Solve Linear Equation: Finding Survey Size Using 8x+4 and Fractions

The number of respondents to a survey is $8x+4$.

$\frac{2}{3}$ of them and another $2x$ responded that they do not like pistachio ice cream, while the remaining 8 participants said that they really liked the flavor.

How many participants were there in the survey?

To solve this problem, we'll follow these steps:

• Step 1: Write an equation based on the information given.
• Step 2: Simplify and solve for $x$.
• Step 3: Compute the total number of participants using the value of $x$.

Step 1: Set up the equation:

$\frac{2}{3}(8x + 4) + 2x + 8 = 8x + 4$

Step 2: Simplify the equation:

First, calculate $\frac{2}{3}(8x + 4)$:

$\frac{2}{3} \cdot 8x + \frac{2}{3} \cdot 4 = \frac{16x}{3} + \frac{8}{3}$

Insert this back into the equation:

$\frac{16x}{3} + \frac{8}{3} + 2x + 8 = 8x + 4$

Multiply the entire equation by 3 to eliminate the fractions:

$16x + 8 + 6x + 24 = 24x + 12$

Simplify:

$22x + 32 = 24x + 12$

Reorganize the equation:

$22x + 32 - 22x = 24x + 12 - 22x$

$32 = 2x + 12$

Solve for $x$:

$32 - 12 = 2x$

$20 = 2x$

$x = 10$

Step 3: Calculate the total number of participants:

Substitute $x = 10$ into $8x + 4$:

$8(10) + 4 = 80 + 4 = 84$

Therefore, the total number of participants is 84.