To compare the fractions \( \frac{6}{7} \) and \( \frac{7}{8} \), using a benchmark can be helpful, but it's also straightforward to compare them directly.
One method is to find a common denominator:
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The least common multiple (LCM) of 7 and 8 is 56.
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Convert both fractions:
\[ \frac{6}{7} = \frac{6 \times 8}{7 \times 8} = \frac{48}{56} \]
\[ \frac{7}{8} = \frac{7 \times 7}{8 \times 7} = \frac{49}{56} \]
Now, you can see that:
\[ \frac{48}{56} < \frac{49}{56} \]
Thus, \( \frac{6}{7} < \frac{7}{8} \).
Alternatively, benchmarks can also be useful. For instance, you might consider common benchmarks like \( \frac{1}{2} \), \( \frac{3}{4} \), or \( \frac{1}{1} \) to judge where these fractions fall. Both \( \frac{6}{7} \) and \( \frac{7}{8} \) are greater than \( \frac{3}{4} \) (or 0.75) but less than 1, meaning they are both close to 1 but you would need to compare their specific values as shown above to see which is greater.
In conclusion, either method can be used, but converting to a common denominator is often the clearest way to compare two fractions directly.
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