A 25% vinegar solution is combined with triple the amount of a 45% vinegar solution and a 5% vinegar solution resulting in 20 milliliters of a 30% vinegar solution.
Write an equation that models this situation and explain what each part represents in the situation. Then solve the equation and find the amount of each of the solutions that were combined.
Please help. We didn't really talk that much about these types of problems and I'm confused.
You are dealing with 2 concepts.
1. The volume of the two liquids. The concentration of vinegar in each one
let the volume of the 45% stuff be x litres
then the volume of the 25% stuff is 3x L
let the volume of the 5% stuff be y L
x+3x+y = 20
y = 20 - 4x
Now to the concentrion:
.45x + .25(3x) + .05(20-4x) = .3(20)
times 100
45x + 25(3x) + 5(20-4x) = 600
solve for x, then back-substitute
notice how much of the 5% vinegar is needed !!
Thanks, been having trouble with that question too.
Let's break down the problem step-by-step and write an equation to model the situation.
1. Let's define the variables:
Let's assume that:
- x represents the amount (in milliliters) of the 25% vinegar solution.
- y represents the amount (in milliliters) of the 45% vinegar solution.
- z represents the amount (in milliliters) of the 5% vinegar solution.
2. Express the given information in terms of the variables:
- The 25% vinegar solution is combined with triple the amount of the 45% vinegar solution: Since the amount of the 45% vinegar solution is tripled, this means that the amount of the 45% vinegar solution is 3y.
- The total amount of vinegar solution is 20 milliliters.
3. Write the equation:
The equation that represents the situation is:
0.25x + 0.45(3y) + 0.05z = 0.30(20)
In this equation:
- 0.25x represents the amount of vinegar (in milliliters) in the 25% vinegar solution, considering that vinegar is 25% of the solution.
- 0.45(3y) represents the amount of vinegar (in milliliters) in the 45% vinegar solution, considering that vinegar is 45% of the solution and the amount of the 45% vinegar solution is tripled.
- 0.05z represents the amount of vinegar (in milliliters) in the 5% vinegar solution, considering that vinegar is 5% of the solution.
- 0.30(20) represents the required amount (in milliliters) of vinegar in the resulting 30% vinegar solution, considering that vinegar is 30% of the solution and the total volume of the resulting solution is 20 milliliters.
4. Solve the equation:
To solve the equation, we need to solve for x, y, and z. However, since there are three variables and only one equation, we cannot find unique values for x, y, and z. There would be an infinite number of possible combinations of x, y, and z that satisfy the equation.
To solve this problem, let's break it down step-by-step and understand the situation.
Let's start by considering the amount of each vinegar solution that was combined.
Let:
- x represent the amount (in milliliters) of the 25% vinegar solution,
- 3x represent the amount (in milliliters) of the tripled 45% vinegar solution, and
- y represent the amount (in milliliters) of the 5% vinegar solution.
According to the problem, a total of 20 milliliters of a 30% vinegar solution was obtained after combining these three solutions.
Now, let's create an equation to model the situation.
Step 1: Considering the total volume of the mixture:
The total volume of the mixture is the sum of the volumes of the three solutions, which can be represented as:
x + 3x + y = total volume of the mixture
Simplifying this expression, we get:
4x + y = total volume equation ---- (Equation 1)
Step 2: Considering the concentration of the mixture:
To determine the concentration, we need to consider the total amount of vinegar in the mixture and divide it by the total volume of the mixture.
The amount of vinegar in the 25% solution is 0.25x.
The amount of vinegar in the tripled 45% solution is 3 * 0.45x = 1.35x.
The amount of vinegar in the 5% solution is 0.05y.
The total amount of vinegar in the mixture is the sum of these amounts, which can be represented as:
0.25x + 1.35x + 0.05y = total amount of vinegar in the mixture
Simplifying this expression, we get:
1.6x + 0.05y = total amount of vinegar equation ---- (Equation 2)
Step 3: Considering the concentration of the mixture:
We know that the final mixture has a concentration of 30% vinegar.
The total amount of vinegar in the mixture can also be represented as a percentage of the total volume of the mixture:
(total amount of vinegar in the mixture / total volume of the mixture) * 100 = 30
Substituting the equations we derived (Equation 1 and Equation 2) into this equation, we can solve for the values of x and y.
Solving the equation system will determine the amount of each vinegar solution that was combined.