Introduction
Benchmark fractions are common fractions that we use as reference points to estimate and compare other fractions. The most common benchmark fractions are halves, thirds, fourths, fifths, and tenths.
Understanding Fractions
A fraction represents a part of a whole. It consists of two numbers separated by a line:
- Numerator: How many parts we have (top number)
- Denominator: How many equal parts the whole is divided into (bottom number)
Example: In \(\frac{3}{4}\), we have 3 parts out of 4 equal parts.
Fraction Conversions
Converting Fractions to Percentages
To convert a fraction to a percentage:
- Divide the numerator by the denominator
- Multiply the result by 100
- Add the % symbol
Example 1: \(\frac{1}{2}\) to percentage
Example 2: \(\frac{3}{4}\) to percentage
Converting Fractions to Decimals
To convert a fraction to a decimal, simply divide the numerator by the denominator:
Benchmark Fractions Conversion Table
| Fraction | Decimal | Percentage |
|---|---|---|
| \(\frac{1}{2}\) | 0.5 | 50% |
| \(\frac{1}{3}\) | 0.333... | 33.33% |
| \(\frac{2}{3}\) | 0.666... | 66.67% |
| \(\frac{1}{4}\) | 0.25 | 25% |
| \(\frac{3}{4}\) | 0.75 | 75% |
| \(\frac{1}{5}\) | 0.2 | 20% |
| \(\frac{2}{5}\) | 0.4 | 40% |
| \(\frac{1}{10}\) | 0.1 | 10% |
Comparing Fractions
Using Benchmarks
We can use benchmark fractions like 0, \(\frac{1}{2}\), and 1 to estimate where other fractions fall:
Closer to 0
Fractions with small numerators compared to denominators:
Closer to \(\frac{1}{2}\)
Fractions that are about halfway:
Closer to 1
Fractions where the numerator is almost equal to the denominator:
Ordering Fractions
To compare and order fractions:
- Convert them to decimals using division
- Compare the decimal values
- Order from smallest to largest
Example: Order \(\frac{2}{3}\), \(\frac{1}{2}\), \(\frac{3}{4}\):
\[ \frac{1}{2} = 0.5, \frac{2}{3} \approx 0.666, \frac{3}{4} = 0.75 \]Order: \(\frac{1}{2}\), \(\frac{2}{3}\), \(\frac{3}{4}\)
Simplifying Fractions
To simplify a fraction, divide both numerator and denominator by their greatest common factor (GCF):
Example: Simplify \(\frac{8}{12}\):
GCF of 8 and 12 is 4:
\[ \frac{8 \div 4}{12 \div 4} = \frac{2}{3} \]Applications
Practical Examples
Understanding benchmark fractions is useful in many real-world situations:
Shopping Discounts
A \(\frac{1}{4}\) discount means 25% off:
Recipe Adjustments
Doubling a recipe that calls for \(\frac{3}{4}\) cup:
Test Scores
Scoring \(\frac{17}{20}\) on a test:
Practice Exercise
Question 1
Convert \(\frac{3}{5}\) to a decimal and percentage
Question 2
Is \(\frac{5}{12}\) closer to 0, \(\frac{1}{2}\), or 1?
\(\frac{5}{12} \approx 0.416\), which is closer to \(\frac{1}{2}\) (0.5) than to 0
Question 3
Simplify \(\frac{9}{15}\)
Question 4
Order these from smallest to largest: \(\frac{2}{5}, \frac{1}{3}, \frac{3}{4}\)
Convert to decimals:
\[ \frac{1}{3} \approx 0.333, \frac{2}{5} = 0.4, \frac{3}{4} = 0.75 \]Order: \(\frac{1}{3}\), \(\frac{2}{5}\), \(\frac{3}{4}\)
Question 5
If you answered \(\frac{9}{10}\) of questions correctly, what percentage is that?
Question 6
Which is larger: \(\frac{5}{8}\) or \(\frac{2}{3}\)?
Convert to decimals:
\[ \frac{5}{8} = 0.625 \] \[ \frac{2}{3} \approx 0.666... \]\(\frac{2}{3}\) is larger
Ready for more challenges?
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Test your skills with benchmark fractions, decimals, and percentages