Question

Simplify 3 1/8 - 7 5/6 + 9 3/4

Answer 1

To simplify the expression \(3 \frac{1}{8} - 7 \frac{5}{6} + 9 \frac{3}{4}\), we first convert each mixed number into an improper fraction.

1. Convert \(3 \frac{1}{8}\) to an improper fraction:
\[
3 \frac{1}{8} = \frac{3 \times 8 + 1}{8} = \frac{24 + 1}{8} = \frac{25}{8}
\]

2. Convert \(7 \frac{5}{6}\) to an improper fraction:
\[
7 \frac{5}{6} = \frac{7 \times 6 + 5}{6} = \frac{42 + 5}{6} = \frac{47}{6}
\]

3. Convert \(9 \frac{3}{4}\) to an improper fraction:
\[
9 \frac{3}{4} = \frac{9 \times 4 + 3}{4} = \frac{36 + 3}{4} = \frac{39}{4}
\]

Now, substituting the improper fractions into the expression gives:
\[
\frac{25}{8} - \frac{47}{6} + \frac{39}{4}
\]

Next, we need to find a common denominator for the fractions. The denominators are \(8\), \(6\), and \(4\). The least common multiple (LCM) of these numbers is \(24\).

Now, we convert each fraction to have a denominator of \(24\):

1. Convert \(\frac{25}{8}\) to a denominator of \(24\):
\[
\frac{25}{8} = \frac{25 \times 3}{8 \times 3} = \frac{75}{24}
\]

2. Convert \(\frac{47}{6}\) to a denominator of \(24\):
\[
\frac{47}{6} = \frac{47 \times 4}{6 \times 4} = \frac{188}{24}
\]

3. Convert \(\frac{39}{4}\) to a denominator of \(24\):
\[
\frac{39}{4} = \frac{39 \times 6}{4 \times 6} = \frac{234}{24}
\]

Now, we can substitute these fractions back into the expression:
\[
\frac{75}{24} - \frac{188}{24} + \frac{234}{24}
\]

Combining the fractions with a common denominator:
\[
\frac{75 - 188 + 234}{24} = \frac{75 - 188 + 234}{24} = \frac{75 - 188 + 234}{24} = \frac{121}{24}
\]

Now, we can convert \(\frac{121}{24}\) back to a mixed number:
\[
121 \div 24 = 5 \quad \text{(with a remainder of } 1\text{)}
\]
So,
\[
\frac{121}{24} = 5 \frac{1}{24}
\]

Thus, the simplified result is:
\[
\boxed{5 \frac{1}{24}}
\]