Problem 92 - Algebra and Trigonometry

Question

The top destinations for U.S. college students studying abroad are the United Kingdom, Italy, and Spain. The number of students studying in the U.K. exceeds the number studying in Spain by 10 thousand. The number of students studying in Italy exceeds the number studying in Spain by 2 thousand. Combined, 93 thousand U.S. students study in the U.K., Italy, and Spain. Determine the number of students studying abroad, in thousands, in the U.K., Italy, and Spain. (Source: Institute of International Education) (Section 1.3, Example 1)

Step-by-Step Solution

Answer

There are 37,000 U.S. students studying in the UK, 29,000 U.S. students studying in Italy, and 27,000 U.S. students studying in Spain.

Step 1: Formulate the equations

From the problem, denote the number of students studying in UK by \( U \), in Italy by \( I \), and in Spain by \( S \). Then, the statements 'the number of students studying in the UK exceeds the number studying in Spain by 10 thousand' and 'the number of students studying in Italy exceeds the number studying in Spain by 2 thousand' can be written as \( U = S + 10 \) and \( I = S + 2 \), respectively. The statement that 'combined, 93 thousand U.S. students study in the UK, Italy, and Spain' can be written as \( U + I + S = 93 \). This gives three equations, in three unknowns.

Step 2: Substitution

Because \( U \) and \( I \) are given in terms of \( S \), the equations can be reformulated to: \( S + 10 + S + 2 + S = 93 \). This simplifies to \( 3S + 12 = 93 \).

Step 3: Solving for S

Subtract 12 from both sides to get \( 3S = 81 \) and then divide by 3 to get \( S = 27 \). This means that there are 27,000 U.S. students studying in Spain.

Step 4: Solving for U and I

Substitute \( S = 27 \) into \( U = S + 10 \), to get \( U = 37 \). This means that there are 37,000 U.S. students studying in the UK. Similarly, substitute \( S = 27 \) into \( I = S + 2 \), to get \( I = 29 \). This means that there are 29,000 U.S. students studying in Italy.

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are essentially mathematical phrases that involve numbers, variables, and operations. They are used extensively in solving various mathematical problems, including systems of equations. In the given exercise, we use algebraic expressions to represent the number of students in different countries.

Each equation involves variables (in this case, the number of students in different countries, represented by letters like \( U \), \( I \), and \( S \)) and operations (such as addition and equality). For example, the expression \( U = S + 10 \) represents the relationship between the number of students in the UK and in Spain.

Being comfortable with algebraic expressions allows you to set up and solve equations that represent real-world situations, as in this exercise.

Linear Equations Explained

Linear equations form the backbone for solving systems of equations, like the ones present in this exercise. A linear equation is one where the highest power of the variable is 1, making it "linear" or straight when graphed.

The given problem provides us with a series of linear equations such as \( U = S + 10 \), \( I = S + 2 \), and \( U + I + S = 93 \). These equations help us express relationships between variables concisely and allow for straightforward arithmetic operations to be performed on them.

Solving linear equations is fundamental in courses focused on algebra, as it helps in understanding more complex concepts later on.

The Role of International Education Statistics

International education statistics provide valuable insight into global educational trends. In this exercise, data about the number of students studying abroad helps form the context for the equations.

These statistics, gathered by organizations like the Institute of International Education, illustrate significant trends in student mobility and preferences for study destinations. Such data can assist educators and policymakers in making informed decisions about international education programs.

Key Problem-Solving Steps

Problem-solving in mathematics often follows a systematic approach, as demonstrated in the exercise. It involves:

• Formulating the problem in terms of equations.
• Using algebraic manipulations like substitution and simplification.
• Solving the equations to find unknown values.
• Interpreting the results in the context of the problem.

Following these steps helps improve both accuracy and understanding, making it easier to tackle more complex problems in the future. Solving these equations not only provides answers but also helps develop critical thinking and analytical skills.