Mixture Problems: Finding the Right Concentration

How can Vince mix up a 17%-concentrated teal dye?

Vince works for a company that manufactures liquid dyes for clothing. He currently wants to mix up some 17%-concentrated teal dye. He has 110 liters of 5%-concentrated teal dye, as well as plenty of 35%-concentrated teal dye. How many liters of the 35%-concentrated teal dye will Vince need to add to the 5%-concentrated teal dye to make a batch with a concentration of 17%?

Answer:

To mix up a 17%-concentrated teal dye from a 5% solution and a 35% solution, Vince needs to add approximately 73.3 liters of the 35% solution to his existing 110 liters of 5% solution.

Explanation: This problem can be solved with a method called mixture problems in algebra. The idea behind it is that the amount of dye in the final mix is the sum of the amount of dye in each ingredient. So, the amount of dye in the 5% solution is 0.05 * 110 = 5.5 liters.

Let's denote the amount of the 35% solution that needs to be added as x liters. Hence, the amount of dye in the 35% solution is 0.35 * x. The total amount of the mix is 110 + x, and the total concentration is supposed to be 17%, which gives the equation 5.5 + 0.35x = 0.17 * (110 + x).

First, distribute the 0.17 on the right side of the equation, getting 5.5 + 0.35x = 18.7 + 0.17x. Move the 0.17x to the left by subtracting 0.17x on both sides to isolate x. Now we have 0.18x = 18.7 - 5.5 = 13.2. Finally, divide by 0.18 to find the amount of the 35% solution that Vince needs to add, which is x = 13.2 / 0.18 ≈ 73.3 liters.