To create a specific shade of green paint, a painter mixes yellow paint and blue paint in … — SAT

To create a specific shade of green paint, a painter mixes yellow paint and blue paint in a ratio of $3:5$ by volume. To create a specific shade of teal paint, the painter mixes yellow paint and blue paint in a ratio of $1:3$ by volume. If the painter mixes equal volumes of the green paint and the teal paint to create a new mixture, what is the ratio of yellow paint to blue paint in the new mixture?

A
1:2
B
17:31
5:11Cevap
D
11:5

Cevap

The ratio of yellow paint to blue paint in the new mixture is 5:11.

The correct answer is the option stating a ratio of 5:11. To find the ratio of yellow paint to blue paint in the final mixture, we first express the composition of each initial paint as fractions of the whole. For the green paint, yellow is

\frac{3}{8}

and blue is

\frac{5}{8}

of the volume. For the teal paint, yellow is

\frac{1}{4}

(or

\frac{2}{8}

) and blue is

\frac{3}{4}

(or

\frac{6}{8}

) of the volume. When equal volumes of the two paints are combined, the total fraction of yellow paint in the mixture is the average of the two individual yellow fractions,

\frac{1}{2} \left(\frac{3}{8} + \frac{2}{8}\right) = \frac{5}{16}

. Similarly, the total fraction of blue paint is

\frac{1}{2} \left(\frac{5}{8} + \frac{6}{8}\right) = \frac{11}{16}

. Thus, the ratio of yellow paint to blue paint in the final mixture is

5:11

Adım Adım Çözüm

Determine the fraction of yellow paint and blue paint in each of the two initial paint mixtures.

For the green paint, the ratio of yellow to blue is

3:5

, meaning yellow paint is

\frac{3}{3+5} = \frac{3}{8}

of the volume, and blue paint is

\frac{5}{8}

of the volume. For the teal paint, the ratio of yellow to blue is

1:3

, meaning yellow paint is

\frac{1}{1+3} = \frac{1}{4}

of the volume, and blue paint is

\frac{3}{4}

of the volume.

Converting part-to-part ratios into part-to-whole fractions is necessary to determine the absolute quantities of each component in a mixture of equal volumes.

Calculate the total fraction of yellow paint and blue paint in the combined mixture.

Since equal volumes

V

of each paint are mixed, the total volume of the new mixture is

2V

. The total volume of yellow paint is

\frac{3}{8}V + \frac{1}{4}V = \frac{3}{8}V + \frac{2}{8}V = \frac{5}{8}V

. The total volume of blue paint is

\frac{5}{8}V + \frac{3}{4}V = \frac{5}{8}V + \frac{6}{8}V = \frac{11}{8}V

Adding the parts of each ingredient from both paint types gives the total volume of each ingredient in the new mixture.

Find the ratio of yellow paint to blue paint in the final mixture by comparing their volumes.

The ratio of yellow paint to blue paint in the combined mixture is

\frac{5}{8}V : \frac{11}{8}V

, which simplifies to

5:11

Simplifying the ratio of the volumes of yellow and blue paint gives the final part-to-part ratio for the new mixture.

Anahtar Kavram

Combining mixtures of different part-to-part ratios by converting them to part-to-whole fractions relative to a common total volume.

Tahmini Süre:1m 30s

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