If the speed of Xtext{ meters per second} is equivalent to the speed of Ytext{ kilometers per hour}, what is...

1. Translate the problem requirements: We need to convert \(\mathrm{X}\) meters per second to \(\mathrm{Y}\) kilometers per hour, expressing \(\mathrm{Y}\) in terms of \(\mathrm{X}\) using the given conversion factor that \(\mathrm{1 \text{ kilometer} = 1000 \text{ meters}}\)
2. Set up the unit conversion relationship: Identify what needs to change - meters to kilometers (divide by \(\mathrm{1000}\)) and seconds to hours (multiply by \(\mathrm{3600}\))
3. Apply conversion factors systematically: Convert \(\mathrm{X}\) meters/second by first changing meters to kilometers, then changing per second to per hour
4. Simplify the resulting expression: Combine the conversion factors to get the final relationship \(\mathrm{Y}\) in terms of \(\mathrm{X}\)

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're asked to do in plain English. We have a speed measured in "\(\mathrm{X}\) meters per second" and we want to find the equivalent speed "\(\mathrm{Y}\) kilometers per hour".

Think of it this way: if a car is moving at \(\mathrm{X}\) meters every second, how many kilometers would it cover in one hour? That's what \(\mathrm{Y}\) represents.

We're given that \(\mathrm{1 \text{ kilometer} = 1000 \text{ meters}}\), which is our conversion factor between distance units.

Process Skill: TRANSLATE - Converting the problem language into a clear mathematical relationship we need to establish

2. Set up the unit conversion relationship

To go from "meters per second" to "kilometers per hour", we need to change two things:

Distance units: We need to convert meters to kilometers
- Since \(\mathrm{1 \text{ kilometer} = 1000 \text{ meters}}\), this means \(\mathrm{1 \text{ meter} = \frac{1}{1000} \text{ kilometers}}\)
- So when we have \(\mathrm{X}\) meters, that equals \(\mathrm{\frac{X}{1000}}\) kilometers

Time units: We need to convert "per second" to "per hour"
- There are \(\mathrm{60}\) seconds in \(\mathrm{1}\) minute
- There are \(\mathrm{60}\) minutes in \(\mathrm{1}\) hour
- So there are \(\mathrm{60 \times 60 = 3600}\) seconds in \(\mathrm{1}\) hour
- This means something happening "per second" happens \(\mathrm{3600}\) times "per hour"

3. Apply conversion factors systematically

Now let's put these conversions together step by step:

Starting with: \(\mathrm{X}\) meters per second

Step 1: Convert meters to kilometers
- \(\mathrm{X}\) meters per second = \(\mathrm{\frac{X}{1000}}\) kilometers per second

Step 2: Convert "per second" to "per hour"
- If we travel \(\mathrm{\frac{X}{1000}}\) kilometers each second
- In \(\mathrm{1}\) hour (\(\mathrm{3600}\) seconds), we travel: \(\mathrm{\frac{X}{1000} \times 3600}\) kilometers
- So our speed is \(\mathrm{\frac{X}{1000} \times 3600}\) kilometers per hour

Therefore: \(\mathrm{Y = \frac{X}{1000} \times 3600}\)

4. Simplify the resulting expression

Let's simplify this expression:

\(\mathrm{Y = \frac{X}{1000} \times 3600}\)
\(\mathrm{Y = \frac{X \times 3600}{1000}}\)
\(\mathrm{Y = \frac{3600X}{1000}}\)

Now we can simplify this fraction by dividing both numerator and denominator by their greatest common factor.

\(\mathrm{3600 ÷ 200 = 18}\)
\(\mathrm{1000 ÷ 200 = 5}\)

So: \(\mathrm{Y = \frac{18X}{5}}\)

Using mathematical notation: \(\mathrm{Y = \frac{18x}{5}}\)

Final Answer

\(\mathrm{Y = \frac{18x}{5}}\)

Looking at our answer choices, this matches choice C: \(\mathrm{\frac{18x}{5}}\).

