Question

A team won 50 percent of its first 60 games in a particular season, and 80 percent of its remaining games. If the team won a total of 60 percent of its games that season, what was the total number of games that the team played? A 180 B 120 C 90 D 85 E 30

Short answer

The total number of games played is 90.

Step-by-step solution

- Analyze the first part of the season

The team won 50% of its first 60 games. Let's calculate the number of games won in this part: \( 0.5 \times 60 = 30 \text{ games} \)

- Define the total number of games

Let \( x \) be the total number of games played in the season. Hence, the remaining games played are \( x - 60 \).

- Calculate remaining games won

The team won 80% of the remaining games. So, the number of games won in the remaining games is: \( 0.8 \times (x - 60) \)

- Express total wins in terms of \( x \)

The total games won is the sum of games won in both parts: \( 30 + 0.8 \times (x - 60) \)

- Set up equation for total wins

The total wins are given as 60% of the total games played. Thus, the equation becomes: \( 30 + 0.8 \times (x - 60) = 0.6x \)

- Simplify the equation

Expanding and simplifying the equation: \( 30 + 0.8x - 48 = 0.6x \) \( 0.8x - 0.6x = 48 - 30 \) \( 0.2x = 18 \)

- Solve for \( x \)

Divide both sides by 0.2: \( x = 90 \)

Key concepts

percentage calculations

Percentage calculations are essential in determining parts of a whole. For instance, in the exercise, we dealt with percentages to find the number of games won by the team.

To calculate a percentage of a number, multiply the percentage (as a decimal) by the total number. For example:

• The team won 50% of its first 60 games, which means 50% of 60 is calculated as:

\(0.5 \times 60 = 30\text{ games}\)

• Similarly, to find 80% of the remaining games, we multiply 80% (or 0.8) by the number of remaining games:

\(0.8 \times (x-60)\)
This approach helps break down complex problems in more manageable steps and is widely used for simplifying and solving various mathematical problems.

algebraic equations

Algebraic equations form the backbone of mathematical problem solving. They help in expressing relationships between variables and constants. In our exercise, we set up an equation to solve for the total number of games played by the team.

The steps involved included defining variables (let \(x\) be the total number of games), expressing conditions using those variables, and then simplifying and solving the equation:

• Step 1: Define the variables and conditions: \(x\) is the total number of games, of which 60 are accounted for initially.
• Step 2: Set up the equation considering all parts: \(30 + 0.8 \times (x-60) = 0.6x\)
• Step 3: Simplify and solve the equation: Expand the terms, combine like terms, and isolate \(x\).

Such equations help us logically step through to find the desired unknown values.

GMAT problem solving

In GMAT problem solving, you'll often encounter multi-step problems that require logical reasoning, clear setup, and systematic calculations.

This exercise is a prime example of GMAT-style questions since it requires understanding percentages, defining variables, setting up equations, and logically solving them:

• Break down the question into parts: First 60 games and the remainder.
• Calculate specific values using given percentages.
• Combine partial results into a comprehensive equation.
• Solve the equation step-by-step to find the correct answer.

Practicing GMAT problems enhances your ability to handle complex word problems efficiently and prepares you for test conditions.

step-by-step mathematical reasoning

Step-by-step mathematical reasoning involves breaking down problems into smaller, manageable parts and solving them sequentially.

In this exercise, our goal was to find the total number of games played by a team based on winning percentages. Here’s how reasoned through the steps:

• Step 1: Determine games won in the first part of the season using percentage calculation.
• Step 2: Define total number of games by introducing a variable \(x\).
• Step 3: Compute the games won in the remaining games using another percentage calculation.
• Step 4: Express total wins using the calculated parts.
• Step 5: Set up an algebraic equation representing total wins.
• Step 6: Simplify and solve the equation to find \(x\).

Breaking problems into steps helps understand and apply fundamental concepts incrementally, making it easier to solve complex problems.