Given,
Solution A contains 2 parts salt to 8 parts water
Solution B contains 3 parts salt to 5 parts water
Target Mixture contains3 parts salt to 7 parts water:
Expressing the salt concentration in decimal form
Solution A: 2/(2 + 8) = 2/10 = 0.20 salt
Solution B: 3/(3 + 5) = 3/8 = 0.375 salt
Mixed Solution: s/(3 + 7) = 3/10 = 0.30 salt
Since target mixture is 280 quarts
If the amount of 0.375 salt present in mixed solution = x
Then, the amount of 0.20 salt, y = 280 – x
Using a typical mixture equation
0.375x + 0.20(280-x) = 0.30(280)
0.375x + 56 - 0.20x = 84
Subtract 56 from both sides of the equation
0.375x - 0.20x + 56 – 56 = 84 – 56
0.375x - 0.20x = 28
0.175x = 28
Divide both sides of the equation by 0.175
0.175x/0.175 = 28/0.175
x = 160
y = 280 – x
y = 280 – 160
y = 120
Therefore,
x = 160 quarts of the solution that contains 3 parts salt to 5 parts water
y = 120 quarts of the solution that contains 2 parts salt to 8 parts water
The volume of the first solution and second solution is [tex]\boxed{{\mathbf{160 quartz\;, 120 quartz\;\;}}}[/tex] respectively.
Further explanation:
Step by step explanation:
Step 1:
It is given that one solution contains 2 parts salt to 8 parts water.
Therefore, the salt concentration in the first solution can be expressed in decimal form as,
[tex]\begin{aligned}{\text{salt concentration}} &= \frac{2}{{2 + 8}} \\&= \frac{2}{{10}} \\&={\text{ 0}}{\text{.20 salt}} \\\end{aligned}[/tex]
Step 2:
The another solution contains 3 parts salt to 5 parts water.
Therefore, the salt concentration in the second solution can be expressed in decimal form as,
[tex]\begin{aligned}{\text{salt concentration}} &= \frac{3}{{3 + 5}} \\&= \frac{3}{8} \\&={\text{ 0}}{\text{.375 salt}} \\\end{aligned}[/tex]
Step 3:
It is given that the volume of the final mixture is 280 quartz of a solution in which 3 parts salt to 7 parts water.
Therefore, the salt concentration in the final solution can be expressed in decimal form as,
[tex]\begin{aligned}{\text{salt concentration}} &= \frac{3}{{3 + 7}} \\&= \frac{3}{{10}} \\&={\text{ 0}}{\text{.30 salt}} \\\end{aligned}[/tex]
Consider [tex]x[/tex] be the volume of the second solution in which [tex]0.375[/tex] salt present in the final solution.
Therefore, the second solution can be expressed as [tex]0.375x[/tex].
The volume of the first solution is [tex]280 - x[/tex] in which [tex]0.20[/tex] salt present in the final solution.
Therefore, the first solution can be expressed as [tex]\left( {280 - x} \right)0.20[/tex].
Step 4:
The final mixture is the sum of the first solution and second solution.
The equation for final mixture can be expressed as,
[tex]\begin{aligned}0.375x + \left( {280 - x} \right)0.20 &= 0.30\left( {280} \right) \hfill \\0.375x + \left( {280 \times 0.20 - 0.20x} \right) &= 0.30\left( {280} \right) \hfill \\0.375x + \left( {56 - 0.20x} \right) &= 84 \hfill \\0.175x &= 28 \hfill \\\end{aligned}[/tex]
Now simplify the further solution.
[tex]\begin{aligned}0.175x &= 28 \\x &= \frac{{28}}{{0.175}} \\x &= 160 \\\end{aligned}[/tex]
Therefore, the volume of the second solution is [tex]160{\text{ quartz}}[/tex].
The volume of the first solution can be calculated as,
[tex]280 - 160 = 120{\text{ quartz}}[/tex]
Therefore, the volume of the first solution is [tex]120{\text{ quartz}}[/tex].
Learn more:
Answer details:
Grade: High school
Subject: Mathematics
Chapter: Linear equation
Keywords: mixture, solution, percent, volume, salt, water, quartz, equation, linear form, fraction, distributive property, denominator, numerator, mixed together, expression
