Problem 73 - Algebra 1

Question

FREIGHT TRAINS A train with 150 freight cars is used to haul two types of grain. Each freight car can haul 97.3 tons of barley or 114 tons of corn. Let \(n\) represent the number of freight cars containing corn. If 90 freight cars contain corn, what is the total weight the train is hauling?

Step-by-Step Solution

Answer

The total weight the train is hauling is 16098 tons.

Step 1: Determine the number of corn freight cars

According to the problem, there are 90 freight cars containing corn.

Step 2: Weight of corn freight cars

Knowing that each freight car of corn hauls 114 tons, the total weight hauled by the corn freight cars is found by multiplying the number of cars by the weight each car can carry: \(90 \times 114 = 10260\) tons.

Step 3: Determine the number of barley freight cars

The train is made up of 150 freight cars in total. Subtracting the 90 freight cars that carry corn leaves 60 freight cars to carry barley.

Step 4: Weight of barley freight cars

Each freight car of barley hauls 97.3 tons so the total weight hauled by the barley freight cars is \(60 \times 97.3 = 5838\) tons.

Step 5: Total weight hauled by the train

The total weight hauled by the train is found by adding up the weights carried by the corn and barley freight cars: \(10260 + 5838 = 16098\) tons.

Key Concepts

Linear Equations

Linear equations are essential tools in algebra that allow us to represent and solve real-world problems using mathematical expressions. In the context of freight car math problems, linear equations can help calculate the total weight hauled by a train. For instance, we have two variables here: one for the number of freight cars carrying corn and another for the freight cars carrying barley.
To solve our freight problem, we create an equation where the total number of cars (both corn and barley) equals the sum of individual car types:

• Let \( n \) represent the number of corn freight cars.
• Total freight cars: 150, so barley cars will be \( 150 - n \).

Using these expressions, we can determine other necessary values, such as weights carried by each type of car. Thus, linear equations provide a way to systematically approach and solve freight car problems by breaking them down into manageable parts.

Weight Calculation

Weight calculation is key to understanding the burden each freight car carries and ultimately determining the overall haulage capacity of the train. In our exercise, two types of cargo are involved: corn and barley, with different weights allocated to each tonnage.
Each corn freight car can haul 114 tons. If there are 90 corn freight cars, the calculation becomes:

• Weight of corn cars: \( 90 \times 114 = 10260 \text{ tons} \)

Similarly, for barley freight cars, each can carry 97.3 tons. Given our train has 60 barley cars, we determine:

• Weight of barley cars: \( 60 \times 97.3 = 5838 \text{ tons} \)

Weight calculation helps to establish a more complete picture of the train's load, highlighting the importance of accurate mathematical representation for practical applications.

Problem-Solving Steps

Problem-solving steps offer a structured framework for tackling math problems systematically. Each exercise must be approached with clear procedures to ensure solutions are accurate and logical.
Here's a breakdown of the problem-solving steps used in the exercise:

• Step 1: Identify given information and assign variables. Here, 90 freight cars carry corn and the total car number is 150.
• Step 2: Calculate the total weight of corn cars by multiplying the number of cars by the weight per car.
• Step 3: Determine the remaining freight cars for barley. Subtract from the total to find that 60 cars carry barley.
• Step 4: Compute the total weight of barley cars similarly to the corn cars.
• Step 5: Add the weights of both types of freight cars to find the total haul.

Breaking down complex problems into smaller, actionable steps makes it easier to comprehend and solve. This structured methodology ensures no component is overlooked, ultimately leading to correct solutions.