# Simplify this expression: [(6-3/5)xx(1/4+2/9-5/12)+3/2xx(9/2-7/4-5/2)]xx2/27+1/4 ?

Mar 9, 2018

$= \frac{3}{10}$

#### Explanation:

Step 1:

Resolve:
$a . \left(6 - \frac{3}{5}\right) = \frac{27}{5}$

$b . \left(\frac{1}{4} + \frac{2}{9} - \frac{5}{12}\right) = \frac{1}{18}$

$c . \left(\frac{9}{2} - \frac{7}{4} - \frac{5}{2}\right) = \frac{1}{4}$

Step 2:

multiply

$a . \left(\frac{27}{5}\right) \cdot \left(\frac{1}{18}\right) = \frac{3}{10}$

$b . \left(\frac{3}{2}\right) \cdot \left(\frac{1}{4}\right) = \frac{3}{8}$

Step 3:

$a . \left(\frac{3}{10}\right) + \left(\frac{3}{8}\right) = \frac{27}{40}$

Step 4:

multiply

$a . \left[\frac{27}{40}\right] \cdot \left(\frac{2}{27}\right) = \frac{1}{20}$

Step 5:

We add the product (again :v)

$a . \frac{1}{20} + \frac{1}{4} = \frac{3}{10}$

The summary is:

$= \left[\left(\frac{27}{5}\right) \cdot \left(\frac{1}{18}\right) + \left(\frac{3}{2}\right) \cdot \left(\frac{1}{4}\right)\right] \cdot \left(\frac{2}{27}\right) + \frac{1}{4}$

$= \left[\left(\frac{3}{10}\right) + \left(\frac{3}{8}\right)\right] \cdot \left(\frac{2}{27}\right) + \frac{1}{4}$

$= \left[\frac{27}{40}\right] \cdot \left(\frac{2}{27}\right) + \frac{1}{4}$

$= \left[\frac{\cancel{27}}{\cancel{40}}\right] \cdot \left(\frac{\cancel{2}}{\cancel{27}}\right) + \frac{1}{4}$

$= \frac{1}{20} + \frac{1}{4}$

$= \frac{1}{20} + \frac{1}{4}$

$= \frac{3}{10}$

Mar 9, 2018

$\frac{3}{10}$

#### Explanation:

Identify the individual terms and then simplify them separately

$\textcolor{b l u e}{\left[\left(6 - \frac{3}{5}\right) \times \left(\frac{1}{4} + \frac{2}{9} - \frac{5}{12}\right) + \frac{3}{2} \times \left(\frac{9}{2} - \frac{7}{4} - \frac{5}{2}\right)\right] \times \frac{2}{27}} \textcolor{red}{\text{ "+" } \frac{1}{4}}$

Within the first term, shown in blue, simplify each bracket separately.

$= \textcolor{b l u e}{\left[\left(5 \frac{2}{5}\right) \times \left(\frac{9 + 8 - 15}{36}\right) + \frac{3}{2} \times \left(\frac{18 - 7 - 10}{4}\right)\right] \times \frac{2}{27}} \textcolor{red}{\text{ "+" } \frac{1}{4}}$

$= \textcolor{b l u e}{\left[\textcolor{g r e e n}{\left(\frac{27}{5}\right) \times \left(\frac{2}{36}\right)} \textcolor{\lim e g r e e n}{+ \frac{3}{2} \times \left(\frac{1}{4}\right)}\right] \times \frac{2}{27}} \textcolor{red}{\text{ "+" } \frac{1}{4}}$

Now cancel where possible

=color(blue)([color(green)(cancel27^3/5xx1/cancel18^2)color(limegreen)( " "+" "3/2xx1/4)]xx2/27) color(red)(" "+" "1/4)

Multiply straight across to get:

$= \textcolor{b l u e}{\left[\textcolor{g r e e n}{\frac{3}{10}} \textcolor{\lim e g r e e n}{+ \frac{3}{8}}\right] \times \frac{2}{27}} \textcolor{red}{\text{ "+" } \frac{1}{4}}$

$= \textcolor{b l u e}{\left[\frac{\textcolor{g r e e n}{12} \textcolor{\lim e g r e e n}{+ 15}}{40}\right] \times \frac{2}{27}} \textcolor{red}{\text{ "+" } \frac{1}{4}}$

$= \textcolor{b l u e}{\frac{27}{40} \times \frac{2}{27}} \textcolor{red}{\text{ "+" } \frac{1}{4}}$

$= \textcolor{b l u e}{\frac{\cancel{27}}{\cancel{40}} ^ 20 \times \frac{\cancel{2}}{\cancel{27}}} \textcolor{red}{\text{ "+" } \frac{1}{4}}$

$= \textcolor{b l u e}{\frac{1}{20}} \textcolor{red}{\text{ "+" } \frac{1}{4}}$

Now add the two terms together,

$= \frac{\textcolor{b l u e}{1} \textcolor{red}{+ 5}}{20}$

$= \frac{6}{20}$

$= \frac{3}{10}$