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.