Question 1204463
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Answers:
<font color=red>15 liters</font> of pure acid
<font color=red>30 liters</font> of the 10% acid solution


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Explanation


There are at least two ways to solve this problem. 
Method 1 is the standard algebraic approach.
Method 2 is what is taught to pharmacy students.




Method 1


Batch A = pure acid = 100% acid
Batch B = 10% acid solution


x = number of liters of batch A
45-x = number of liters of batch B
These two amounts combine to 45 liters total.


x = amount of pure acid from batch A
0.10(45-x) = 4.5 - 0.10x = amount of pure acid from batch B
x+(4.5 - 0.10x) = 0.90x+4.5 = total amount of pure acid


Divide this over the 45 total liters, and the goal is to arrive at a 40% concentration.


(amount of pure acid)/(total) = concentration
(0.90x+4.5)/(45) = 0.40
0.90x+4.5 = 45*0.40
0.90x+4.5 = 18
0.90x = 18-4.5
0.90x = 13.5
x = 13.5/0.90
x = <font color=red>15</font>
which leads to
45-x = 45-15 = <font color=red>30</font>


Therefore, we need <font color=red>15 liters</font> of pure acid to add to the <font color=red>30 liters</font> of the 10% acid solution.


In other words, we need <font color=red>15 liters</font> of batch A and <font color=red>30 liters</font> of batch B.


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Method 2


Batch A = 100% acid
Batch B = 10% acid
target solution = 40%


Gap from target to 100% is 60%
Gap from 10% to target is 30%


The ratio of 60% to 30% is 60:30 = 2:1


Meaning we'll need twice as much one batch compared to the other.


2:1 scales up to 2k:1k for some positive real number k.
2k+1k = 45
3k = 45
k = 45/3
k = 15
So 2k = 2*15 = 30


The amounts for batches A and B are 15 and 30. The order isn't clear just yet.


If we had 30 L of batch A, then we'll have 15*0.10 = 1.5 L of pure acid from batch B.
The total amount of pure acid would be 30+1.5 = 31.5
But notice that 31.5/45 = 0.70 = 70% which isn't the 40% target we're after.


If we had 15 L of batch A, then we'll have 30*0.10 = 3 L of pure acid from batch B.
15+3 = 18 L of pure acid total out of 45 L total
18/45 = 0.40 = 40%
We've reached the goal.


For more information, search out "Alligation Method Pharmacy". 
The spelling of "Alligation" is correct. There isn't an "e" but an "i" instead. Think "alligator". 
It's unfortunate the two words sound identical, and are spelled almost the same.


Here is a useful calculator
<a href="https://calculatormaths.com/alligation-calculator">https://calculatormaths.com/alligation-calculator</a>
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