Problem 74 - 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 72 freight cars contain barley, what is the total weight the train is hauling?

Step-by-Step Solution

Answer

The total amount of weight the train is hauling is 15897.6 tons.

Step 1: Determine the number of freight cars carrying corn

Given that there are 150 freight cars in total and 72 of these carry barley, we need to calculate the rest of the cars carrying corn. This is done by subtracting the number of barley cars from the total number. The calculation is as follows: \( n = 150 - 72 = 78 \). So, there are 78 freight cars carrying corn.

Step 2: Calculate the total amount of barley

Next, the total weight of barley is calculated by multiplying the number of cars that carry barley by the weight per car. Since each car carries 97.3 tons of barley, and there are 72 cars, the total weight of barley is: \(Weight_{barley} = 97.3 \times 72 = 7005.6 \text{ tons}\)

Step 3: Calculate the total amount of corn

Similarly, the total weight of corn is calculated by multiplying the number of cars that carry corn (n) by the weight per car. Each car carries 114 tons of corn, and we have 78 cars, the total weight of corn is: \( Weight_{corn} = 114 \times 78 = 8892 \text{ tons} \)

Step 4: Calculate total weight hauled by the train

Lastly, to get the total weight hauled by the train, add up the total weights of barley and corn. The calculation is as follows: \( Total Weight = Weight_{barley} + Weight_{corn} = 7005.6 + 8892 = 15897.6 \text{ tons}\)

Key Concepts

Understanding Linear Equations in Real-World Problems

Linear equations are equations where the highest exponent of the variable is one. In the freight train problem, we see an application of simple arithmetic combined in a single linear equation to solve the problem of determining unknown quantities. Here, the variable represents the number of freight cars carrying corn.

• We know the total number of freight cars is 150.
• We are given the number for barley-carrying cars as 72.

By forming a linear equation: \[ n = 150 - 72 \] This is a straightforward equation with just one variable \( n \), the unknown number of cars carrying corn. Solving it involves simple subtraction, illustrating the basic principle of linear equations: balancing all elements on either side of the equals sign. This process is crucial in dividing complex real-world scenarios like logistics problems into manageable mathematical tasks.

Weight Calculation of Freight Cars

Weight calculation is the process of determining the total weight based on known quantities and weights per item. In the context of the freight train problem, weight calculation involves understanding the total cargo capacity for the cars carrying either barley or corn.

• Each barley-carrying freight car holds 97.3 tons.
• Each corn-carrying freight car holds 114 tons.

Using these numbers:Barley:- Multiply the number of barley cars (72) by the weight each car carries (97.3 tons): \[ Weight_{barley} = 97.3 \times 72 = 7005.6 \text{ tons} \]Corn:- Similarly, multiply the number of corn cars (78) by the weight each car carries (114 tons): \[ Weight_{corn} = 114 \times 78 = 8892 \text{ tons} \]Summing these results gives us the total weight hauled by the train. This method is excellent for practicing multiplication and addition while understanding weight distribution in logistical operations.

Introducing Algebraic Expressions

Algebraic expressions simplify and abstract real-world problems using variables and operations. In this exercise, we encounter algebraic expressions in calculating the total weight.An algebraic expression does not always have to be complex. Here, the expressions used are: - Calculating the number of corn-carrying cars: \( n = 150 - 72 \)- Determining total barley and corn weight: \( Weight_{barley} = 97.3 \times 72 \) \( Weight_{corn} = 114 \times 78 \)These expressions represent real quantities and operations, allowing us to understand the problem better. Algebra is not just about solving for \( x \) or \( n \); it’s also about transforming various problem components into mathematical forms to find practical solutions. Embracing these fundamentals helps us work through everyday challenges, like optimizing cargo loads in freight transportation.