Converting Between Fractions and Decimals
Translate between fraction and decimal forms using division, place value, and repeating decimal notation.
Prices, measurements, and statistics almost always appear as decimals — but discounts and proportions often appear as fractions. A sales associate who can't quickly convert \(\frac{1}{8}\) to 0.125 (12.5% off) will give customers wrong discount prices. Fluent conversion between forms makes you faster, more accurate, and more versatile in any number-heavy role.
- Convert any fraction to a decimal by dividing the numerator by the denominator
- Convert terminating decimals to fractions using place value, then simplify
- Recognize common benchmark fractions and their decimal equivalents from memory
Fraction → Decimal: divide the numerator by the denominator. \(\frac{3}{4} = 3 \div 4 = 0.75\). It's that direct — a fraction bar means division.
Decimal → Fraction: read the place value of the last digit, write it over that power of 10, then simplify. Example: 0.6 = \(\frac{6}{10} = \frac{3}{5}\).
Repeating decimals arise from fractions whose denominators have prime factors other than 2 and 5 (like thirds and sevenths). \(\frac{1}{3} = 0.333...\) — it never ends, but we write it as \(0.\overline{3}\).
Benchmark Fraction ↔ Decimal Reference
Common Benchmarks to Memorize
| Fraction | Decimal | Division Check |
|---|---|---|
| \(\frac{1}{2}\) | 0.5 | 1 ÷ 2 = 0.5 |
| \(\frac{1}{4}\) | 0.25 | 1 ÷ 4 = 0.25 |
| \(\frac{3}{4}\) | 0.75 | 3 ÷ 4 = 0.75 |
| \(\frac{1}{3}\) | \(0.\overline{3}\) ≈ 0.333 | 1 ÷ 3 = 0.333… |
| \(\frac{2}{3}\) | \(0.\overline{6}\) ≈ 0.667 | 2 ÷ 3 = 0.666… |
| \(\frac{1}{5}\) | 0.2 | 1 ÷ 5 = 0.2 |
| \(\frac{1}{8}\) | 0.125 | 1 ÷ 8 = 0.125 |
| \(\frac{1}{10}\) | 0.1 | 1 ÷ 10 = 0.1 |
Convert \(\frac{7}{8}\) to a decimal.
Divide: 7 ÷ 8 = 0.875.
\[ \frac{7}{8} = 0.875 \]Check: 8 × 0.875 = 7.000 ✓
Convert 0.36 to a fraction in simplest form.
- 0.36 reads "36 hundredths" → \(\frac{36}{100}\).
- GCF(36, 100) = 4. Divide both: \(\frac{9}{25}\).
- Check: GCF(9, 25) = 1 ✓ — fully simplified.
Sales associate Jordan is marking down items. A sign says "Take \(\frac{3}{8}\) off the original price." A customer asks, "So what percent off is that?" Jordan needs the decimal equivalent fast.
- \(\frac{3}{8} = 3 \div 8 = 0.375\).
- As a percent: 0.375 × 100 = 37.5% off.
Jordan can confidently answer: "That's 37.5% off" — without a calculator, using a known benchmark.
Test yourself:
- Convert \(\frac{5}{8}\) to a decimal.
- Convert 0.45 to a fraction in simplest form.
- Is \(\frac{1}{6}\) a terminating or repeating decimal? What is it approximately equal to?
▶ Show Answers
- \(5 \div 8 = \) 0.625.
- 0.45 = \(\frac{45}{100}\). GCF=5 → \(\frac{9}{20}\).
- Repeating: \(\frac{1}{6} = 0.1\overline{6} \approx 0.167\).
- Dividing denominator by numerator: ❌ \(\frac{3}{4}\) → 4 ÷ 3 = 1.333 — wrong. ✅ Always divide top by bottom: 3 ÷ 4 = 0.75.
- Wrong place value when converting decimals: 0.3 is "3 tenths" = \(\frac{3}{10}\), NOT \(\frac{3}{100}\). Count decimal places carefully.
- Rounding repeating decimals too aggressively: \(\frac{2}{3} \approx 0.667\), not 0.67 — rounding too early causes compounding errors in multi-step problems.
- Fraction to decimal: divide numerator ÷ denominator.
- Decimal to fraction: name the place value, write over power of 10, simplify.
- Repeating decimals come from fractions that can't fully divide — write them with a bar notation: \(0.\overline{3}\).
- Memorize 8–10 benchmarks (½, ¼, ¾, ⅓, ⅛ etc.) for instant estimation in real situations.
Sales professionals, buyers, and merchandisers constantly switch between fraction-based markdowns (25% off = ¼ off) and decimal multipliers (multiply by 0.75 to find the sale price). A buyer who can instantly convert between forms can negotiate faster, verify discounts mentally, and spot pricing errors before they hit the register.
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Converting Between Fractions and Decimals — Quiz
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