Problem 19 - Beginning and Intermediate Algebra

Question

The top three apple growers in the world - China, the United States, and Turkey - grew a total of about 74 billion lb of apples in a recent year. China produced 44 billion lb more than the combined production of the United States and Turkey. The United States produced twice as many pounds of apples as Turkey. (Source: U.S. Apple Association) Find the number of pounds of apples produced by each country. (THE IMAGES CANNOT COPY)

Step-by-Step Solution

Answer

The apple production in each country is as follows: China produced 74 billion lb, the United States produced 20 billion lb, and Turkey produced 10 billion lb of apples.

Step 1: Write the system of linear equations

Based on the given information, we can write the following system of linear equations: 1. \(x + y + z = 74\) 2. \(x = y + z + 44\) 3. \(y = 2z\)

Step 2: Substitute equation 3 into equations 1 and 2

Substitute the value of y (2z) from equation 3 into equations 1 and 2: 1. \(x + 2z + z = 74\) 2. \(x = 2z + z + 44\)

Step 3: Simplify equations

Simplify the equations after substituting the value of y: 1. \(x + 3z = 74\) (This will be our new equation 1) 2. \(x = 3z + 44\) (This will be our new equation 2)

Step 4: Solve the system of linear equations

We can solve the equations 1 and 2 by the substitution or elimination method. Let's use the elimination method here. We can see that both equations have x with the same coefficient, so we don't need to modify any equations: New equation 1: \(x + 3z = 74\) New equation 2: \(x = 3z + 44\) Subtract equation 2 from equation 1: \((x + 3z) - x = 74 - (3z + 44)\) Simplifying, we get: \(3z - 3z = 30\) This means \(z = 10\) billion lb (Turkey's apple production).

Step 5: Find the apple production of the United States and China

Now we can use the value of z to find the values of x and y: From equation 3: \(y = 2z\) \(y = 2 * 10\) \(y = 20\) billion lb (United States' apple production) From equation 2: \(x = 3z + 44\) \(x = 3 * 10 + 44\) \(x = 30 + 44\) \(x = 74\) billion lb (China's apple production)

Step 6: Present the answer

The apple production in each country is as follows: 1. China: 74 billion lb 2. United States: 20 billion lb 3. Turkey: 10 billion lb

Key Concepts

Apple Production

Apple production is a significant agricultural activity worldwide. In this context, understanding how countries like China, the United States, and Turkey contribute to global apple production is essential. The challenge is to determine how many pounds of apples each of these countries produced, using a mathematical framework. Knowing the total production allows us to understand the scale and distribution of apple growing in these leading countries. The information suggests that in a particular year, these top apple growers collectively produced about 74 billion pounds of apples. China, being the largest producer, contributed significantly more than the United States and Turkey. Studying apple production data helps us appreciate the importance of mathematics in real-world applications such as agriculture, showcasing how figures and equations can provide insights into production dynamics.

Mathematical Modeling

Mathematical modeling involves creating representations of real-world systems using mathematical language and concepts. In this problem, we translate the apple production scenario into a system of linear equations. This technique allows us to describe relationships and constraints mathematically. For instance, we use mathematical equations to represent:

• The total production of apples as 74 billion pounds
• China’s production as being 44 billion pounds more than the combined output of the US and Turkey
• The US producing twice as many apples as Turkey

By forming these equations, mathematical modeling helps us solve complex problems in a structured manner, translating verbal information into mathematical language that is easier to manipulate and understand.

Problem Solving

Problem solving in mathematics is a systematic process where we start with a problem statement and apply logical steps to reach a solution. This exercise is a great example of problem solving through the use of linear equations. The process begins with identifying the known and unknown variables — in this case, each country's apple production. We then form equations based on the relationships described in the problem. Using techniques such as substitution, we simplify and solve the equations, leading us to the solution. This structured approach ensures that each calculation is based on sound reasoning, eventually revealing the apple production figures for each country. Problem solving not only boosts numerical acuity but sharpens analytical thinking and reasoning skills.

Substitution Method

The substitution method is a popular technique for solving a system of linear equations. It involves solving one equation for one variable and then substituting that expression into another equation. In this apple production problem, we use the substitution method effectively:

• First, we solve one of the equations to express one variable in terms of another.
• In this case, we use the equation relating the US and Turkey's production to express Turkey's production in terms of the US.
• Substitute this expression into the other equations to simplify and solve.

By substituting the equations, we reduce complexity, ultimately finding the amounts of apples produced by each country. This method is especially useful when dealing with larger systems or when one variable can be easily expressed in terms of another. It's a critical tool in mathematical problem solving, helping break down problems into simpler, manageable parts.