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 #?

Step 1:

Resolve:
#a. (6-3/5) = 27/5#

#b. (1/4+ 2/9 -5/12) = 1/18#

#c. (9/2 -7/4-5/2) = 1/4#

Step 2:

multiply

#a.(27/5) * (1/18) = 3/10#

#b. (3/2) * (1/4) = 3/8#

Step 3:

We add the product

#a. (3/10)+(3/8) = 27/40#

Step 4:

multiply

#a. [27/40] * (2/27) = 1/20#

Step 5:

We add the product (again :v)

#a. 1/20 + 1/4 = 3/10#

The summary is:

#= [(27/5) * (1/18)+(3/2) * (1/4)] * (2/27) + 1/4#

#= [(3/10)+(3/8)] * (2/27) + 1/4#

#= [27/40] * (2/27) + 1/4#

#= [cancel(27)/cancel(40)] * (cancel(2)/cancel(27)) + 1/4#

#= 1/20 + 1/4#

#= 1/20 + 1/4#

#= 3/10#

Identify the individual terms and then simplify them separately

#color(blue)([(6-3/5)xx(1/4+2/9-5/12) +3/2xx(9/2-7/4-5/2)]xx2/27) color(red)(" "+" "1/4) #

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

#=color(blue)([(5 2/5)xx((9+8-15)/36) +3/2xx((18-7 -10)/4)]xx2/27) color(red)(" "+" "1/4) #

#=color(blue)([color(green)((27/5)xx((2)/36)) color(limegreen)(+3/2xx((1)/4))]xx2/27) color(red)(" "+" "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:

#=color(blue)([ color(green)(3/10)color(limegreen)(+3/8)]xx2/27) color(red)(" "+" "1/4) #

#=color(blue)([(color(green)(12)color(limegreen)(+15))/40]xx2/27) color(red)(" "+" "1/4) #

#=color(blue)(27/40xx2/27) color(red)(" "+" "1/4) #

#=color(blue)(cancel27/cancel40^20xxcancel2/cancel27) color(red)(" "+" "1/4) #

#=color(blue)(1/20) color(red)(" "+" "1/4) #

Now add the two terms together,

#=(color(blue)(1)color(red)(+5))/20#

#=6/20#

#=3/10#