Have two tanks of 100 liters each, with salt solution. One is at 9% and 4% other. How many liters at least must be removed from each container, so that upon mixing which is removed to obtain a 7% solution of salt concentration?
Mathmate showed this clever solution some time ago. If I can do it right it's like this.
......4%........7%........9%
......|.7-4=3...||..9-7=2..|
So you take 2 part of 4% and mix with 3 parts of 9% to give you 5 parts of 7%.
.09*H+.04L=.07(H+L)
Where H is the number liters high concentration, and L is number of liters of low concentraton.
Now, H<100, and L<100,
because the problem does not say if they have to be mixed in one of the 100 L contaners....I think it meant that.
If it meant that, then 100=H+L
H=100-L
.09(100-L)+.04L=.07*100
-.05L=7-9
L= 40 H=60
so you remove 60 liters of Low concentration, and pour from the high concentration 40 liters to fill the orignally low concentration, and you have 7 percent solution.
Now if you mix the two you didn't use in the other barrel, you have
.04*60+.09*40=X*100
X=(2.4+3.6)/100=six percent solution in that barrel.
To determine how many liters at least must be removed from each container, we can follow these steps:
Step 1: Assign variables
Let's assign variables to the unknown quantities:
- Let x be the amount (in liters) removed from the 9% solution tank.
- Let y be the amount (in liters) removed from the 4% solution tank.
Step 2: Calculate the amount of salt in each tank after removal
After removing x liters from the 9% tank, there will be (100 - x) liters remaining in the 9% tank.
Similarly, after removing y liters from the 4% tank, there will be (100 - y) liters remaining in the 4% tank.
The amount of salt in the 9% solution tank after removal is 0.09 * (100 - x) liters.
And the amount of salt in the 4% solution tank after removal is 0.04 * (100 - y) liters.
Step 3: Setting up the equation
For the resulting solution to have a salt concentration of 7%, the total amount of salt in the mixture should be equal to 7% of the total volume.
The total amount of salt in the mixture is given by:
0.09 * (100 - x) + 0.04 * (100 - y)
The total volume after mixing is given by:
100 - x + 100 - y = 200 - (x + y)
Setting up the equation:
0.09 * (100 - x) + 0.04 * (100 - y) = 0.07 * (200 - (x + y))
Step 4: Solve the equation
Simplify the equation:
9(100 - x) + 4(100 - y) = 7(200 - (x + y))
900 - 9x + 400 - 4y = 1400 - 7x - 7y
Combining like terms:
-9x - 4y = -7x - 7y + 1000
Simplifying further:
2x - 3y = 1000
Step 5: Determine the minimum amount to be removed
Now, we need to find the minimum values of x and y that satisfy the equation 2x - 3y = 1000.
One approach is to find the least common multiple (LCM) of the coefficients 2 and -3 (which is 6) and start with the smallest possible values of x and y.
Let's consider x = 0 and y = 0 first:
2(0) - 3(0) = 0 - 0 = 0 ≠ 1000
Since 0 is not equal to 1000, we need to increase the values of x and y.
Let's try x = 6 and y = 2:
2(6) - 3(2) = 12 - 6 = 6 ≠ 1000
Again, 6 is not equal to 1000, so we need to increase the values further.
Now, let's try x = 500 and y = 332, which is a multiple of 6:
2(500) - 3(332) = 1000 - 996 = 4 ≠ 1000
So, x = 500 and y = 332 do not satisfy the equation.
Since we are looking for the minimum amount to be removed, we need to continue exploring the possible values for x and y. However, it is not possible to reduce the salt concentration to exactly 7% by removing a specific amount of solution because there is no solution that satisfies the equation 2x - 3y = 1000.
Therefore, there is no combination of liters to be removed that will result in a 7% solution of salt concentration.