When dividing fractions with the same denominator, the reason you can find the quotient by dividing the numerators is due to the property of equivalence in fractions. Since both fractions share a common denominator, the division focuses solely on the numerators.
Explanation:
When you divide one fraction by another with the same denominator, you can express the division as follows:
\[ \frac{a}{c} \div \frac{b}{c} \]
Here, \(a\) and \(b\) are the numerators, while \(c\) is the shared denominator. According to the rules of fraction division, we can rewrite the division of these fractions as:
\[ \frac{a}{c} \div \frac{b}{c} = \frac{a}{c} \times \frac{c}{b} \]
When multiplying by the reciprocal, the \(c\) in the numerator and the \(c\) in the denominator will cancel out:
\[ = \frac{a \cdot c}{c \cdot b} = \frac{a}{b} \]
Thus, the division simplifies down to just dividing the numerators \(a\) and \(b\), while the common denominator \(c\) doesn't affect the result.
Example with Mixed Numbers:
Let’s consider the mixed numbers \(2 \frac{1}{4}\) and \(1 \frac{3}{4}\).
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First, convert the mixed numbers to improper fractions:
- \(2 \frac{1}{4} = \frac{2 \times 4 + 1}{4} = \frac{9}{4}\)
- \(1 \frac{3}{4} = \frac{1 \times 4 + 3}{4} = \frac{7}{4}\)
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Now, we can perform the division: \[ \frac{9}{4} \div \frac{7}{4} \]
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Since both fractions have the same denominator, we can simply divide the numerators: \[ \frac{9}{4} \div \frac{7}{4} = \frac{9}{7} \]
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Finally, we can convert \(\frac{9}{7}\) back to a mixed number if desired:
- \(9 \div 7 = 1\) remainder \(2\), so \(\frac{9}{7} = 1 \frac{2}{7}\).
Conclusion:
Thus, when dividing fractions with the same denominator, it is valid to divide the numerators, giving us a straightforward method to find the quotient. In our example, the division of \(2 \frac{1}{4}\) by \(1 \frac{3}{4}\) resulted in \( \frac{9}{7} \) or \( 1 \frac{2}{7} \).
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