Question

Simplify the expression. Write your answers using integers or improper fractions.
minus, start fraction, 2, divided by, 3, end fraction, left parenthesis, one third, g, minus, start fraction, 3, divided by, 4, end fraction, right parenthesis, minus, 10, g
−
3
2
​
(
3
1
​
g−
4
3
​
)−10g

Answer 1

Let's simplify the expression step by step.

The expression given is:

\[
-\frac{2}{3}\left(\frac{1}{3}g - \frac{3}{4}\right) - 10g
\]

1. **Distribute** \(-\frac{2}{3}\) to the terms inside the parentheses:

\[
-\frac{2}{3} \cdot \frac{1}{3}g + \frac{2}{3} \cdot \frac{3}{4}
\]

Calculating \(-\frac{2}{3} \cdot \frac{1}{3}g\):

\[
-\frac{2}{9}g
\]

Now calculating \(\frac{2}{3} \cdot \frac{3}{4}\):

\[
\frac{2 \cdot 3}{3 \cdot 4} = \frac{6}{12} = \frac{1}{2}
\]

So, the expression becomes:

\[
-\frac{2}{9}g + \frac{1}{2} - 10g
\]

2. **Combine the \(g\) terms**:

To combine \(-\frac{2}{9}g\) with \(-10g\), we need a common denominator. The common denominator for 9 and 1 (denominator of 10) is 9.

Express \(-10g\) with a denominator of 9:

\[
-10g = -\frac{90}{9}g
\]

Now we can combine:

\[
-\frac{2}{9}g - \frac{90}{9}g = -\frac{92}{9}g
\]

3. **Put it all together**:

Now, combine \(-\frac{92}{9}g\) with \(\frac{1}{2}\):

The expression is now:

\[
-\frac{92}{9}g + \frac{1}{2}
\]

To combine these two terms, we need a common denominator for \(\frac{1}{2}\) and \(-\frac{92}{9}g\). The common denominator is 18.

Convert \(\frac{1}{2}\):

\[
\frac{1}{2} = \frac{9}{18}
\]

Thus, the final expression is:

\[
-\frac{92}{9}g + \frac{9}{18} = -\frac{92}{9}g + \frac{9}{18}
\]

This cannot be further simplified meaningfully since they are not like terms.

Therefore, the final expression is:

\[
-\frac{92}{9}g + \frac{9}{18}
\]

This can also be represented as:

\[
-\frac{92}{9}g + \frac{1}{2}
\]

So, the simplified result is:

\[
-\frac{92}{9}g + \frac{1}{2}
\]