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?

# 3 Answers

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: **

### V_{f,anti} =V_{f,water}

We have two different initial solutions that will be mixed:

8 litters with 70 % of antifeeze

and

**X** liters with 40 % of antifreeze.
* ( X is the variable we want to calculate)*

**translating to an equations we have:**

*for the 8 liter:*

**V _{i,ant i}= 8 x 0.7 = 5.6 liters of antifreeze and
**

**V _{i,water }= 8 x 0.3 =2.4 liters of Water **

*for the 40%*

**V _{add, anti }= X x 0.4 liters of antifreeze **

**V _{add, water = }X x 0.6 liters of water**

*The final total volume of water and antifreeze that we have afer mixter is *

** V _{f,ant i}= V_{add,anti} + V_{i,anti}**

** V _{f,water} = V_{add,water} + V_{i}**

_{,water}

*sbstutuing the variables*

**V _{f,anti} = 8 x 0.7 + X x 0.4**

**V _{f,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

X(0.4-0.6)= 2.4-5.6

Multiplying the equation by -1

0.2X=3.2

X=3.2/0.2

X=16 liters

## Comments

THANK YOU VERY MUCH!

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