Let's go through each question one by one.
Question 1: The recipe requires 3 cups of flour for every 2 cups of sugar. To find out how many cups of flour are needed for 8 cups of sugar, we can set up a ratio or a proportion.
\[ \text{Ratio of flour to sugar} = \frac{3 \text{ cups of flour}}{2 \text{ cups of sugar}} = \frac{x \text{ cups of flour}}{8 \text{ cups of sugar}} \]
Cross-multiplying gives: \[ 3 \cdot 8 = 2 \cdot x \] \[ 24 = 2x \] \[ x = \frac{24}{2} = 12 \]
So, the answer is 12.
Question 2: If \( y = 5 \) when \( x = 2 \), then the constant of proportionality \( k \) can be found: \[ k = \frac{y}{x} = \frac{5}{2} \]
Now to find \( y \) when \( x = 6 \): \[ y = kx = \frac{5}{2} \cdot 6 = 15 \]
So the answer is 15.
Question 3: If \( y \) is directly proportional to \( x \) and \( y = 12 \) when \( x = 4 \), we again find the constant of proportionality \( k \): \[ k = \frac{y}{x} = \frac{12}{4} = 3 \]
Hence, the constant of proportionality is 3.
Question 4: To solve the equation \( 3x - 4 = 11 \): \[ 3x = 11 + 4 \] \[ 3x = 15 \] \[ x = \frac{15}{3} = 5 \]
Therefore, the solution is 5.
Question 5: To solve the equation \( 3x + 9 = 0 \): \[ 3x = -9 \] \[ x = \frac{-9}{3} = -3 \]
Thus, the value of \( x \) is -3.
Here are the final answers:
- 12
- 15
- 3
- 5
- -3