To solve this problem, let the five consecutive even integers be represented as $x, x+2, x+4, x+6, x+8$, where $x$ is the smallest.

Given Condition:

The smallest integer is $\frac{3}{4}$ of the largest integer. The largest integer is $x+8$.

Thus, we have the equation: $x = \frac{3}{4}(x+8)$

Step 1: Solve for $x$

Multiply through by 4 to eliminate the fraction: $4x = 3(x + 8)$

Simplify: $4x = 3x + 24$

Subtract $3x$ from both sides: $x = 24$

Step 2: Find the integers

If $x = 24$, then the five integers are: $24, 26, 28, 30, 32$

Final Answer:

From least to greatest, the numbers are: $24, 26, 28, 30, \text{and } 32$

Let me know if you'd like a detailed explanation of the solution! Here are five related questions to expand understanding:

1. How can this problem be adjusted to use consecutive odd integers instead?
2. What happens if the smallest number is a fraction of the middle integer?
3. How would you solve this problem if six integers were involved instead of five?
4. Can you write a similar problem where the condition relates to the second smallest integer instead?
5. What is the general process to solve for $n$ consecutive integers with a condition?

Tip: For problems involving consecutive integers, always use expressions like $x, x+2, x+4, \dots$ for even numbers to set up equations systematically.