A really hard math question?

Question

Several; liters of 8%, 10% and 20% sodium hydroxide will be mixed together to produce 8 liters of a 12.5% concentration of sodium hydroxide. The amount of 20% solution must be 2 liters less thatn the amount of 8% solution used. How many liters of the 20% sodium hydroxide solution must be used.
P.s put this in three equations.

Answer 1

Let x = no of Litres of 8% solution
y = no of Litres of 10% solution
and z = no of Litres of 20% solution

Then x + y + z = 8 ........................ (1)
0.08x + 0.10y + 0.20z = 0.125 * 8 = 1
ie 0.08x + 0.10y + 0.20z = 1 ..............(2)
z = x - 2 ............................... (3)

Substitute equation (3) into equations (1) and (2)
x + y + x - 2 = 8
So 2x + y = 10 ...................... (4)
0.08x + 0.10y + 0.20 (x - 2) = 1
ie 0.28x + 0.10y = 1.4
ie 14x + 5y = 70 .................... (5)
Equation (5) - 5 x equation (4)
4x = 20
Thus x = 5
From equation (4) y = 0
From equation (3) z = 3

Check x + y + z = 8

0.08x + 0.10y + 0.20z = 0.4 + 0 + 0.6 = 1

Thus 3 litres of 20% sodium hydrroxide are used (as well as 5 litres of the 8% solution and none of the 10% solution)

Answer 2

there are three solutions, so say 8% has total vol of a L, 10% has 'b' L, and 20% = c L

so we know
8% of a + 10% of b + 20% of c = 12.5% of 8
the first equation, got it from this data:
"Several; liters of 8%, 10% and 20% sodium hydroxide will be mixed together to produce 8 liters of a 12.5% concentration of sodium hydroxide"

and then c= a-2
this equation, i got from:
"The amount of 20% solution must be 2 liters less thatn the amount of 8% solution used"

have you provided enough data?

because i can't figure out how to get the third equation....

Answer 3

Based on the question it appears it can be solved with just (2) equations - see below:
Let 'x' equal the number of liters of the 8% solution required.
Let 'y' equal the number of liters of 10% solution required.
Let 'x-2' equal the number of liters of the 20% solution required.
So: .08x + .1y + .2(x-2) = .125(8)
and x + y + (x-2) = 8
Now distribute the first equation: .08x+.1y+.2x-.4= 1 so....
.28x+.1y-.4=1 so .28x+.1y=1.4
Now distribute the second equation: x+y+x-2=8 so
2x+y-2=8 so 2x+y=10
Now process the two equations as simultaneous equations by multiplying the second one by( -.1) and you get -.2x-.1y=-1
then add the two equations together and you get
.08x=.4 or x=.4/.08 or x=5
If x=5 [number of 8% solution required], then x-2=3 [number of 20% solution required]. None of the 10% solution need be used to get the desired result.

Answer 4

let x be number of liters of 8%
let y be number of liters of 10%
let z be number of liters of 12%

you have x+y+z = 8 (1)

8x +10y +20z = 8*12.5 = 100 or 4x +5y +10 z =50 (2)

x =z+2 (3) put this in 1 and 2

y+z+2 = 8 4z+8 +5y +10z = 50

result x = 10/3 y=14/3 z=4/3

Answer 5

x = no. of liters of the 8% solution
y = " " 10% "
z = " " 20% "

Equations:

1: x + y + z = 8
2: 8x + 10y + 20z = 100
3: x = z + 2

substitute 3 in 1 & 2 to get:

1: y + 2z = 6
2: 10y + 28z = 84

multiply equation 1 with -10 to get:

1. -10y - 20z = -60
2. 10y + 28z = 84

add equations 1 and 2 to get:

8z = 24; z = 3

from equation 3, we get x = 5 and using equation 1, y = 0

So answer is: 3L of 20% solution

Answer 6

Let's start by assigning definitions:

x = liters of the 8% NaOH solution
y = liters of the 10% NaOH solution
z = liters of the 20% NaOH solution

We have three variables, therefore we need three equations to solve this problem.

First equation: x + y + z = 8 (total of 8 liters)
=> Another way to write this is y = 8 - z - x
Second equation: x = z + 2 (amount of 20% sol'n must be 2 less than 8% sol'n)
Third equation: x(0.08) + y(0.10) + z(0.20) = 8(0.125)

Solving the third equation for z (the amount of 20% solution you need, which is what your looking for), you get:

0.08x + 0.1y + 0.2z = 1
0.08(z + 2) + 0.1(8 - z - x) + 0.2z =1
0.08(z + 2) + 0.1(8 - z - (z + 2)) + 0.2z =1
(0.08z + 0.16) + (0.8 - 0.1z - 0.1z - 0.2) + 0.2z = 1
0.08z + 0.76 = 1
8z + 76 = 100
8z = 24
z = 3

Therefore, you need 3 liters of 20% solution (and 5 liters of 8% solution and 0 liters of 10% solution).

Check your answer:

5(0.08) + 0(0.10) + 3(0.20) = 8(0.125)
0.4 + 0 + 0.6 = 1? yes, therefore correct!

Answer 7

The third equation is

a + b + c = 8

or

x + y + z = 8

So we have

.08x + .10y + .20z = (.125)(8)
z = x - 2
x + y + z = 8

or

.08x + .10y + .20(x - 2) = 1
x + y + x - 2 = 8

Then

.28x + .10y = 1.4
2x + y = 10

Multiply the first equation by -10

-2.8 - y = -14
2x + y = 10

Add the two equations

-.8x = -4
x = 5
z = 3
y = 0

Several of us have arrived at the same, correct, answer by using three equations. Choose as best whichever method is clearest to you.

Answer 8

.08x + .10y + .20z = (.125)(8)
z = x + 2

Sorry, I don't see where a third equation is used.

Answer 9

Well, I know you would start off with this...

.08x + .1y + .2z = 12.5

But I'm not sure where to go from there.