A custom paint blend, Mixture Alpha, contains blue pigment, yellow pigment, and white base in the proportion 1:4:5. A second blend, Mixture Beta, contains yellow pigment, white base, and clear binder in the proportion 2:3:5. If identical masses of these two blends are mixed together to create a final paint mixture, how much clear binder, in kilograms, is present per kilogram of the resulting mixture?

The correct answer is 0.25 kg.

To find the mass of clear binder per kilogram of the resulting mixture, we calculate the proportion of clear binder contributed by each blend when equal masses are combined.

Step 1: Determine clear binder in Mixture Alpha
Mixture Alpha contains blue pigment, yellow pigment, and white base in the ratio 1 : 4 : 5.
Total parts in Mixture Alpha = 1 + 4 + 5 = 10 parts.
Since Mixture Alpha contains no clear binder, the fraction of clear binder in Mixture Alpha is 0.

$\text{Fraction of clear binder in Alpha}=0$

Step 2: Determine clear binder in Mixture Beta
Mixture Beta contains yellow pigment, white base, and clear binder in the ratio 2 : 3 : 5.
Total parts in Mixture Beta = 2 + 3 + 5 = 10 parts.
Clear binder accounts for 5 out of 10 parts of Mixture Beta.

$\text{Fraction of clear binder in Beta}=\frac{5}{10}=0.5$

Step 3: Calculate total clear binder in the combined mixture
Assume we mix 1 kg of Mixture Alpha with 1 kg of Mixture Beta to make a total of 2 kg of paint mixture.
Mass of clear binder from Mixture Alpha = 1 kg × 0 = 0 kg.
Mass of clear binder from Mixture Beta = 1 kg × 0.5 = 0.5 kg.
Total mass of clear binder in 2 kg of the resulting mixture = 0 kg + 0.5 kg = 0.5 kg.

Step 4: Calculate mass of clear binder per kilogram of final mixture
Dividing the total mass of clear binder by the total mass of the mixture:

$\text{Mass per kg}=\frac{0.5\text{kg}}{2\text{kg}}=0.25\text{kg}$

Therefore, 0.25 kg of clear binder is present per kilogram of the resulting mixture.