Chapter 10: Problem 4
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
Expert verified
The total number of games played is 90.
Step by step solution
01
- 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: \( 0.5 \times 60 = 30 \text{ games} \)
02
- Define the total number of games
Let \( x \) be the total number of games played in the season. Hence, the remaining games played are \( x - 60 \).
03
- Calculate remaining games won
The team won 80% of the remaining games. So, the number of games won in the remaining games is: \( 0.8 \times (x - 60) \)
04
- Express total wins in terms of \( x \)
The total games won is the sum of games won in both parts: \( 30 + 0.8 \times (x - 60) \)
05
- Set up equation for total wins
The total wins are given as 60% of the total games played. Thus, the equation becomes: \( 30 + 0.8 \times (x - 60) = 0.6x \)
06
- Simplify the equation
Expanding and simplifying the equation: \( 30 + 0.8x - 48 = 0.6x \) \( 0.8x - 0.6x = 48 - 30 \) \( 0.2x = 18 \)
07
- Solve for \( x \)
Divide both sides by 0.2: \( x = 90 \)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
This approach helps break down complex problems in more manageable steps and is widely used for simplifying and solving various mathematical problems.
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:
- Similarly, to find 80% of the remaining games, we multiply 80% (or 0.8) by the number of remaining games:
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:
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 \(x\).
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:
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.
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:
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 \(x\).