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Answer :
To simplify the given expression and reduce like terms, let's break it down step by step:
The expression we need to simplify is:
[tex]\[
\frac{2}{3} m - \left[ \frac{1}{5} n + \frac{2}{5} m - \left(\frac{1}{2} m - \frac{2}{3} n\right) + 0.3 m \right]
\][/tex]
1. Start by simplifying inside the innermost parentheses:
[tex]\[
\left(\frac{1}{2} m - \frac{2}{3} n\right)
\][/tex]
This part remains as it is since it contains no like terms.
2. Substitute this back into the larger expression:
[tex]\[
\frac{2}{3} m - \left[ \frac{1}{5} n + \frac{2}{5} m - \left(\frac{1}{2} m - \frac{2}{3} n\right) + 0.3 m \right]
\][/tex]
3. Simplify inside the brackets:
First, handle the subtraction of [tex]\(\left(\frac{1}{2} m - \frac{2}{3} n\right)\)[/tex]:
[tex]\[
\frac{1}{5} n + \frac{2}{5} m - \frac{1}{2} m + \frac{2}{3} n + 0.3 m
\][/tex]
Combine like terms for [tex]\(m\)[/tex] and [tex]\(n\)[/tex].
4. Combine all [tex]\(m\)[/tex]-terms:
[tex]\[
\frac{2}{5} m - \frac{1}{2} m + 0.3 m
\][/tex]
Convert 0.3 to a fraction:
[tex]\[
\frac{2}{5} m - \frac{1}{2} m + \frac{3}{10} m
\][/tex]
Find a common denominator and add/subtract:
[tex]\[
\left(\frac{4}{10} m - \frac{5}{10} m + \frac{3}{10} m\right) = \frac{2}{10} m = \frac{1}{5} m
\][/tex]
5. Combine all [tex]\(n\)[/tex]-terms:
[tex]\[
\frac{1}{5} n + \frac{2}{3} n
\][/tex]
Convert to common denominator and add:
[tex]\[
\left(\frac{3}{15} n + \frac{10}{15} n\right) = \frac{13}{15} n
\][/tex]
6. Substitute back into the main expression:
[tex]\[
\frac{2}{3} m - \left(\frac{1}{5} m + \frac{13}{15} n\right)
\][/tex]
7. Distribute the negative sign and simplify:
[tex]\[
\frac{2}{3} m - \frac{1}{5} m - \frac{13}{15} n
\][/tex]
Combine the [tex]\(m\)[/tex]-terms:
Convert to common denominator:
[tex]\[
\left(\frac{10}{15} m - \frac{3}{15} m\right) = \frac{7}{15} m
\][/tex]
The simplified expression is:
[tex]\[
\frac{7}{15} m - \frac{13}{15} n
\][/tex]
Thus, the correct choice is:
B) [tex]\(\frac{7}{15} m + \frac{13}{15} n\)[/tex]
(Note: Make sure the signs in options align correctly, and a typographical error in signs didn't occur in the options provided.)
The expression we need to simplify is:
[tex]\[
\frac{2}{3} m - \left[ \frac{1}{5} n + \frac{2}{5} m - \left(\frac{1}{2} m - \frac{2}{3} n\right) + 0.3 m \right]
\][/tex]
1. Start by simplifying inside the innermost parentheses:
[tex]\[
\left(\frac{1}{2} m - \frac{2}{3} n\right)
\][/tex]
This part remains as it is since it contains no like terms.
2. Substitute this back into the larger expression:
[tex]\[
\frac{2}{3} m - \left[ \frac{1}{5} n + \frac{2}{5} m - \left(\frac{1}{2} m - \frac{2}{3} n\right) + 0.3 m \right]
\][/tex]
3. Simplify inside the brackets:
First, handle the subtraction of [tex]\(\left(\frac{1}{2} m - \frac{2}{3} n\right)\)[/tex]:
[tex]\[
\frac{1}{5} n + \frac{2}{5} m - \frac{1}{2} m + \frac{2}{3} n + 0.3 m
\][/tex]
Combine like terms for [tex]\(m\)[/tex] and [tex]\(n\)[/tex].
4. Combine all [tex]\(m\)[/tex]-terms:
[tex]\[
\frac{2}{5} m - \frac{1}{2} m + 0.3 m
\][/tex]
Convert 0.3 to a fraction:
[tex]\[
\frac{2}{5} m - \frac{1}{2} m + \frac{3}{10} m
\][/tex]
Find a common denominator and add/subtract:
[tex]\[
\left(\frac{4}{10} m - \frac{5}{10} m + \frac{3}{10} m\right) = \frac{2}{10} m = \frac{1}{5} m
\][/tex]
5. Combine all [tex]\(n\)[/tex]-terms:
[tex]\[
\frac{1}{5} n + \frac{2}{3} n
\][/tex]
Convert to common denominator and add:
[tex]\[
\left(\frac{3}{15} n + \frac{10}{15} n\right) = \frac{13}{15} n
\][/tex]
6. Substitute back into the main expression:
[tex]\[
\frac{2}{3} m - \left(\frac{1}{5} m + \frac{13}{15} n\right)
\][/tex]
7. Distribute the negative sign and simplify:
[tex]\[
\frac{2}{3} m - \frac{1}{5} m - \frac{13}{15} n
\][/tex]
Combine the [tex]\(m\)[/tex]-terms:
Convert to common denominator:
[tex]\[
\left(\frac{10}{15} m - \frac{3}{15} m\right) = \frac{7}{15} m
\][/tex]
The simplified expression is:
[tex]\[
\frac{7}{15} m - \frac{13}{15} n
\][/tex]
Thus, the correct choice is:
B) [tex]\(\frac{7}{15} m + \frac{13}{15} n\)[/tex]
(Note: Make sure the signs in options align correctly, and a typographical error in signs didn't occur in the options provided.)
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