should benchmarks be used to compare 6/7 and 7/8

Answer 1

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:

• The least common multiple (LCM) of 7 and 8 is 56.

• 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.