Question

Evaluate the expression shown below and write your answer as a fraction or mixed number in simplest form.
left parenthesis, start fraction, 8, divided by, 7, end fraction, plus, start fraction, 2, divided by, 35, end fraction, right parenthesis, dot, start fraction, 2, divided by, 9, end fraction
(
7
8
​
+
35
2
​
)⋅
9
2
​

Answer 1

To evaluate the expression

\[
\left( \frac{8}{7} + \frac{2}{35} \right) \cdot \frac{2}{9}
\]

we start by simplifying the expression inside the parentheses.

First, we need to add the fractions \(\frac{8}{7}\) and \(\frac{2}{35}\). To do this, we need a common denominator. The least common multiple (LCM) of 7 and 35 is 35.

Next, we convert \(\frac{8}{7}\) to an equivalent fraction with a denominator of 35:

\[
\frac{8}{7} = \frac{8 \cdot 5}{7 \cdot 5} = \frac{40}{35}
\]

Now we can add the fractions:

\[
\frac{40}{35} + \frac{2}{35} = \frac{40 + 2}{35} = \frac{42}{35}
\]

Next, we simplify \(\frac{42}{35}\):

Both numerator and denominator can be divided by 7:

\[
\frac{42 \div 7}{35 \div 7} = \frac{6}{5}
\]

Now, we substitute this back into the original expression:

\[
\left( \frac{6}{5} \right) \cdot \frac{2}{9}
\]

This can be simplified by multiplying the fractions:

\[
\frac{6 \cdot 2}{5 \cdot 9} = \frac{12}{45}
\]

Next, we simplify \(\frac{12}{45}\):

The greatest common divisor (GCD) of 12 and 45 is 3. Thus, we divide both the numerator and denominator by 3:

\[
\frac{12 \div 3}{45 \div 3} = \frac{4}{15}
\]

Thus, the final answer in simplest form is:

\[
\boxed{\frac{4}{15}}
\]