Question

Deon has 12 milliliters of 5% solution. He also has a solution that has a concentration of 30%. How many milliliters of the 30% solution does Deon need to add to the 5% solution to obtain a 20% solution

Answer 1

if he adds both solutions, the "average conentration" should be 20%

average: a% + b% /2

we have a "weighted average", instead of just counting "a" and "b" once,
and divide by two, we count the exact millilitres, and divide by total ml

weighted average:
x * 5% +y*30% = 20% /(x+y)

if x and y were =1, then it would be the usual average. instead they are
now weighted.

Answer 2

we can solve the equation by letting the 5% solution =12 ml
and re-arranging the terms:

12 * 5 +y*30 = 20 /(12+y) | *(12+y)
60 (12+y) + 30y (12+y) = 20
720 + 60y + 360 y + 30y² = 20

30 y ² + 380 y + 720 = 20 <--- we can set that to =0, and use pq

Answer 3

30 y ² + 380 y + 700 = 0

change to standard form for quadratic formula

y^2 + 12.67 y + 23.33

b=12.67
c=23.33
\[x = -\frac{-b\pm\sqrt{b^2-4c}}{2a} \]\[x = -\frac{-12.67\pm\sqrt{12.67^2-4\times 23.33}}{2} \]
\[x_1 = -\frac{-12.67\pm\sqrt{67}}{2} \]\[x_2 = -\frac{-12.67\pm\sqrt{253}}{2} \]

Answer 4

\[x_1 = -\frac{-12.67+\sqrt{160}}{2} = \frac{-12.67+12.64}{2} \approx0\]\[x_2 = -\frac{-12.67-\sqrt{160}}{2} = -\frac{-12.67-12.64}{2}=-12.65.\]yeah well, that can't be right. the ml of the second solution has to be positive

Answer 5

Don't you have to do a table

Answer 6

can you help me do a table

Answer 7

the formula should be: 60 +30y = 20 /(12+y)
I'm not sure where to start...

Answer 8

maybe we should start with second solution = 0 ml ? :)

Answer 9

function:
concentration =(x*5 + 30y) /(x+y) with x=12 ml
= (60 + 30y) /(12+y)

second solution (ml) | concentration level (%)
-----------------------------------------
0 | 5%
1 | 6.92%
2 | 8.57%
3 | 10%
4 | 11.25%
12 | 17.5%
13 | 18.0%
14 |
15 | 18.88%
16 | 19.28%
17 | 19.65%
18 | 20.00% <-------------------
19 | 20.32%

Answer 10

it's also easy to find the result using algebra, if you do it right.
my algebra above was broken.

correct way:
\[\frac{12\times 5\% +y\times 30\% }{12+y} =20\% \]then, multiply both sides with 12+y (I had it divided, that was the error)
\[60 +30y =20 (12+y) \]this can be simplified to:
\[y=18 ml.\]