1. Let the speed of the turtle be \(t\) m/hr.
2. The speed of the snail is \(t - 32\) m/hr.
3. The snail travels for \(1\) hour and \(36\) minutes, which is \(1.6\) hours.
4. The turtle travels for \(52 / t\) hours.
5. Since both arrive at the same time:
   \[1.6 + \frac { 52} { t} = \frac { 52} { t - 32} \]
6. Solve the equation:
   \[1.6( t - 32) + 52 = 52\]
   \[1.6t - 51.2 + 52 = 52\]
   \[1.6t + 0.8 = 52\]
   \[1.6t = 51.2\]
   \[t = 32\]
7. The speed of the snail:
   \[t - 32 = 32 - 32 = 8\]

Supplemental Knowledge

When solving word problems involving distance, rate, and time, it's essential to understand the relationship between these three variables. The fundamental formula used is:
\[\text { Distance} = \text { Rate} \times \text { Time} \]
In this problem:

• We need to determine the speed (rate) of the snail.
• We know the snail's speed is 32 m/hr slower than the turtle's speed.
• Both travel the same distance but start at different times and arrive simultaneously.
  To solve such problems:

1. Define variables for unknown quantities.
2. Set up equations based on given information.
3. Solve the system of equations to find the unknowns.

Concepts to Actions

Imagine organizing a race between two friends; one on a bicycle and the other running. One begins first while their racer begins after some time but eventually catches up at the finish line. By knowing their speeds and start times, you can calculate exactly how fast each participant must move for them both to finish together at once.
This scenario shows how understanding rates and times is useful when planning events or logistics where timing and speed play a vital part.

Mastering word problems involving distance, rate, and time is a valuable skill that applies to various real-life situations. At UpStudy, we offer comprehensive resources to help you tackle these challenges with confidence.
For detailed guidance on solving algebraic word problems,explore UpStudy’s Algebra Equations Calculator today!