for any real number y let {y} refer to the frational part of y. suppose x sarisfies 3x+ {x} = 100 then find 4x 

Given the equation \( 3x + \{x\} = 100 \), where \(\{x\}\) refers to the fractional part of \(x\), we need to find \(4x\).

### Step 1: Express \(x\) as the sum of its integer and fractional parts
Let:
\[
x = n + f
\]
where \(n\) is the integer part of \(x\) and \(f = \{x\}\) is the fractional part of \(x\) such that \(0 \leq f < 1\).

### Step 2: Substitute into the equation
Substituting \(x = n + f\) into the given equation:
\[
3(n + f) + f = 100
\]
Simplify it:
\[
3n + 3f + f = 100
\]
\[
3n + 4f = 100
\]

### Step 3: Solve for \(f\)
Since \(n\) is an integer and \(0 \leq f < 1\), we have:
\[
4f = 100 - 3n
\]
\[
f = \frac{100 - 3n}{4}
\]
For \(f\) to be a valid fractional part, \(0 \leq \frac{100 - 3n}{4} < 1\).

### Step 4: Determine \(n\)
The inequality \(0 \leq \frac{100 - 3n}{4} < 1\) implies:
\[
0 \leq 100 - 3n < 4
\]
\[
0 \leq 100 - 3n < 4
\]
\[
96 \leq 3n \leq 100
\]
\[
32 \leq n \leq \frac{100}{3} \approx 33.33
\]
Since \(n\) is an integer, \(n\) must be either 32 or 33.

### Step 5: Calculate \(f\) and check validity
**Case 1: \(n = 32\)**
\[
f = \frac{100 - 3(32)}{4} = \frac{100 - 96}{4} = \frac{4}{4} = 1
\]
\(f = 1\) is not valid since \(f\) must be less than 1.

**Case 2: \(n = 33\)**
\[
f = \frac{100 - 3(33)}{4} = \frac{100 - 99}{4} = \frac{1}{4} = 0.25
\]
Here, \(f = 0.25\) is valid.

### Step 6: Find \(4x\)
If \(n = 33\) and \(f = 0.25\), then:
\[
x = 33 + 0.25 = 33.25
\]
Thus:
\[
4x = 4 \times 33.25 = 133
\]

The final answer is \(\boxed{133}\).

Do you have any questions or need further details?

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Here are 5 related questions:
1. What is the fractional part of \( x \) in the equation \( x = n + f \) where \( n \) is the integer part?
2. How does the value of \( n \) affect the range of \( f \) in the equation \( 3x + \{x\} = 100 \)?
3. What is the significance of the inequality \( 0 \leq \frac{100 - 3n}{4} < 1 \) in solving the equation?
4. Can \( f = 1 \) ever be a valid fractional part in such problems? Why or why not?
5. How would the solution change if the equation were \( 2x + \{x\} = 50 \)?

**Tip:** Remember that the fractional part of a number is always non-negative and less than 1.

Explore how to solve equations involving fractional parts, specifically solving 3x + {x} = 100 to find 4x. This problem requires understanding equations and the concept of fractional parts. Suitable for high school students.

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