The problem asks to find the value of $x$, given that the square and equilateral triangle have the same perimeter. The perimeter expressions are provided as:

• The square's perimeter: $2\frac{1}{2}x + 30$
• The triangle's perimeter: $5\frac{1}{3}x$

Step 1: Convert Mixed Numbers to Improper Fractions

• $2\frac{1}{2}x = \frac{5}{2}x$
• $5\frac{1}{3}x = \frac{16}{3}x$

Step 2: Set the Perimeters Equal

Since the square and equilateral triangle have the same perimeter: $\frac{5}{2}x + 30 = \frac{16}{3}x$

Step 3: Solve for $x$

1. Clear the fractions by finding the least common denominator (LCD), which is 6. Multiply every term by 6: $6\left(\frac{5}{2}x + 30\right) = 6\left(\frac{16}{3}x\right)$ Simplifying each term: $15x + 180 = 32x$

2. Move all terms involving $x$ to one side: $180 = 32x - 15x$ Simplifying: $180 = 17x$

3. Solve for $x$: $x = \frac{180}{17}$

   So, $x \approx 10.59$.

Let me know if you'd like further explanation or have any other questions!

Related Questions:

1. How would the problem change if the triangle had a different number of sides, like a pentagon?
2. Can you solve this equation by graphing both sides of the equation?
3. How do you handle similar problems involving circles and polygons with different perimeters?
4. What if the perimeter of the square was given in terms of a different variable?
5. Can this be solved by substituting specific values for $x$?

Tip:

Always convert mixed numbers to improper fractions to make solving equations easier.