Answer :
Answer:
Step-by-step explanation:
To solve this problem, we need to determine the amounts of the 5% hydrochloric acid solution and the 45% hydrochloric acid solution that Rajan should mix to get 1 liter of a 15% hydrochloric acid solution. Let \( x \) be the amount of 5% solution and \( y \) be the amount of 45% solution.
We have two conditions to satisfy:
1. The total volume of the mixture should be 1 liter.
2. The final concentration of hydrochloric acid in the mixture should be 15%.
First, let's set up the equation for the total volume:
\[ x + y = 1 \]
Next, let's set up the equation for the concentration of hydrochloric acid. The amount of pure hydrochloric acid in the final mixture must be equal to the sum of the amounts of pure hydrochloric acid from each of the two solutions. The pure hydrochloric acid from each solution can be calculated by multiplying the volume by the concentration:
\[ 0.05x + 0.45y = 0.15 \times 1 \]
Simplify the second equation:
\[ 0.05x + 0.45y = 0.15 \]
Now we have the system of linear equations:
\[
\begin{cases}
x + y = 1 \\
0.05x + 0.45y = 0.15
\end{cases}
\]
We can solve this system using substitution or elimination. We'll use substitution.
From the first equation:
\[ y = 1 - x \]
Substitute \( y = 1 - x \) into the second equation:
\[ 0.05x + 0.45(1 - x) = 0.15 \]
Distribute the 0.45:
\[ 0.05x + 0.45 - 0.45x = 0.15 \]
Combine like terms:
\[ 0.05x - 0.45x = 0.15 - 0.45 \]
\[ -0.40x = -0.30 \]
Solve for \( x \):
\[ x = \frac{-0.30}{-0.40} \]
\[ x = 0.75 \]
Now, substitute \( x = 0.75 \) back into the first equation to find \( y \):
\[ y = 1 - 0.75 \]
\[ y = 0.25 \]
Therefore, Rajan should mix 0.75 liters of the 5% hydrochloric acid solution with 0.25 liters of the 45% hydrochloric acid solution to obtain 1 liter of a 15% hydrochloric acid solution.
Answer:
- 0.75 L of 5%
- 0.25 L of 45%
Step-by-step explanation:
You want the number of liters of 45% acid and of 5% acid that must be combined to make 1 liter of 15% acid.
Setup
If x represents the number of liters of 45% acid, the amount of acid in the 1-liter solution will be ...
0.45x +0.05(1 -x) = 0.15(1)
Solution
Simplifying the equation gives ...
0.45x -0.05x +0.05 = 0.15
0.40x = 0.10 . . . . . . . . . . . . . . . . subtract 0.05
x = 0.10/0.40 = 1/4 = 0.25 . . . . . divide by the coefficient of x
1-x = 1 -0.25 = 0.75 . . . . . . . . . . . amount of 5% acid
Rajan should combine 0.25 liters of 45% acid with 0.75 liters of 5% acid to make 1 liter of 15% acid solution.
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Additional comment
Mixture problems are nicely solved by letting a single variable represent the quantity of the most potent contributor.
