2.2 Decimals and Conversions
Terminating Decimals
Terminating decimals are decimals that end — they have a finite number of digits after the decimal point.
Recurring decimals have a pattern of digits that repeats forever. We use dots above the digits to show which part repeats.
There's a clever algebraic method to convert any recurring decimal into a fraction.
Terminating decimals are decimals that end — they have a finite number of digits after the decimal point.
Examples of Terminating Decimals:
Recurring Decimals
• $\frac{1}{5} = 0.2$
• $\frac{1}{4} = 0.25$
• $\frac{3}{8} = 0.375$
• $\frac{7}{20} = 0.35$
• $\frac{1}{4} = 0.25$
• $\frac{3}{8} = 0.375$
• $\frac{7}{20} = 0.35$
Recurring decimals have a pattern of digits that repeats forever. We use dots above the digits to show which part repeats.
Notation for Recurring Decimals:
Converting Recurring Decimals to Fractions
• A single dot means that digit repeats: $0.\dot{3} = 0.3333...$
• Dots at the start and end show the repeating block: $0.\dot{1}4285\dot{7} = 0.142857142857...$
• $\frac{1}{3} = 0.\dot{3}$
• $\frac{1}{7} = 0.\dot{1}4285\dot{7}$
• $\frac{2}{11} = 0.\dot{1}\dot{8} = 0.181818...$
• Dots at the start and end show the repeating block: $0.\dot{1}4285\dot{7} = 0.142857142857...$
• $\frac{1}{3} = 0.\dot{3}$
• $\frac{1}{7} = 0.\dot{1}4285\dot{7}$
• $\frac{2}{11} = 0.\dot{1}\dot{8} = 0.181818...$
There's a clever algebraic method to convert any recurring decimal into a fraction.
Example 1: Convert $0.\dot{7}4\dot{5}$ (i.e., $0.745745745...$) to a fraction
Step 1: Let $x = 0.745745745...$
Step 2: Multiply by 1000 (to shift one complete repeating block left of the decimal)
$$1000x = 745.745745...$$
Step 3: Subtract the original equation
$$1000x - x = 745.745745... - 0.745745...$$ $$999x = 745$$
Step 4: Solve for $x$
$$x = \frac{745}{999}$$
Step 2: Multiply by 1000 (to shift one complete repeating block left of the decimal)
$$1000x = 745.745745...$$
Step 3: Subtract the original equation
$$1000x - x = 745.745745... - 0.745745...$$ $$999x = 745$$
Step 4: Solve for $x$
$$x = \frac{745}{999}$$
Example 2: Convert $0.6\dot{8}$ (i.e., $0.68888...$) to a fraction
Step 1: Let $x = 0.6888...$
Step 2: We need two equations where the recurring part lines up
$$10x = 6.888...$$ $$100x = 68.888...$$
Step 3: Subtract to eliminate the recurring part
$$100x - 10x = 68.888... - 6.888...$$ $$90x = 62$$
Step 4: Solve and simplify
$$x = \frac{62}{90} = \frac{31}{45}$$
Step 2: We need two equations where the recurring part lines up
$$10x = 6.888...$$ $$100x = 68.888...$$
Step 3: Subtract to eliminate the recurring part
$$100x - 10x = 68.888... - 6.888...$$ $$90x = 62$$
Step 4: Solve and simplify
$$x = \frac{62}{90} = \frac{31}{45}$$
💡 The opposite: Converting Fractions to Recurring Decimals
If you can convert the denominator to all 9s, the numerator gives you the recurring decimal!
Example: $\frac{21}{33}$
Multiply by $\frac{3}{3}$: $\frac{21 \times 3}{33 \times 3} = \frac{63}{99}$
So $\frac{21}{33} = 0.\dot{6}\dot{3}$
Example: $\frac{39}{111}$
Multiply by $\frac{9}{9}$: $\frac{39 \times 9}{111 \times 9} = \frac{351}{999}$
So $\frac{39}{111} = 0.\dot{3}5\dot{1}$
Example: $\frac{21}{33}$
Multiply by $\frac{3}{3}$: $\frac{21 \times 3}{33 \times 3} = \frac{63}{99}$
So $\frac{21}{33} = 0.\dot{6}\dot{3}$
Example: $\frac{39}{111}$
Multiply by $\frac{9}{9}$: $\frac{39 \times 9}{111 \times 9} = \frac{351}{999}$
So $\frac{39}{111} = 0.\dot{3}5\dot{1}$
🧮 Recurring Decimal Converter
Enter a recurring decimal (e.g., 0.333 for $0.\dot{3}$, or 0.142857142857 for $0.\dot{1}4285\dot{7}$):
🎯 Recognition: Identify the Decimal Type
Real Life Uses:
• Measurements: Converting between metric and imperial often gives recurring decimals
• Computing: Understanding why computers sometimes have tiny rounding errors
• Sharing: Dividing things equally (3 people sharing 10 cookies = 3.333... each)
• Measurements: Converting between metric and imperial often gives recurring decimals
• Computing: Understanding why computers sometimes have tiny rounding errors
• Sharing: Dividing things equally (3 people sharing 10 cookies = 3.333... each)
The Conversion Triangle
Fractions, decimals, and percentages are three different ways of expressing the same value. Here's how to convert between them:
Simply divide the numerator by the denominator.
