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.