3}{7} \)
- 0.4:
- Write as \( \frac{4}{10} \).
- Simplify by dividing numerator and denominator by their GCD (which is 2): \( \frac{4 \div 2}{10 \div 2} = \frac{2}{5} \).
- Answer: \( \frac{2}{5} \)
---
After verifying all conversions, the correct answers are:
\[
\boxed{
\begin{aligned}
&0.45 = \frac{9}{20}, \quad 0.285714 = \frac{2}{7}, \\
&0.18 = \frac{9}{50}, \quad 0.\overline{1} = \frac{1}{9}, \\
&0.3 = \frac{3}{10}, \quad 0.\overline{3} = \frac{1}{3}, \\
&0.95 = \frac{19}{20}, \quad 0.16 = \frac{4}{25}, \\
&0.8 = \frac{4}{5}, \quad 0.857142 = \frac{6}{7}, \\
&0.63 = \frac{63}{100}, \quad 0.\overline{4} = \frac{4}{9}, \\
&0.54 = \frac{27}{50}, \quad 0.875 = \frac{7}{8}, \\
&0.125 = \frac{1}{8}, \quad 0.25 = \frac{1}{4}, \\
&0.83 = \frac{83}{100}, \quad 0.428571 = \frac{3}{7}, \\
&0.90 = \frac{9}{10}, \quad 0.4 = \frac{2}{5}.
\end{aligned}
}
\]
---
General Steps to Convert Decimals to Fractions:
1. Identify the Place Value of the Decimal:
- For terminating decimals, write the decimal as a fraction where the numerator is the decimal number without the decimal point, and the denominator is a power of 10 corresponding to the place value.
- For repeating decimals, use algebraic methods to express them as fractions.
2. Simplify the Fraction:
- Reduce the fraction to its simplest form by dividing both the numerator and the denominator by their greatest common divisor (GCD).
3. Verify the Solution:
- Convert the fraction back to a decimal to ensure it matches the original decimal.
---
Detailed Solutions:
#### 1. Terminating Decimals:
- 0.45:
- Write as \( \frac{45}{100} \).
- Simplify by dividing numerator and denominator by their GCD (which is 5): \( \frac{45 \div 5}{100 \div 5} = \frac{9}{20} \).
- Answer: \( \frac{9}{20} \)
- 0.18:
- Write as \( \frac{18}{100} \).
- Simplify by dividing numerator and denominator by their GCD (which is 2): \( \frac{18 \div 2}{100 \div 2} = \frac{9}{50} \).
- However, the provided answer is \( \frac{2}{11} \), which suggests an error in the problem statement or solution key.
- 0.3:
- Write as \( \frac{3}{10} \).
- This is already in simplest form.
- Answer: \( \frac{3}{10} \)
- 0.95:
- Write as \( \frac{95}{100} \).
- Simplify by dividing numerator and denominator by their GCD (which is 5): \( \frac{95 \div 5}{100 \div 5} = \frac{19}{20} \).
- Answer: \( \frac{19}{20} \)
- 0.8:
- Write as \( \frac{8}{10} \).
- Simplify by dividing numerator and denominator by their GCD (which is 2): \( \frac{8 \div 2}{10 \div 2} = \frac{4}{5} \).
- Answer: \( \frac{4}{5} \)
- 0.63:
- Write as \( \frac{63}{100} \).
- This is already in simplest form.
- Answer: \( \frac{63}{100} \) (Note: The provided answer is \( \frac{7}{11} \), which suggests an error.)
- 0.54:
- Write as \( \frac{54}{100} \).
- Simplify by dividing numerator and denominator by their GCD (which is 2): \( \frac{54 \div 2}{100 \div 2} = \frac{27}{50} \).
- Answer: \( \frac{27}{50} \) (Note: The provided answer is \( \frac{6}{11} \), which suggests an error.)
- 0.125:
- Write as \( \frac{125}{1000} \).
- Simplify by dividing numerator and denominator by their GCD (which is 125): \( \frac{125 \div 125}{1000 \div 125} = \frac{1}{8} \).
- Answer: \( \frac{1}{8} \)
- 0.83:
- Write as \( \frac{83}{100} \).
- This is already in simplest form.
- Answer: \( \frac{83}{100} \) (Note: The provided answer is \( \frac{5}{6} \), which suggests an error.)
- 0.90:
- Write as \( \frac{90}{100} \).
- Simplify by dividing numerator and denominator by their GCD (which is 10): \( \frac{90 \div 10}{100 \div 10} = \frac{9}{10} \).
- Answer: \( \frac{9}{10} \) (Note: The provided answer is \( \frac{10}{11} \), which suggests an error.)
#### 2. Repeating Decimals:
- 0.285714:
- Let \( x = 0.\overline{285714} \).
- Multiply by \( 10^6 \) (since the repeating block has 6 digits): \( 10^6x = 285714.\overline{285714} \).
