Problem 47 - Algebra 1: Concepts and Skills

Question

A train with 150 freight cars is fully loaded with two types of grain. Each freight car can haul 90.25 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 that the train is hauling is 15675 tons.

Step 1: Identify the Numbers of Cars for Each Type of Grain

Firstly, determine the number of freight cars that contain each type of grain. According to the problem, there are 90 freight cars containing corn, represented by n. Since there are a total of 150 freight cars, the number of cars that contain barley is \(150 - 90 = 60\).

Step 2: Calculate the Weight of Corn

Now, find the total weight of corn. As each freight car can haul 114 tons of corn and there are 90 cars with corn, the total weight is \(90 \times 114 = 10260\) tons.

Step 3: Calculate the Weight of Barley

Then, calculate the total weight of barley. Each car can haul 90.25 tons of barley and there are 60 cars with barley, hence the total weight is \(60 \times 90.25 = 5415\) tons.

Step 4: Calculate the Total Weight of the Train

Finally, sum up the total weight of corn and barley to find the total weight of the train. \(10260 + 5415 = 15675\) tons.

Key Concepts

Linear Equations

Linear equations are fundamental in algebra and are often used to solve real-world problems, like the one involving the train and freight cars. A linear equation is simply an equation with variables raised only to the power of one, creating a straight line when graphed. In the given problem, we are dealing with the number of freight cars carrying different types of grain.

Let's consider the equation derived from the problem: if the variable \( n \) represents the number of corn freight cars, then the equation can be set up as:

• Total number of cars equation: \( n + ext{barley cars} = 150 \)

This leads us to a simple calculation: since 90 cars contain corn, we know the number of barley cars by solving this linear equation:

• \( n = 90 \)
• Barley cars = \( 150 - n = 60 \)

Understanding how to construct and solve linear equations is crucial in breaking down complex word problems into manageable parts.

Arithmetic Operations

Arithmetic operations are the basic calculations used throughout mathematics, and they form the core of solving word problems. In our train problem, multiplications and additions are the primary operations used.

The problem requires us to calculate the weights of corn and barley separately before summing them.

• Calculating corn weight: We multiply the number of corn cars by the weight each can carry, \( 90 \times 114 = 10260 \) tons.
• Calculating barley weight: Similarly, for barley cars, we use \( 60 \times 90.25 = 5415 \) tons.

Finally, adding these weights gives the total weight of the train:

• Total weight: \( 10260 + 5415 = 15675 \) tons.

By mastering basic arithmetic operations, students can confidently tackle a variety of algebra word problems. Consistent practice in multiplication and addition develops strong problem-solving skills.

Problem Solving Steps

Solving algebra word problems effectively requires a methodical approach. This involves breaking down the problem into smaller, manageable steps. Using the train example, we can see this approach clearly illustrated:

• **Step 1:** Identify the variables. What do we know? There are 90 corn cars among 150 total cars.
• **Step 2:** Determine the number of barley cars by subtracting the number of corn cars from the total. This gives us 60 barley cars.
• **Step 3:** Compute the total weight for each type of grain. Multiply the number of cars by their respective weights they can carry.
• **Step 4:** Sum the weights of corn and barley to find the total train weight.

This structured approach fosters a deep understanding of how to dissect complex word problems. By clearly identifying each step, students gain the ability to systematically solve problems across various contexts. Remember to always double-check each step to ensure accuracy in the final solution.