Let's break down this problem and solve it step-by-step.

Problem Analysis

1. We know the family covers a total distance to reach the gate of Magic Kingdom.
2. The family:
   • Walks $\frac{1}{4}$ mile.
   • Takes a tram for $\frac{1}{3}$ of the total distance.
   • Takes a monorail for $\frac{1}{2}$ of the total distance.

Let the total distance be $D$ miles.

Solution Steps

1. The distance walked is explicitly given as $\frac{1}{4}$ mile.
2. The tram distance is $\frac{1}{3}$ of the total distance $D$, which can be expressed as $\frac{1}{3}D$.
3. The monorail distance is $\frac{1}{2}$ of the total distance $D$, which can be expressed as $\frac{1}{2}D$.

According to the problem, the sum of these distances should equal the total distance $D$. Therefore, we can set up the equation:

$\frac{1}{4} + \frac{1}{3}D + \frac{1}{2}D = D$

Solving the Equation

1. Move all terms involving $D$ to one side of the equation:

   $\frac{1}{4} = D - \frac{1}{3}D - \frac{1}{2}D$

2. Factor out $D$ on the right side:

   $\frac{1}{4} = D \left(1 - \frac{1}{3} - \frac{1}{2}\right)$

3. Calculate $1 - \frac{1}{3} - \frac{1}{2}$:

   • Find a common denominator (6) for $\frac{1}{3}$ and $\frac{1}{2}$: $\frac{1}{3} = \frac{2}{6}, \quad \frac{1}{2} = \frac{3}{6}$
   • Subtract these values from 1: $1 - \frac{2}{6} - \frac{3}{6} = 1 - \frac{5}{6} = \frac{1}{6}$

   So, we have:

   $\frac{1}{4} = D \cdot \frac{1}{6}$

4. Solve for $D$ by multiplying both sides by 6:

   $D = \frac{1}{4} \times 6 = \frac{6}{4} = 1.5 \text{ miles}$

Conclusion

The total distance from their car to the gate of Magic Kingdom is 1.5 miles.

Would you like more details or have any further questions?

Here are some related questions for further practice:

1. What fraction of the total distance did the family cover by tram?
2. How much of the distance was covered by walking and by tram combined?
3. If they had walked the entire way, how long would it have taken if they walked at 3 miles per hour?
4. How much time would they spend on each segment if the tram and monorail have fixed speeds?
5. How would the equation change if the distances for tram and monorail were different fractions?

Tip: Always verify your total fractions sum up to 1 when breaking down problems like these!