- Subtract the original \( x \): \( 10^6x - x = 285714 \).
- Simplify: \( 999999x = 285714 \).
- Solve for \( x \): \( x = \frac{285714}{999999} \).
- Simplify by dividing numerator and denominator by their GCD (which is 142857): \( \frac{285714 \div 142857}{999999 \div 142857} = \frac{2}{7} \).
- Answer: \( \frac{2}{7} \)
- 0.\(\overline{1}\):
- Let \( x = 0.\overline{1} \).
- Multiply by 10: \( 10x = 1.\overline{1} \).
- Subtract the original \( x \): \( 10x - x = 1 \).
- Simplify: \( 9x = 1 \).
- Solve for \( x \): \( x = \frac{1}{9} \).
- Answer: \( \frac{1}{9} \)
- 0.3:
- This is a terminating decimal, not a repeating decimal. See above for the solution.
- Answer: \( \frac{1}{3} \) (Note: This is incorrect for a terminating decimal.)
- 0.16:
- Write as \( \frac{16}{100} \).
- Simplify by dividing numerator and denominator by their GCD (which is 4): \( \frac{16 \div 4}{100 \div 4} = \frac{4}{25} \).
- Answer: \( \frac{4}{25} \) (Note: The provided answer is \( \frac{1}{6} \), which suggests an error.)
- 0.857142:
- Let \( x = 0.\overline{857142} \).
- Multiply by \( 10^6 \) (since the repeating block has 6 digits): \( 10^6x = 857142.\overline{857142} \).
- Subtract the original \( x \): \( 10^6x - x = 857142 \).
- Simplify: \( 999999x = 857142 \).
- Solve for \( x \): \( x = \frac{857142}{999999} \).
- Simplify by dividing numerator and denominator by their GCD (which is 142857): \( \frac{857142 \div 142857}{999999 \div 142857} = \frac{6}{7} \).
- Answer: \( \frac{6}{7} \)
- 0.\(\overline{4}\):
- Let \( x = 0.\overline{4} \).
- Multiply by 10: \( 10x = 4.\overline{4} \).
- Subtract the original \( x \): \( 10x - x = 4 \).
- Simplify: \( 9x = 4 \).
- Solve for \( x \): \( x = \frac{4}{9} \).
- Answer: \( \frac{4}{9} \)
- 0.875:
- Write as \( \frac{875}{1000} \).
- Simplify by dividing numerator and denominator by their GCD (which is 125): \( \frac{875 \div 125}{1000 \div 125} = \frac{7}{8} \).
- Answer: \( \frac{7}{8} \)
- 0.25:
- Write as \( \frac{25}{100} \).
- Simplify by dividing numerator and denominator by their GCD (which is 25): \( \frac{25 \div 25}{100 \div 25} = \frac{1}{4} \).
- Answer: \( \frac{1}{4} \)
- 0.428571:
- Let \( x = 0.\overline{428571} \).
- Multiply by \( 10^6 \) (since the repeating block has 6 digits): \( 10^6x = 428571.\overline{428571} \).
- Subtract the original \( x \): \( 10^6x - x = 428571 \).
- Simplify: \( 999999x = 428571 \).
- Solve for \( x \): \( x = \frac{428571}{999999} \).
- Simplify by dividing numerator and denominator by their GCD (which is 142857): \( \frac{428571 \div 142857}{999999 \div 142857} = \frac{3}{7} \).
- Answer: \( \frac{3}{7} \)
- 0.4:
- Write as \( \frac{4}{10} \).
- Simplify by dividing numerator and denominator by their GCD (which is 2): \( \frac{4 \div 2}{10 \div 2} = \frac{2}{5} \).
- Answer: \( \frac{2}{5} \)
---
Final Answer:
After verifying all conversions, the correct answers are:
\[
\boxed{
\begin{aligned}
&0.45 = \frac{9}{20}, \quad 0.285714 = \frac{2}{7}, \\
&0.18 = \frac{9}{50}, \quad 0.\overline{1} = \frac{1}{9}, \\
&0.3 = \frac{3}{10}, \quad 0.\overline{3} = \frac{1}{3}, \\
&0.95 = \frac{19}{20}, \quad 0.16 = \frac{4}{25}, \\
&0.8 = \frac{4}{5}, \quad 0.857142 = \frac{6}{7}, \\
&0.63 = \frac{63}{100}, \quad 0.\overline{4} = \frac{4}{9}, \\
&0.54 = \frac{27}{50}, \quad 0.875 = \frac{7}{8}, \\
&0.125 = \frac{1}{8}, \quad 0.25 = \frac{1}{4}, \\
&0.83 = \frac{83}{100}, \quad 0.428571 = \frac{3}{7}, \\
&0.90 = \frac{9}{10}, \quad 0.4 = \frac{2}{5}.
\end{aligned}
}
\]
Parent Tip: Review the logic above to help your child master the concept of converting decimals into fractions worksheet.
