Question 1204664
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Batch A: 10% morphine
Batch B: 50% morphine
Target concentration: 20% morphine


The gap from 10% to 20% is 10
The gap from 20% to 50% is 30
The ratio 10:30 reduces to 1:3


The nurse will need 3 times as much of one solution compared to the other.
The ratio 1:3 scales up to 1x:3x for some positive real number x.
Add up those parts and set the sum equal to the goal of 50 mL.
1x+3x = 50
4x = 50
x = 50/4
x = 12.5


She'll need x = 12.5 mL of one batch and 3x = 3*12.5 = 37.5 mL of the other batch.


The question is: which values go where?


Let's make a table where we have 12.5 mL of batch A and 37.5 mL of batch B
<table border = "1" cellpadding = "5"><tr><td></td><td>Amount of solution</td><td>Amount of pure morphine</td></tr><tr><td>A</td><td>12.5</td><td>0.10*12.5 = 1.25</td></tr><tr><td>B</td><td>37.5</td><td>0.50*37.5 = 18.75</td></tr><tr><td>Total</td><td>50</td><td>1.25+18.75 = 20</td></tr></table>
The takeaway from that table is we have 20 mL of pure morphine out of 50 mL total
20/50 = 0.40 = 40% is the final concentration.


We wanted a 20% concentration instead, so we must flip the values.
<table border = "1" cellpadding = "5"><tr><td></td><td>Amount of solution</td><td>Amount of pure morphine</td></tr><tr><td>A</td><td>37.5</td><td>0.10*37.5 = 3.75</td></tr><tr><td>B</td><td>12.5</td><td>0.50*12.5 = 6.25</td></tr><tr><td>Total</td><td>50</td><td>3.75+6.25 = 10</td></tr></table>
10 mL of pure morphine out of 50 mL total
10/50 = 0.20 = 20% is the final concentration, which is the target we're after.



Answers:
<font color=red>37.5 mL</font> of the 10% solution
<font color=red>12.5 mL</font> of the 50% solution


More practice with a similar question
<a href="https://www.algebra.com/algebra/homework/word/mixtures/Mixture_Word_Problems.faq.question.1204463.html">https://www.algebra.com/algebra/homework/word/mixtures/Mixture_Word_Problems.faq.question.1204463.html</a>
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