A biologist has three salt solutions: some 5% solution, some 15% solution, and some 25% solution. She needs to mix some of each to get 48 liters of 18.75% solution. She wants the number of liters of the 25% solution to be equal to one liter less than the sum of twice the number of liters of the 5% solution and the number of liters of the 15% solution. How many liters of each solution should she use?
I've been trying this for hours, I just don't know where to starts. :(
If the amounts are x,y,z then we have
x+y+z = 48
.05x+.15y+.25z = .1875*48
z = 2x+y-1
now just solve for x,y,z
To solve this problem, let's break it down step by step:
1. Assign variables: Let the number of liters of the 5% solution be x, the number of liters of the 15% solution be y, and the number of liters of the 25% solution be z.
2. Set up equations based on the given information:
a) The total volume of the mixture is 48 liters: x + y + z = 48.
b) The concentration of the mixture is 18.75%: (0.05x + 0.15y + 0.25z) / 48 = 0.1875.
c) The number of liters of the 25% solution is one less than the sum of twice the number of liters of the 5% solution and the number of liters of the 15% solution: z = 2x + y - 1.
3. Simplify and solve the equations:
a) Rearrange equation (c) to obtain y = z - 2x + 1.
b) Substitute equations (a) and (b) into equation (c) to get z = 3x + 3.
c) Substitute equations (a) and (c) into equation (b) to obtain (0.05x + 0.15y + 0.25z) / 48 = 0.1875.
d) Simplify equation (c) to (0.05x + 0.15(z - 2x + 1) + 0.25(3x + 3)) / 48 = 0.1875.
e) Further simplify equation (c) to (0.05x + 0.15z - 0.3x + 0.15 + 0.75x + 0.75) / 48 = 0.1875.
f) Combine like terms, resulting in (0.5x + 0.9z + 0.9) / 48 = 0.1875.
g) Multiply both sides by 48 to eliminate the denominator: 0.5x + 0.9z + 0.9 = 48 * 0.1875.
h) Simplify further to 0.5x + 0.9z + 0.9 = 9.
i) Rearrange equation (h) to obtain 0.5x + 0.9z = 9 - 0.9.
j) Simplify to 0.5x + 0.9z = 8.1.
4. Now we have a system of two equations with two variables:
a) x + y + z = 48.
b) 0.5x + 0.9z = 8.1.
5. We can solve this system of equations using substitution or elimination methods. Let's use the substitution method:
a) Solve equation (a) for y: y = 48 - x - z.
b) Substitute y = 48 - x - z into equation (b): 0.5x + 0.9z = 8.1.
c) Simplify equation (b) to get 0.5x + 0.9z = 8.1.
d) Now we have a single equation with two variables: 0.5x + 0.9z = 8.1.
6. Solve equation (d) for x in terms of z:
a) Rearrange equation (d) to obtain x = (8.1 - 0.9z) / 0.5.
7. Substitute the value of x obtained in step 6 into equation (a) to find the corresponding values of y and z.
By following these steps, you should be able to find the number of liters of each solution that the biologist should use.