Converting Forms
Strategically
Master the art of switching between percents, fractions, and decimals to solve problems faster and with more confidence.
Pre-Algebra
The Three Forms of Every Number
Every rational number can be written in three equivalent ways. Recognizing and converting between them is a foundational skill for algebra, statistics, and everyday life.
Fraction
A ratio of two integers: \(\frac{3}{4}\)
Decimal
Base-ten notation: \(0.75\)
Percent
Parts per hundred: \(75\%\)
The Conversion Map
Why Convert Strategically?
Not every form is equally convenient for every task. Choosing the right form before you compute saves time and prevents errors.
Fractions
Best for exact arithmetic: multiplying, dividing, and simplifying ratios.
Decimals
Best for comparing, ordering, and using a calculator.
Percents
Best for describing parts of a whole: tips, taxes, discounts, and data.
Fraction ↔ Decimal
1
Fraction → Decimal: Divide the numerator by the denominator. \(\frac{3}{8} = 3 \div 8 = 0.375\)
2
Decimal → Fraction: Read the decimal's place value, write as a fraction, then simplify. \(0.6 = \frac{6}{10} = \frac{3}{5}\)
Decimal ↔ Percent
1
Decimal → Percent: Multiply by \(100\) (move the decimal point two places right). \(0.45 \times 100 = 45\%\)
2
Percent → Decimal: Divide by \(100\) (move the decimal point two places left). \(12.5\% \div 100 = 0.125\)
Fraction ↔ Percent
1
Fraction → Percent: Convert to decimal first, then multiply by \(100\). \(\frac{2}{5} = 0.4 = 40\%\)
2
Percent → Fraction: Write over \(100\), then simplify. \(65\% = \frac{65}{100} = \frac{13}{20}\)
3
Shortcut: If the denominator divides evenly into 100, scale directly. \(\frac{3}{20} = \frac{15}{100} = 15\%\)
Choosing the Right Form
| Task | Best Form | Why |
|---|---|---|
| Compare two quantities | Decimal | Easy to see which is larger |
| Multiply or divide exactly | Fraction | Cancel common factors |
| Describe a rate or change | Percent | Universal "out of 100" scale |
| Add or subtract mixed values | Decimal | Align place values easily |
| Simplify a ratio | Fraction | GCF reduction is natural |
The Strategic Conversion Checklist
1
Read the problem. Identify what operation you need: comparing, multiplying, finding a part of a whole?
2
Pick the best form. Use the table from the previous slide as your guide.
3
Convert. Apply the correct rule: divide, multiply by \(100\), or write over \(100\).
4
Solve. Perform the calculation in the chosen form.
5
Convert back if the problem asks for a specific form in the answer.
Example 1: Quick Conversion
Convert \(\frac{7}{8}\) to a decimal and a percent.
1
Fraction → Decimal: \(7 \div 8 = 0.875\)
2
Decimal → Percent: \(0.875 \times 100 = 87.5\%\)
Example 2: Choosing Wisely
A store offers \(30\%\) off a \(\$60\) jacket. What is the discount?
1
Best form? We need to find a part of a whole: multiplying is easier with a fraction or decimal.
2
Convert: \(30\% = \frac{30}{100} = \frac{3}{10}\)
3
Solve: \(\frac{3}{10} \times 60 = \frac{180}{10} = 18\)
Example 3: Comparing Values
Which is greater: \(\frac{5}{9}\) or \(54\%\)?
1
Best form? Decimals are easiest for comparing.
2
Convert both: \(\frac{5}{9} = 5 \div 9 \approx 0.5556\) and \(54\% = 0.54\)
3
Compare: \(0.5556 > 0.54\), so \(\frac{5}{9}\) is greater.
Benchmark Conversions to Memorize
Handling Repeating Decimals
Some fractions produce decimals that never terminate. These are called repeating decimals and we write them with a bar over the repeating digits.
\(\frac{1}{3}\)
\(= 0.\overline{3} = 0.333\ldots\)
\(\frac{1}{6}\)
\(= 0.1\overline{6} = 0.1666\ldots\)
\(\frac{2}{11}\)
\(= 0.\overline{18} = 0.1818\ldots\)
Common Mistakes to Avoid
✕ Decimal Point Drift
Moving the decimal the wrong number of places. \(0.4\) becomes \(4\%\) instead of the correct \(40\%\). Always move exactly two places.
✕ Forgetting to Simplify
Writing \(\frac{25}{100}\) as the final answer instead of simplifying to \(\frac{1}{4}\). Always check for a common factor.
✕ Percent ≠ Decimal
Using \(40\) instead of \(0.40\) in calculations. Remember: \(40\% = 0.40\), not \(40\).
Quick Reference Conversion Table
| From → To | Method | Example |
|---|---|---|
| Fraction → Decimal | Divide: \(a \div b\) | \(\frac{3}{8} = 0.375\) |
| Decimal → Fraction | Place value, simplify | \(0.6 = \frac{3}{5}\) |
| Decimal → Percent | Multiply by \(100\) | \(0.45 = 45\%\) |
| Percent → Decimal | Divide by \(100\) | \(12.5\% = 0.125\) |
| Fraction → Percent | Divide, then \(\times 100\) | \(\frac{2}{5} = 40\%\) |
| Percent → Fraction | Write over \(100\), simplify | \(65\% = \frac{13}{20}\) |
Question 1 of 5
What is \(\frac{3}{5}\) expressed as a percent?
\(\frac{3}{5} = 3 \div 5 = 0.6\), and \(0.6 \times 100 = 60\%\).
Question 2 of 5
Convert \(0.875\) to a fraction in simplest form.
\(0.875 = \frac{875}{1000}\). Dividing numerator and denominator by \(125\) gives \(\frac{7}{8}\).
Question 3 of 5
A shirt costs \(\$40\). It is on sale for \(15\%\) off. What form is most strategic for computing the discount mentally?
\(15\% = \frac{15}{100} = \frac{3}{20}\). Then \(\frac{3}{20} \times 40 = \frac{120}{20} = 6\). The fraction form made the multiplication exact and simple. Choice C is a valid method, but often harder to do mentally than using the simplified fraction.
Question 4 of 5
Which of the following is greatest?
Convert all to decimals: \(\frac{3}{8} = 0.375\), \(0.36\), \(37\% = 0.37\), and \(\frac{2}{5} = 0.40\). Since \(0.40\) is the largest, \(\frac{2}{5}\) is greatest.
Question 5 of 5
Express \(0.\overline{3}\) as a simplified fraction.
\(0.\overline{3} = 0.333\ldots = \frac{1}{3}\). Choice D, \(\frac{3}{9}\), is equivalent but not simplified: \(\frac{3}{9} = \frac{1}{3}\).