I know that the answer is 16, I just dont know how that answer is found
how many liters of 40% antifreeze solution must be mixed with 8 liters 70% antifreeze solution to get a 50% antifreeze?
Let x equal the number of liters of 40% solution, because that is the unknown value we want to find.
We can write the equation as follows (I'm writing the percentages as decimal values - e.g. 40% = 0.4)
(x)(0.4) + (8)(0.7) = (x+8)(0.5)
The total number of liters of solution (x+8) is the sum of the two individual amounts (x liters of 40% and 8 liters of 70%)
Multiply the values together and distribute the 0.5 into the parentheses on the right:
0.4x + 5.6 = 0.5x + 4.0
Subtract 0.4x from both sides to get all the terms with an x on the right side:
0.4x - 0.4x + 5.6 = 0.5x - 0.4x + 4.0
5.6 = 0.1x + 4.0
Subtract 4.0 from both sides to get all the constant terms on the left side:
5.6 - 4.0 = 0.1x + 4.0 - 4.0
1.6 = 0.1x
Divide both sides by 0.1 to get x by itself:
1.6/0.1 = (0.1x)/0.1
16 = x
So you need 16 liters of the 40% solution.
Use the method of Alligation:
40% 70% Write down the concentration of each "component" of the mixture
50% Write down the concentration of the resuling solution
20 % 10% Go across,find the difference of the values of components/result
(70-50 = 20) (50 - 40 =10)
Then the 40% AF is 2 times (=20/10) the 70 % AF so the answer is 16 Litres.
You have to think:
What we want?
50% antifreeze solution.
In other words
“to obtain final solution that contains an equal volume of antifreeze and water”
We can translate this to an equation:
We have two different initial solutions that will be mixed:
8 litters with 70 % of antifeeze
X liters with 40 % of antifreeze. (X is the variable we want to calculate)
translating to an equations we have:
for the 8 liter:
Vi,ant i= 8 x 0.7 = 5.6 liters of antifreeze and
Vi,water = 8 x 0.3 =2.4 liters of Water
for the 40%
Vadd, anti = X x 0.4 liters of antifreeze
Vadd, water = X x 0.6 liters of water
The final total volume of water and antifreeze that we have afer mixter is
Vf,ant i= Vadd,anti + Vi,anti
Vf,water = Vadd,water + Vi,water
sbstutuing the variables
Vf,anti = 8 x 0.7 + X x 0.4
Vf,water= 8 x 0.3 + X x0.6
So now we determinated the Variables os the first equation:
subtituing the variable in the first equation we have:
8 x 0.7 + X x 0.4 = 8 x 0.3 + X x 0.6
Isolate the X variable.
Xx0.4 - Xx0.6=8*0.3-8x0.7
Multiplying the equation by -1