To find a number such that $\frac{1}{4}$ of the number is 50 less than $\frac{2}{3}$ of the number, let's represent the number by $x$.

The problem can be translated into the equation:

$\frac{1}{4}x = \frac{2}{3}x - 50$

Now, let's solve this equation step by step.

1. Eliminate the fractions: Multiply every term by 12 (the least common multiple of 4 and 3) to eliminate the fractions.

$12 \times \frac{1}{4}x = 12 \times \left(\frac{2}{3}x - 50\right)$

This simplifies to:

$3x = 8x - 600$

2. Isolate $x$: Subtract $8x$ from both sides.

$3x - 8x = -600$

This simplifies to:

$-5x = -600$

3. Solve for $x$: Divide both sides by $-5$.

$x = \frac{-600}{-5} = 120$

Thus, the number is $\boxed{120}$.

Would you like more details on this solution, or do you have any questions?

---

Here are 5 related questions for further practice:

1. Find a number such that $\frac{1}{3}$ of the number is 40 less than $\frac{5}{6}$ of the number.
2. Find a number such that $\frac{1}{5}$ of the number is 30 more than $\frac{1}{2}$ of the number.
3. If $\frac{2}{5}$ of a number is equal to 60, what is the number?
4. Solve for a number where $\frac{3}{4}$ of it is equal to $\frac{2}{3}$ of 90.
5. Find a number such that $\frac{1}{6}$ of the number is 20 less than $\frac{3}{4}$ of the number.

Tip: When solving equations with fractions, multiplying through by the least common multiple of the denominators can simplify the process.