Calculating Commute Distance: A Fun Challenge for Fire Academy Students

If Greg drives home from fire academy training at 58 mph and returns at 53 mph, taking an additional 30 minutes, how long is his commute?

• A. 261 miles
• B. 292.5 miles
• C. 301.7 miles
• D. 307.4 miles

Answer: (D)

To determine the length of Greg's commute, we can use the relationship between distance, speed, and time. The total distance he travels is the same for both trips, so we denote the distance as \(d\). First, we can express the time taken for each trip. The time for the commute to the academy at 58 mph can be calculated as \( \frac{d}{58} \), and the return trip at 53 mph can be calculated as \( \frac{d}{53} \). The problem states that the return trip takes an additional 30 minutes (or 0.5 hours) compared to the trip to the academy. Therefore, we can set up the equation: \[ \frac{d}{53} = \frac{d}{58} + 0.5 \] To eliminate the fractions, we can first find a common denominator, which will facilitate solving for \(d\). The lowest common multiple of 53 and 58 is 3074, so we can multiply through by this to clear the denominators: \[ 3074 \left(\frac{d}{53}\right) = 3074\left(\frac{d}{58} + 0.5

Ever find yourself stuck trying to unravel a math problem that seems more like a riddle? If you're prepping for the firefighter exam, you'll encounter questions that blend real-life situations with math challenges — and this one is a prime example! Greg's daily commute is a tale of speeds and times that cleverly nudges you to think critically about distance. So, let’s roll up our sleeves and tackle it!

Here’s the gist: Greg drives home from fire academy training at 58 mph and returns at 53 mph. There's a twist — the return journey takes him an extra 30 minutes. Sounds simple, right? But let's add some numbers to the mix!

Breaking Down the Problem: Distance, Speed, and Time

To figure out how long Greg's commute is, we're going to rely on the age-old relationship: distance = speed × time. But first, let’s denote his road trip distance as (d). This distance stays the same whether he’s on his way to the academy or cruising back home.

Now, how do we calculate the time for each trip?

• For the trip to the academy, the formula is:

[

\text{Time} = \frac{d}{58}

]

• For the return trip, it’s:

[

\text{Time} = \frac{d}{53}

]

Here's where we add the twist: the return trip takes an additional 30 minutes or 0.5 hours longer than the trip to the academy. So, we come up with the equation:

[

\frac{d}{53} = \frac{d}{58} + 0.5

]

Sounds complicated? Not really! Solving it just requires a little bit of algebra.

Cleaning Up That Equation

To simplify matters, let’s eliminate those pesky fractions. We can find a common denominator — and in this case, it’s 3074 (the lowest common multiple of 53 and 58). Multiplying everything by 3074 helps clear the fractions, leading us to:

[

3074 \left(\frac{d}{53}\right) = 3074\left(\frac{d}{58} + 0.5\right)

]

Now isn't that much cleaner? Without getting too lost in the math weeds, we can solve for (d) pretty straightforwardly.

Finding the Distance

With a little algebra, you will eventually arrive at the distance that Greg commutes. You would find out that the total distance is 307.4 miles. Yup, you read that right! It’s not just about moving from one spot to another; it's about understanding the journey — both literally and figuratively.

Why This Matters for Firefighters

Now, you might be wondering, "Why should I care about this scenario?" Well, it’s about more than just numbers. Understanding these basic principles sharpens your problem-solving skills, which, believe it or not, holds a lot of weight in firefighting scenarios. Whether you're estimating response times or calculating hose lengths at a fire scene, these math skills become incredibly valuable.

So, next time you find yourself driving home after a grueling day at the academy (or simply trying to figure out the quickest route to your favorite pizza joint), remember Greg’s tale — and those speeds and times. They’re part of the essential toolkit that will prepare you for the challenges ahead.

So, next train of thought: Are you ready? Stay sharp, practice your math, and keep that firefighter spirit strong! We all have a journey to make, and being prepared can make all the difference in saving lives and making a lasting impact.