To compare the mixed fractions [tex]\( 5 \frac{3}{4} \)[/tex] and [tex]\( 6 \frac{1}{4} \)[/tex] and to determine the result of their subtraction, let’s go through the step-by-step process to convert these mixed fractions into improper fractions and then perform the subtraction.
1. Convert Mixed Fractions to Improper Fractions:
- For [tex]\( 5 \frac{3}{4} \)[/tex]:
[tex]\[
5 \frac{3}{4} = 5 + \frac{3}{4} = \frac{20}{4} + \frac{3}{4} = \frac{23}{4}
\][/tex]
- For [tex]\( 6 \frac{1}{4} \)[/tex]:
[tex]\[
6 \frac{1}{4} = 6 + \frac{1}{4} = \frac{24}{4} + \frac{1}{4} = \frac{25}{4}
\][/tex]
2. Subtract the Improper Fractions:
Now we need to subtract [tex]\(\frac{25}{4}\)[/tex] from [tex]\(\frac{23}{4}\)[/tex]:
[tex]\[
\frac{23}{4} - \frac{25}{4} = \frac{23 - 25}{4} = \frac{-2}{4} = -\frac{1}{2}
\][/tex]
3. Compare the Result with Zero:
The result of the subtraction is [tex]\(-\frac{1}{2}\)[/tex].
Since [tex]\(-\frac{1}{2}\)[/tex] is less than 0, we can conclude:
[tex]\[
5 \frac{3}{4} - 6 \frac{1}{4} < 0
\][/tex]
Therefore, the correct comparison using the symbols [tex]\(<\)[/tex], [tex]\(>\)[/tex], or [tex]\(=\)[/tex] is:
[tex]\[
5 \frac{3}{4} - 6 \frac{1}{4} < 0
\][/tex]
Hence,
[tex]\(\boxed{<}\)[/tex]
