solve: floor(0.9Xi)+floor(0.1Xi)=[62,63,184] where B=[150,0,255], M=[255,150,0], W=[0,255,150]

The problem involves solving an equation with floor functions and a sum across an index \( i \), based on certain values given. Specifically, the equation to solve is:

\[
\lfloor 0.9 X_i \rfloor + \lfloor 0.1 X_i \rfloor = [62, 63, 184]
\]

Additionally, we are given:

- \( B = [150, 0, 255] \)
- \( M = [255, 150, 0] \)
- \( W = [0, 255, 150] \)

To break down and solve the equation:

### 1. Understanding the equation

The floor function \( \lfloor x \rfloor \) gives the greatest integer less than or equal to \( x \). The equation suggests that for each \( X_i \), when you compute \( \lfloor 0.9 X_i \rfloor + \lfloor 0.1 X_i \rfloor \), the result equals a vector \( [62, 63, 184] \), which represents individual elements to be solved for each index \( i \).

Thus, we have:

\[
\lfloor 0.9 X_1 \rfloor + \lfloor 0.1 X_1 \rfloor = 62
\]
\[
\lfloor 0.9 X_2 \rfloor + \lfloor 0.1 X_2 \rfloor = 63
\]
\[
\lfloor 0.9 X_3 \rfloor + \lfloor 0.1 X_3 \rfloor = 184
\]

Now, we need to solve for \( X_1 \), \( X_2 \), and \( X_3 \) individually.

### 2. Solving for \( X_1 \), \( X_2 \), and \( X_3 \)

Let's take each equation step by step.

#### For \( X_1 \):
We have:
\[
\lfloor 0.9 X_1 \rfloor + \lfloor 0.1 X_1 \rfloor = 62
\]

Let \( X_1 = n_1 + f_1 \), where \( n_1 \) is the integer part and \( f_1 \) is the fractional part of \( X_1 \).

This breaks down into two components:
- \( \lfloor 0.9 X_1 \rfloor = \lfloor 0.9 n_1 + 0.9 f_1 \rfloor \)
- \( \lfloor 0.1 X_1 \rfloor = \lfloor 0.1 n_1 + 0.1 f_1 \rfloor \)

We need to analyze similar equations for \( X_2 \) and \( X_3 \).

I'll proceed to solve for each \( X_i \).The values of \( X_1 \), \( X_2 \), and \( X_3 \) that satisfy the given equations are:

\[
X_1 = 63, \quad X_2 = 64, \quad X_3 = 185
\]

This solves the equation \( \lfloor 0.9 X_i \rfloor + \lfloor 0.1 X_i \rfloor = [62, 63, 184] \).

### Would you like further details or clarifications?

Here are five related questions to expand on this concept:

1. How does the floor function affect the solution in similar equations?
2. Can the method be extended to solve other floor function problems involving different coefficients?
3. What is the behavior of the floor function when applied to negative numbers?
4. Could we solve this problem using inequalities instead of the floor function?
5. How do these solutions compare if we use the ceiling function instead of the floor function?

**Tip:** When working with the floor function, always consider the fractional part of the number and how it affects rounding down to the nearest integer.

solve: floor(0.9Xi) + floor(0.1Xi) = [62, 63, 184] where B = [150, 0, 255], M = [255, 150, 0], W = [0, 255, 150]

This page explores how to solve equations involving the floor function with specific coefficients. Learn step-by-step how to approach floor(0.9Xi) + floor(0.1Xi) = [62, 63, 184] and how the floor function affects rounding and integer values. Suitable for students in Grades 10-12.

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