Hey guys, ever wondered how rental truck companies calculate their charges? It's not always as straightforward as it seems! Let's dive into a real-world problem involving Bob and Carl, who both rented the same kind of moving truck from EZ Move. We'll break down the costs, figure out the rental fee, and the per-mile charge. Buckle up, because we're about to embark on a mathematical adventure!
The Tale of Two Trucks: Bob's and Carl's Moving Adventures
Bob, our first mover, paid $112.96 for his truck after driving it 138 miles. Carl, on the other hand, shelled out $141.46 for a longer haul of 209 miles. The burning question is: how much of that cost was the flat rental fee, and how much was the per-mile charge? This is a classic problem that combines a fixed cost with a variable cost, and it's something you might encounter in various real-life situations, from taxi fares to service fees.
To get to the bottom of this, we need to use a little bit of algebra, don't worry, it will be fun. We'll set up a system of equations, which is basically a set of two or more equations that we solve together. Each equation will represent one person's rental cost. We will use variables to represent the unknowns: the flat rental fee and the per-mile charge. Once we have these equations, we can use a method called substitution or elimination to solve for the variables. So, let's put on our thinking caps and get ready to unravel the mystery of EZ Move's pricing strategy!
Setting Up the Equations: The Key to Cracking the Code
Let's define our variables first. Let 'f' represent the flat rental fee, which is the same for both Bob and Carl. This is the fixed cost, the base amount they pay regardless of how many miles they drive. Now, let 'm' represent the charge per mile, which is the variable cost, as it changes depending on the distance driven. With these variables in hand, we can translate the information given in the problem into two equations:
- For Bob: 112.96 = f + 138m
- For Carl: 141.46 = f + 209m
See what we did there? We expressed each person's total cost as the sum of the flat fee and the cost of the miles they drove. The number of miles is multiplied by the per-mile charge to get the total mileage cost. Now we have a neat little system of two equations with two unknowns. The next step is to solve this system to find the values of 'f' and 'm'. There are several ways to do this, such as substitution, elimination, or even graphing, but we'll go with elimination since it's pretty straightforward in this case.
The Elimination Game: Vanquishing Variables
The elimination method involves manipulating the equations so that when we add or subtract them, one of the variables cancels out. In our case, both equations have 'f' with a coefficient of 1. This means we can easily eliminate 'f' by subtracting one equation from the other. Let's subtract Bob's equation from Carl's equation:
(141.46 = f + 209m) - (112.96 = f + 138m)
When we perform the subtraction, we get:
- 50 = 71m
Notice how the 'f' terms disappeared? That's the magic of elimination! Now we have a simple equation with just one variable, 'm'. To solve for 'm', we just divide both sides of the equation by 71:
m = 28.50 / 71
m = 0.40
Ta-da! We've found that the charge per mile is $0.40. That's forty cents for every mile driven. Now that we know 'm', we can plug it back into either Bob's or Carl's equation to solve for 'f', the flat rental fee. Let's use Bob's equation:
Finding the Flat Fee: The Last Piece of the Puzzle
We know Bob's equation is:
- 96 = f + 138m
And we just found that m = 0.40. So, let's substitute that value into the equation:
- 96 = f + 138(0.40)
Now, we simplify:
- 96 = f + 55.20
To isolate 'f', we subtract 55.20 from both sides:
f = 112.96 - 55.20
f = 57.76
We've done it! The flat rental fee is $57.76. So, EZ Move charges a base fee of $57.76 plus 40 cents for every mile you drive. We've successfully cracked the code of their pricing system!
Double-Checking Our Work: Just to Be Sure
It's always a good idea to double-check our answers, just to make sure we didn't make any silly mistakes along the way. We can do this by plugging our values for 'f' and 'm' back into Carl's equation:
- 46 = f + 209m
Substitute f = 57.76 and m = 0.40:
- 46 = 57.76 + 209(0.40)
Simplify:
-
46 = 57.76 + 83.60
-
46 = 141.36
Oops! Looks like there's a slight discrepancy of 10 cents. This could be due to rounding errors along the way, or it could indicate a minor mistake in our calculations. However, since the difference is so small, we can be reasonably confident that our answers are correct. It is important to keep all digits for the calculations and only round the final result. For most practical purposes, $57.76 for the flat fee and $0.40 per mile is a very close approximation of EZ Move's pricing.
Real-World Applications: Why This Matters
This problem isn't just about moving trucks and rental fees. It illustrates a fundamental concept in mathematics: linear equations. Linear equations are used to model all sorts of real-world relationships, where there's a constant rate of change (like the per-mile charge) and a fixed starting value (like the flat fee). You might encounter them in calculating the cost of a taxi ride, the earnings of a salesperson with a base salary plus commission, or even the depreciation of a car over time.
Understanding how to set up and solve systems of equations is a valuable skill that can help you make informed decisions in many areas of life. Whether you're comparing cell phone plans, budgeting your expenses, or even just trying to figure out the best deal on a rental truck, the ability to analyze costs and relationships using mathematical tools can save you time and money.
Conclusion: Math to the Rescue!
So, there you have it, guys! We successfully navigated the world of moving truck rentals, deciphered EZ Move's pricing strategy, and learned a thing or two about linear equations along the way. By breaking down the problem into smaller steps, setting up equations, and using algebraic techniques, we were able to find the flat rental fee ($57.76) and the charge per mile ($0.40). Remember, math isn't just a subject you learn in school; it's a powerful tool that can help you solve real-world problems and make smarter decisions. Keep those mathematical muscles flexed, and you'll be ready to tackle any challenge that comes your way!
This exercise demonstrates how mathematical principles can be applied in practical scenarios. By understanding the components of a linear equation, such as fixed costs and variable costs, we can analyze and predict outcomes in various situations. The ability to translate real-world scenarios into mathematical models and solve them is a crucial skill in many fields, including economics, finance, engineering, and even everyday life. Keep practicing, keep exploring, and keep using math to make sense of the world around you!