Ellie, whose company makes liquid dyes for use in dyeing clothing, currently needs to prepare 260 liters of 38%-concentrated pink dye. The company stocks 26%-concentrated pink dye and 46%-concentrated pink dye. How much of each dye should she mix together?

Question

Write your answers as whole numbers or as decimals rounded to the nearest tenth.
liters of 26% dye
liters of 46% dye

Answer 1

To solve for the amounts of 26%-concentrated dye (\(x\)) and 46%-concentrated dye (\(y\)) that Ellie needs to mix to obtain 260 liters of 38%-concentrated dye, we can set up a system of equations based on the volume and concentration.

1. The total volume equation: \[ x + y = 260 \]

2. The total dye concentration equation: \[ 0.26x + 0.46y = 0.38(260) \]

Now, we first calculate the total amount of dye in the desired concentration: \[ 0.38(260) = 98.8 \text{ liters of dye} \]

So our equations are:

1. \(x + y = 260\)
2. \(0.26x + 0.46y = 98.8\)

Now, we can solve the first equation for \(y\): \[ y = 260 - x \]

Next, we substitute \(y\) in the second equation: \[ 0.26x + 0.46(260 - x) = 98.8 \]

Expanding: \[ 0.26x + 119.6 - 0.46x = 98.8 \] Combining like terms: \[ -0.20x + 119.6 = 98.8 \] \[ -0.20x = 98.8 - 119.6 \] \[ -0.20x = -20.8 \] Dividing by \(-0.20\): \[ x = \frac{-20.8}{-0.20} = 104 \]

Now substitute \(x = 104\) back into the first equation to find \(y\): \[ y = 260 - 104 = 156 \]

Thus, the amounts of dye needed are:

• Liters of 26% dye: \( \mathbf{104} \)
• Liters of 46% dye: \( \mathbf{156} \)

Final result:

• 26% dye: 104 liters
• 46% dye: 156 liters