Problem 61 - Algebra for College Students

Question

Solve each problem. How much pure dye must be added to 4 gal of a \(25 \%\) dye solution to increase the solution to \(40 \% ?\) (Hint: Pure dye is \(100 \%\) dye.).

Step-by-Step Solution

Answer

1 gallon of pure dye must be added.

Step 1 - Identify the Initial Amount of Dye

Start by identifying the amount of dye in the initial solution. The solution is 4 gallons of a 25% dye solution.

Step 2 - Calculate the Initial Amount of Pure Dye

Calculate the amount of pure dye by multiplying the volume by the percentage: : : : Can do it Calculate it as follows: Initial Amount of Pure Dye = 4 gallons * 0.25 = 1 gallon.

Step 3 - Set Up the Equation for the Final Solution

Let the amount of pure dye to be added be x gallons. The final solution must be 40% dye. Therefore, set up the equation: 1 (gallon of existing dye) + x (gallons of pure dye) = 0.40 (4 + x) (total gallons of the new solution).

Step 4 - Solve the Equation

Solve the equation: 1 + x = 0.40 * (4 + x) Distribute the 0.40: 1 + x = 1.6 + 0.40x Subtract 0.40x from both sides: 1 + 0.60x = 1.6 Subtract 1 from both sides: 0.60x = 0.6 Divide both sides by 0.60: x = 1.

Step 5 - State Your Answer

Therefore, 1 gallon of pure dye must be added to the 4 gallons of 25% dye solution to increase the concentration to 40%.

Key Concepts

Percentage Concentration

Understanding percentage concentration is crucial for solving mixture problems.
The percentage concentration tells us the proportion of one substance within a total mixture. It is given as a percentage.
For example, a 25% dye solution means there are 25 parts of dye for every 100 parts of the solution.
To convert this percentage to a decimal for calculations, divide by 100. For instance, 25% becomes 0.25.
In the exercise, we start with 4 gallons of solution, containing 25% dye. This means there are 4 gallons * 0.25 = 1 gallon of pure dye in the initial mixture.
Keep in mind: percentage concentration is always relative to the total volume of the solution.

Algebraic Equations

Algebraic equations are essential to solving mixture problems like this. They help us express relationships between quantities mathematically.
For our problem, we need to find how much pure dye to add to reach a 40% concentration.
Let’s break it down.
We know:

• Initial pure dye: 1 gallon
• New volume after adding x gallons of pure dye: 4 + x gallons
• Desired concentration: 40% or 0.40 in decimal

The equation combines all these elements:
1 (gallon of pure dye) + x (pure dye to be added) = 0.40 * (4 + x) (total volume).
This equation represents the balance we are trying to achieve with the added dye.

Mixture Problems

Mixture problems often involve finding the right balance of substances to achieve a desired concentration.
They require careful setup of the problem and usage of algebra to solve them.
In our exercise, we need to add pure dye to increase the concentration.
This involves understanding:

• The initial state of the solution (4 gallons with 25% dye)
• The desired state (40% dye concentration)

The equation we formed is solved step-by-step:

• 1 + x = 0.40 * (4 + x)
• Distribute: 1 + x = 1.6 + 0.40x
• Group like terms: 1 + 0.60x = 1.6
• Isolate x: 0.60x = 0.6, then x = 1

This tells us we need to add 1 gallon of pure dye to achieve the desired concentration. Understanding the logical flow from problem setup to final solution is key to mastering mixture problems.