Multiply by 100 (or move the decimal point 2 places right).
Divide by 100 (or move the decimal point 2 places left).
Use place value to write the decimal as a fraction, then simplify.
Fractions, decimals, and percentages are three different ways of expressing the same value. Here's how to convert between them:
💡 Conversion Guide:
Fraction to Decimal
Fraction → Decimal: Divide the numerator by the denominator
Decimal → Percentage: Multiply by 100
Percentage → Decimal: Divide by 100
Decimal → Fraction: Use place value (tenths, hundredths, etc.)
Fraction → Percentage: Convert to decimal first, then × 100
Decimal → Percentage: Multiply by 100
Percentage → Decimal: Divide by 100
Decimal → Fraction: Use place value (tenths, hundredths, etc.)
Fraction → Percentage: Convert to decimal first, then × 100
Simply divide the numerator by the denominator.
Examples:
Decimal to Percentage
$$\frac{3}{4} = 3 \div 4 = 0.75$$
$$\frac{5}{8} = 5 \div 8 = 0.625$$
$$\frac{3}{7} = 3 \div 7 = 0.42857...$$
Multiply by 100 (or move the decimal point 2 places right).
Examples:
Percentage to Decimal
$$0.75 \times 100 = 75\%$$
$$0.125 \times 100 = 12.5\%$$
$$0.42857 \times 100 \approx 42.9\%$$
Divide by 100 (or move the decimal point 2 places left).
Examples:
Decimal to Fraction (Terminating Decimals)
$$45\% \div 100 = 0.45$$
$$7.5\% \div 100 = 0.075$$
$$125\% \div 100 = 1.25$$
Use place value to write the decimal as a fraction, then simplify.
Examples:
$0.6$ = 6 tenths = $\frac{6}{10} = \frac{3}{5}$
$0.65$ = 65 hundredths = $\frac{65}{100} = \frac{13}{20}$
$0.625$ = 625 thousandths = $\frac{625}{1000} = \frac{5}{8}$
$0.375$ = 375 thousandths = $\frac{375}{1000} = \frac{3}{8}$
$0.65$ = 65 hundredths = $\frac{65}{100} = \frac{13}{20}$
$0.625$ = 625 thousandths = $\frac{625}{1000} = \frac{5}{8}$
$0.375$ = 375 thousandths = $\frac{375}{1000} = \frac{3}{8}$
💡 Common Conversions
| Fraction | Decimal | Percentage |
|---|---|---|
| $\frac{1}{2}$ | 0.5 | 50% |
| $\frac{1}{4}$ | 0.25 | 25% |
| $\frac{3}{4}$ | 0.75 | 75% |
| $\frac{1}{5}$ | 0.2 | 20% |
| $\frac{1}{3}$ | $0.\dot{3}$ | 33.3...% |
| $\frac{1}{8}$ | 0.125 | 12.5% |
🧮 Conversion Calculator Conversion Calculator
Fraction → Decimal → Percentage:
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Decimal → Fraction & Percentage:
Percentage → Decimal & Fraction:
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🎯 Conversion Practice
Real Life Uses:
• Shopping: Calculating discounts (20% off = multiply by 0.8)
• Grades: Converting test scores (42/50 = 84%)
• Finance: Interest rates, tax calculations, tips
• Statistics: Survey results, probability, data analysis
• Cooking: Scaling recipes up or down
• Shopping: Calculating discounts (20% off = multiply by 0.8)
• Grades: Converting test scores (42/50 = 84%)
• Finance: Interest rates, tax calculations, tips
• Statistics: Survey results, probability, data analysis
• Cooking: Scaling recipes up or down