How do you multiply #7\frac { 1} { 2} \cdot ( - 7\frac { 1} { 5} )#?

First, convert each mixed number to an improper fraction:

#7 1/2 * (-7 1/5) =>#

#(7 + 1/2) * (-(7 + 1/5)) =>#

#([2/2 * 7] + 1/2) * (-([5/5 * 7] + 1/5)) =>#

#(14/2 + 1/2) * (-(35/5 + 1/5)) =>#

#(14 + 1)/2 * (-(35 + 1)/5) =>#

#15/2 * (-36/5)#

Now, multiply the numerators over the denominators multiplied:

#-(15 * 36)/(2 * 5) =>#

#-(color(red)(cancel(color(black)(15)))3 * color(green)(cancel(color(black)(36)))18)/(color(green)(cancel(color(black)(2)))1 * color(red)(cancel(color(black)(5)))1) =>#

#-(3 * 18)/(1 * 1) =>#

#-54/1 =>#

#-54#

To convert to an improper fraction, you need to multiply the whole number (#7# in both fractions) with the denominator. The answer you get is then ADDED to the numerator.

The problem should look like this:

#7 1/2 * (- 7 1/5) = (7 * 2 + 1)/2 * (- (7 ( 5 + 1))/5)#

# = 15/2 * (-36/5)#

After you do this, you can then multiply across: numerator times numerator, and denominator times denominator.

# = (15 * (-36))/(2 * 5) = 5 * (-18)#

Because the second fraction is a negative, the answer will be negative.

Convert it into standard form.
#7*1/2= ((2*7)+1)/2=15/2#

#-7*1/5= -((5*7)+1)/2=-36/5#

Now, Multiply:
#15/2*-36/5=(15*(-36))/(2*5)#
#=-540/10#

#=color(blue)(-54)#