Certainly! Let's solve the given question step by step.
### Problem 4: How much greater is the difference of [tex]\(8 \frac{3}{4}\)[/tex] and [tex]\(4 \frac{1}{2}\)[/tex] than the sum of [tex]\(1 \frac{2}{3}\)[/tex] and [tex]\(1 \frac{5}{6}\)[/tex]?
#### Step 1: Convert mixed fractions to improper fractions.
- [tex]\(8 \frac{3}{4}\)[/tex]:
[tex]\[
8 \frac{3}{4} = 8 + \frac{3}{4} = \frac{32}{4} + \frac{3}{4} = \frac{32 + 3}{4} = \frac{35}{4}
\][/tex]
- [tex]\(4 \frac{1}{2}\)[/tex]:
[tex]\[
4 \frac{1}{2} = 4 + \frac{1}{2} = \frac{8}{2} + \frac{1}{2} = \frac{8 + 1}{2} = \frac{9}{2}
\][/tex]
#### Step 2: Find the difference between [tex]\(8 \frac{3}{4}\)[/tex] and [tex]\(4 \frac{1}{2}\)[/tex].
Convert [tex]\(\frac{9}{2}\)[/tex] to have the same denominator as [tex]\(\frac{35}{4}\)[/tex]:
[tex]\[
\frac{9}{2} = \frac{9 \times 2}{2 \times 2} = \frac{18}{4}
\][/tex]
Now, subtract:
[tex]\[
\frac{35}{4} - \frac{18}{4} = \frac{35 - 18}{4} = \frac{17}{4} = 4.25
\][/tex]
#### Step 3: Convert the other mixed fractions to improper fractions.
- [tex]\(1 \frac{2}{3}\)[/tex]:
[tex]\[
1 \frac{2}{3} = 1 + \frac{2}{3} = \frac{3}{3} + \frac{2}{3} = \frac{3 + 2}{3} = \frac{5}{3}
\][/tex]
- [tex]\(1 \frac{5}{6}\)[/tex]:
[tex]\[
1 \frac{5}{6} = 1 + \frac{5}{6} = \frac{6}{6} + \frac{5}{6} = \frac{6 + 5}{6} = \frac{11}{6}
\][/tex]
#### Step 4: Find the sum of [tex]\(1 \frac{2}{3}\)[/tex] and [tex]\(1 \frac{5}{6}\)[/tex].
Convert [tex]\(\frac{5}{3}\)[/tex] to have the same denominator as [tex]\(\frac{11}{6}\)[/tex]:
[tex]\[
\frac{5}{3} = \frac{5 \times 2}{3 \times 2} = \frac{10}{6}
\][/tex]
Now, add:
[tex]\[
\frac{10}{6} + \frac{11}{6} = \frac{10 + 11}{6} = \frac{21}{6} = 3.5
\][/tex]
#### Step 5: Find how much greater the difference [tex]\(4.25\)[/tex] is than the sum [tex]\(3.5\)[/tex].
Subtract the sum from the difference:
[tex]\[
4.25 - 3.5 = 0.75
\][/tex]
Hence, the difference of [tex]\(8 \frac{3}{4}\)[/tex] and [tex]\(4 \frac{1}{2}\)[/tex] is 0.75 greater than the sum of [tex]\(1 \frac{2}{3}\)[/tex] and [tex]\(1 \frac{5}{6}\)[/tex].