To verify: If \(\mathrm{x = 5}\) meters/second, then \(\mathrm{Y = \frac{18 \times 5}{5} = 18}\) km/hr. We can check this makes sense: \(\mathrm{5}\) meters/second \(\mathrm{\times 3600}\) seconds/hour = \(\mathrm{18000}\) meters/hour = \(\mathrm{18}\) km/hour ✓

Answer: C

Common Faltering Points

Errors while devising the approach

Faltering Point 1: Misunderstanding the direction of conversion
Students may confuse which direction the conversion should go. The question asks for \(\mathrm{Y}\) (kilometers per hour) in terms of \(\mathrm{X}\) (meters per second), but students might incorrectly try to convert from kilometers per hour to meters per second instead. This leads them to flip the conversion factors and get the wrong relationship.

Faltering Point 2: Forgetting that both distance and time units need conversion
Many students focus only on converting distance units (meters to kilometers) and forget that the time component also needs conversion (per second to per hour). They might only apply the \(\mathrm{1000}\)-meter conversion factor and miss the \(\mathrm{3600}\)-second conversion, leading to an incomplete setup.

Faltering Point 3: Setting up conversion factors in the wrong direction
Students often get confused about whether to multiply or divide by conversion factors. For example, they might think since \(\mathrm{1 \text{ km} = 1000 \text{ m}}\), they should multiply by \(\mathrm{1000}\) when converting meters to kilometers, when they actually need to divide by \(\mathrm{1000}\).

Errors while executing the approach

Faltering Point 1: Arithmetic errors when calculating seconds in an hour
When converting time units, students need to calculate \(\mathrm{60 \times 60 = 3600}\) seconds in one hour. Common errors include calculating this as \(\mathrm{120}\) (\(\mathrm{60 + 60}\)) or \(\mathrm{360}\) (forgetting one zero), leading to incorrect final expressions.

Faltering Point 2: Mistakes in fraction simplification
When simplifying \(\mathrm{\frac{3600X}{1000}}\), students may make errors finding the greatest common factor or in the division process. They might incorrectly simplify to get ratios like \(\mathrm{\frac{36X}{10}}\) or \(\mathrm{\frac{360X}{100}}\), missing the final step to reach \(\mathrm{\frac{18X}{5}}\).

Faltering Point 3: Incorrect handling of the variable X
Students might treat \(\mathrm{X}\) as a specific number rather than a variable, or make errors in maintaining \(\mathrm{X}\) throughout their calculations, sometimes dropping it entirely or placing it in the wrong position in their final expression.

Errors while selecting the answer

Faltering Point 1: Choosing an answer that represents the inverse conversion
Students who mixed up the conversion direction might arrive at \(\mathrm{\frac{5X}{18}}\) (choice A), which is actually the conversion factor for going from kilometers per hour to meters per second, the opposite of what's asked.

Faltering Point 2: Not recognizing equivalent expressions
Some students might arrive at mathematically correct expressions like \(\mathrm{\frac{18 \times X}{5}}\) but fail to recognize this matches choice C: \(\mathrm{\frac{18x}{5}}\), thinking their answer doesn't appear among the choices and selecting a different option.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a convenient value for X
Let's choose \(\mathrm{X = 5}\) meters per second (selected because it will give us clean numbers when we apply the conversion factors)

Step 2: Convert meters to kilometers
Since \(\mathrm{1 \text{ kilometer} = 1000 \text{ meters}}\):
\(\mathrm{5}\) meters per second = \(\mathrm{\frac{5}{1000}}\) kilometers per second = \(\mathrm{0.005}\) kilometers per second

Step 3: Convert seconds to hours
Since \(\mathrm{1 \text{ hour} = 3600 \text{ seconds}}\):
\(\mathrm{0.005}\) kilometers per second = \(\mathrm{0.005 \times 3600}\) kilometers per hour = \(\mathrm{18}\) kilometers per hour

Step 4: Establish the relationship
We found that when \(\mathrm{X = 5}\), \(\mathrm{Y = 18}\)
So \(\mathrm{Y = 18}\) and \(\mathrm{X = 5}\), which means \(\mathrm{Y = \frac{18}{5} \times X}\)

Step 5: Verify with answer choices
\(\mathrm{Y = \frac{18}{5}X}\) matches answer choice C: \(\mathrm{\frac{18x}{5}}\)

Why this smart number works: \(\mathrm{X = 5}\) was chosen strategically because \(\mathrm{5 \times 3600 ÷ 1000 = 18}\), giving us clean integer results that make the pattern easy to identify in the answer choices.