Mathematical Calculation of Mixed Numbers
The problem in the image is:
\[ 4 \frac{1}{2} (5 \frac{5}{8} - 15\frac{5}{8}) + (1 \frac{1}{2} - 3\frac{1}{4}) \]
First, let's convert the mixed numbers into improper fractions:
For \( 4 \frac{1}{2} \), we have \( 4 \times 2 + 1 = 9/2 \).
For \( 5 \frac{5}{8} \), we have \( 5 \times 8 + 5 = 45/8 \).
For \( 15 \frac{5}{8} \), we have \( 15 \times 8 + 5 = 125/8 \).
For \( 1 \frac{1}{2} \), we have \( 1 \times 2 + 1 = 3/2 \).
For \( 3 \frac{1}{4} \), we have \( 3 \times 4 + 1 = 13/4 \).
Then the expression becomes:
\[ \frac{9}{2} \left( \frac{45}{8} - \frac{125}{8} \right) + \left( \frac{3}{2} - \frac{13}{4} \right) \]
Now we'll do the operations within the parentheses:
\[ \frac{45}{8} - \frac{125}{8} = -\frac{80}{8} = -10 \]
\[ \frac{3}{2} - \frac{13}{4} = \frac{6}{4} - \frac{13}{4} = -\frac{7}{4} \]
Now our expression is:
\[ \frac{9}{2} \times -10 + -\frac{7}{4} \]
Multiplying \( \frac{9}{2} \) by -10:
\[ \frac{9}{2} \times -10 = \frac{9 \times -10}{2} = -\frac{90}{2} = -45 \]
We now have:
\[ -45 - \frac{7}{4} \]
To subtract these, we'll express -45 as a fraction with a denominator of 4:
\[ -45 = -\frac{180}{4} \]
Now subtract:
\[ -\frac{180}{4} - \frac{7}{4} = -\frac{187}{4} \]
The answer is \(-\frac{187}{4}\), which can be left as an improper fraction or converted back into a mixed number:
\[ -\frac{187}{4} = -46 \frac{3}{4} \]
So the final answer is \( -46 \frac{3}{4} \).
