12. Writing to Explain Explain how you know that \( \frac{7}{12} \) is greater than \( \frac{1}{3} \) but less than \( \frac{2}{3} \) ?

To determine why \( \frac{7}{12} \) is greater than \( \frac{1}{3} \) but less than \( \frac{2}{3} \), we can compare these fractions by converting them to a common denominator or by converting them to decimal form. Here's a step-by-step explanation using both methods: ### Method 1: Common Denominator 1. **Find a Common Denominator** - The denominators of the fractions are 3 and 12. The least common denominator (LCD) for these is 12. 2. **Convert Each Fraction to the Common Denominator** - \( \frac{1}{3} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12} \) - \( \frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12} \) - \( \frac{7}{12} \) already has the denominator 12. 3. **Compare the Numerators** - \( \frac{4}{12} < \frac{7}{12} < \frac{8}{12} \) - This shows that \( \frac{7}{12} \) is greater than \( \frac{1}{3} \) (\( \frac{4}{12} \)) and less than \( \frac{2}{3} \) (\( \frac{8}{12} \)). ### Method 2: Decimal Conversion 1. **Convert Each Fraction to a Decimal** - \( \frac{1}{3} \approx 0.333 \) - \( \frac{7}{12} \approx 0.583 \) (since \( 7 \div 12 = 0.583 \)) - \( \frac{2}{3} \approx 0.666 \) 2. **Compare the Decimals** - \( 0.333 < 0.583 < 0.666 \) - This confirms that \( \frac{7}{12} \) is between \( \frac{1}{3} \) and \( \frac{2}{3} \). ### Visual Representation: Number Line Imagine a number line where: - \( \frac{0}{12} = 0 \) - \( \frac{4}{12} = \frac{1}{3} \) - \( \frac{7}{12} \) is somewhere between \( \frac{4}{12} \) and \( \frac{8}{12} \) - \( \frac{8}{12} = \frac{2}{3} \) - \( \frac{12}{12} = 1 \) Placing these on the number line clearly shows that \( \frac{7}{12} \) falls between \( \frac{1}{3} \) and \( \frac{2}{3} \). ### Conclusion By either converting to a common denominator, turning the fractions into decimals, or visualizing them on a number line, it's evident that \( \frac{7}{12} \) is greater than \( \frac{1}{3} \) and less than \( \frac{2}{3} \).