Problem 57 - Algebra A Combined Function

Question

In the 2008 Summer Olympics, Korea won more gold medals than Germany, which won more gold medals than Australia. If the numbers of gold medals won by these three countries are three consecutive integers whose sum is \(21,\) find the number of gold medals won by each. (Source: Beijing 2008 Olympics)

Step-by-Step Solution

Answer

Australia won 6 medals, Germany 7 medals, and Korea 8 medals.

Step 1: Understand the Problem

We are given that the gold medals won by Korea, Germany, and Australia are three consecutive integers. If Korea won the most and Australia the least, the problem states that their sum equals 21.

Step 2: Define Variables

Let the number of gold medals won by Australia be \( n \). Then the number of medals won by Germany is \( n+1 \) and by Korea is \( n+2 \), since they are consecutive integers with Korea winning the most.

Step 3: Set Up the Equation

According to the problem, the sum of the gold medals is 21: \[ n + (n+1) + (n+2) = 21 \]

Step 4: Simplify the Equation

Combine like terms in the equation: \[ 3n + 3 = 21 \]

Step 5: Solve for n

Subtract 3 from both sides to isolate the terms with \( n \): \[ 3n = 18 \] Divide both sides by 3: \[ n = 6 \]

Step 6: Determine the Number of Medals

Since \( n = 6 \), Australia won 6 medals, Germany won \( 6 + 1 = 7 \) medals, and Korea won \( 6 + 2 = 8 \) medals.

Key Concepts

Consecutive Integers

Consecutive integers are numbers that follow each other in order without any gaps. In simpler terms, they are numbers that have a difference of 1 between each pair. For example, numbers like 4, 5, and 6 are consecutive integers. Similarly, you could also have negative consecutive integers like -2, -1, and 0.

In algebra word problems like the one from the 2008 Summer Olympics, understanding consecutive integers is crucial. When the problem mentions three consecutive integers, it suggests a relationship between the numbers. The smallest number can be represented by a variable, say \( n \). The next consecutive integer will be \( n + 1 \), and the one after that will be \( n + 2 \).

This setup allows us to connect consecutive numbers easily in equations. This concept makes these types of problems more approachable because it transforms them into simple algebra equations that can be solved step by step.

Equation Solving

Equation solving is a key concept in algebra that involves finding the value of variables that satisfy a given equation. In the problem of finding the number of gold medals won by different countries, setting up and solving equations becomes essential.

In this particular instance, we start by letting the smallest of the three consecutive numbers (the number of gold medals won by Australia) be \( n \). Since the medals are consecutive, the equation becomes:

• Australia: \( n \)
• Germany: \( n + 1 \)
• Korea: \( n + 2 \)

The equation representing their total medals: \[ n + (n + 1) + (n + 2) = 21 \]

To solve this, we first simplify the equation by combining like terms: \[ 3n + 3 = 21 \]

Next, isolate \( n \) by subtracting 3 from both sides to get: \[ 3n = 18 \]

Finally, divide both sides by 3 to solve for \( n \): \[ n = 6 \]

This process shows how logical steps can help in breaking down equations, making it easier to solve seemingly complex problems.

Sum of Integers

The sum of integers refers to the total obtained when individual integers are added together. In algebra word problems, it is common to find the sum because it provides a direct relationship among quantities.

In the Olympics exercise, it is stated that the total number of gold medals is 21. This sum involves three consecutive integers, which adds an additional layer as it directly ties into how these numbers relate sequentially. By expressing these integers as \( n \), \( n+1 \), and \( n+2 \), obtaining their sum can be formalized in an algebraic expression:

To solve for this expression, add each part:

• \( n \)
• \( n+1 \)
• \( n+2 \)

This gives: \[ n + (n+1) + (n+2) = 21 \]

Summing them simplifies to \( 3n + 3 \), emphasizing the importance of combining like terms. Solving such sums in equation form illustrates the principle that the sum of intently positioned numbers can be efficiently dealt with using algebraic methods. It provides a structured approach, ensuring clarity and precision when tackling mathematical word problems.