# 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?

May 26, 2018

$80 L$ of the first, $120 L$ of the second

#### Explanation:

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

42% of $200$ is the same as $0.42 \cdot 200$, which is $84$.

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

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

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

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

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

this can be written as $0.3 a + 0.5 b = 84$
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
$0.6 a + b = 168$

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

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

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

we can then substitute the value for $a$ into the equation for $b$.
$0.6 a + b = 168$
$0.6 \cdot 80 = 48$
$48 + b = 168$

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

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

30% of $80 L = 0.3 \cdot 80 L = 24 L$
50%# of $120 L = 0.5 \cdot 120 L = 60 L$
$24 L + 60 L = 84 L$