A solution containing 30% insecticide is to be mixed with a solution containing 50% insecticide to make 200 L of a solution containingg 42% insecticide. How much of each solution should be used?

the final mixture will have a capacity of #200L#, and it will be made up of #42%# insecticide.

#42%# of #200# is the same as #0.42 * 200#, which is #84#.

this means that the final mixture will have #84L# of insecticide.

the first solution has a certain capacity which we can denote as #a# litres (#aL#), while the second has a capacity of #b# litres (#bL#).

we know that the capacity of insecticide in the first solution is #30%# of the solution's capacity, which is the same as #0.3 *# the solution's capacity (#aL#).
this can be written as #0.3a#.

we know that the capacity of insecticide in the second solution is #50%# of the solution's capacity, which is the same as #0.3 *# the solution's capacity (#bL#).
this can be written as #0.5b#.

we then know that #0.3a + 0.5b# is the same as #42%# of #200L#, which is #84L#.

this can be written as #0.3a + 0.5b = 84#
we also have the equation #a + b = 200#, from the overall capacities of the solutions.

we can then make the #b# term equal in both equations by multiplying all terms of the first one by #2#, to give
#0.6a + b = 168#

using the method of elimination for simultaneous equations (where you subtract the two):

#a + b = 200#
subtracted with
#0.6a + b = 168#
gives #0.4a + 0 = 32#
or #0.4a = 32#

multiplying both sides by #2.5# gives #a = 80#
meaning that #80L# of the first solution will be used.

we can then substitute the value for #a# into the equation for #b#.
#0.6a + b = 168#
#0.6 * 80 = 48#
#48 + b = 168#

#b = 120#, meaning that #120L# of the second solution will be used.

to check, you can find the capacities of each insecticide and see whether they add to #84L# (#42% of #200L#):

#30%# of #80L = 0.3 * 80L = 24L#
#50%# of #120L = 0.5 * 120L = 60L#
#24L + 60L = 84L#
so the answers make sense.