Questions: How much pure dye must be added to 3 gal of a 25% dye solution to increase the solution to 50% (Hint: pure dye is 100% dye.) To obtain a 50% dye solution, gal of pure dye must be added to 3 gal of a 25% dye solution. (Round to the nearest tenth as needed.)

How much pure dye must be added to 3 gal of a 25% dye solution to increase the solution to 50% (Hint: pure dye is 100% dye.)

To obtain a 50% dye solution, gal of pure dye must be added to 3 gal of a 25% dye solution. (Round to the nearest tenth as needed.)

Transcript text: How much pure dye must be added to 3 gal of a $25 \%$ dye solution to increase the solution to $50 \%$ (Hint: pure dye is $100 \%$ dye.) To obtain a $50 \%$ dye solution, $\square$ gal of pure dye must be added to 3 gal of a $25 \%$ dye solution. (Round to the nearest tenth as needed.)

Solution

Solution Steps

Step 1: Define the variables

Let $ x $ be the amount of pure dye (in gallons) that needs to be added to the solution.

Step 2: Set up the equation

The total amount of dye in the final solution is the sum of the dye in the original solution and the pure dye added. The final solution should be $ 50\% $ dye. Therefore: \[ 0.25 \times 3 + 1 \times x = 0.50 \times (3 + x) \]

Step 3: Solve the equation

Simplify the equation: \[ 0.75 + x = 1.5 + 0.5x \] Subtract $ 0.5x $ from both sides: \[ 0.75 + 0.5x = 1.5 \] Subtract $ 0.75 $ from both sides: \[ 0.5x = 0.75 \] Divide both sides by $ 0.5 $: \[ x = 1.5 \]

Step 4: Round the result

The value of $ x $ is $ 1.5 $ gallons, which is already rounded to the nearest tenth.