An artist makes a certain shade of green paint by mixing blue and yellow inn a ratio of 3:4. She makes orange by mixing red and yellow in the ratio of 2:3. If on one day she mixes both green and orange and uses equal amounts of blue and red paint, what fractional part of the paint that she uses is yellow?

Options

A. `(7)/(12)`
B. `(17)/(29)`
C. `(7)/(5)`
D. `(12)/(17)`

Answer

To solve the problem step by step, we will analyze the paint mixtures and calculate the amount of yellow paint used.

### Step 1: Determine the amounts of blue and yellow in green paint
The ratio of blue to yellow in green paint is 3:4.
Let the amount of blue paint be \(3x\) and the amount of yellow paint be \(4x\).

### Step 2: Determine the amounts of red and yellow in orange paint
The ratio of red to yellow in orange paint is 2:3.
Let the amount of red paint be \(2y\) and the amount of yellow paint be \(3y\).

### Step 3: Set the amounts of blue and red paint equal
The problem states that equal amounts of blue and red paint are used.
Let’s assume the amount of blue paint used is 6 liters. Therefore, the amount of red paint used is also 6 liters.

### Step 4: Calculate the value of \(x\) for green paint
From the blue paint ratio:
\[ 3x = 6 \implies x = 2 \]
Now, substitute \(x\) back to find the yellow paint in green:
\[ 4x = 4 \times 2 = 8 \text{ liters of yellow paint} \]

### Step 5: Calculate the value of \(y\) for orange paint
From the red paint ratio:
\[ 2y = 6 \implies y = 3 \]
Now, substitute \(y\) back to find the yellow paint in orange:
\[ 3y = 3 \times 3 = 9 \text{ liters of yellow paint} \]

### Step 6: Calculate the total amounts of yellow paint used
The total amount of yellow paint from both mixtures is:
\[ \text{Total yellow} = 8 \text{ (from green)} + 9 \text{ (from orange)} = 17 \text{ liters} \]

### Step 7: Calculate the total amount of paint used
The total amount of paint used (blue, red, and yellow) is:
\[ \text{Total paint} = 6 \text{ (blue)} + 6 \text{ (red)} + 17 \text{ (yellow)} = 29 \text{ liters} \]

### Step 8: Calculate the fractional part of yellow paint
The fractional part of the paint that is yellow is:
\[ \text{Fraction of yellow} = \frac{\text{Total yellow}}{\text{Total paint}} = \frac{17}{29} \]

### Final Answer
The fractional part of the paint that she uses which is yellow is \(\frac{17}{29}\).

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