Question

Team \(\mathrm{X}\) won 40 basketball games. What percent of its basketball games did Team \(\mathrm{X}\) win? (1) Team X played the same number of basketball games as Team Y. (2) Team Y won 45 games, representing 2.5 percent of the basketball games it played.

Short answer

Team X won 2.22% of its basketball games.

Step-by-step solution

Understand the problem

Determine what percentage of the basketball games Team X won based on the given information.

Analyze given information from Statement (1)

Team X played the same number of games as Team Y. However, the number of games played by Team Y is not directly given.

Analyze given information from Statement (2)

Team Y won 45 games, and these wins represent 2.5% of the games played by Team Y. From this, calculate the total number of games played by Team Y.

Calculate the total games played using Statement (2)

Let the number of games played by Team Y be represented by the variable \( G \). Using the percentage formula: \[ 45 = 0.025 \times G \] Solve for \( G \): \[ G = \frac{45}{0.025} = 1800 \]

Determine the total games played by Team X

Since Team X played the same number of games as Team Y, Team X played 1800 games.

Calculate the percentage of games won by Team X

Team X won 40 games out of 1800 games. The percentage of games won by Team X is: \[ \text{Percentage} = \frac{40}{1800} \times 100 \] Simplify the fraction: \[ \text{Percentage} = \frac{40 \times 100}{1800} = \frac{4000}{1800} = 2.22\text{%} \]

Key concepts

percentage calculation

Percentage calculations are critical in many mathematical problems, including those on the GMAT. Understanding how to calculate the percentage can help you solve questions effectively and efficiently. In this problem, we need to find out what percent of its basketball games Team X won. Understanding the formula for percentage is key: use the formula \( \text{Percentage} = \frac{\text{Part}}{\text{Total}} \times 100 \). Here, the part refers to the number of games won by Team X, and the total refers to the total number of games played by Team X. Applying this directly to the given exercise requires some algebraic manipulation and data sufficiency reasoning.

data sufficiency

Data sufficiency is a particular question type in the GMAT. It asks you to determine whether the provided data is enough to answer the question. In this exercise, we have two statements. Statement (1) tells us that Team X played the same number of games as Team Y. Statement (2) tells us that Team Y won 45 games, which represent 2.5% of the games it played. We need to check if each statement alone or both together can answer the question. First, let's consider statement (2). It helps us calculate the total number of games played by Team Y using the formula \( \text{Part} = \text{Percentage} \times \text{Total} \). Rearrange the formula to find the total when you know the part and the percentage: \[ Total = \frac{\text{Part}}{\text{Percentage}} = \frac{45}{0.025} = 1800 \]. Then, using statement (1), since Team X played the same number of games as Team Y, we know both played 1800 games. Combining both statements allows us to answer the question about Team X's winning percentage.

algebraic manipulation

Algebraic manipulation involves rearranging equations to isolate variables and solve for unknowns. In this problem, we used algebraic manipulation to find the total number of games played by Team Y. From statement (2), the equation is \( 45 = 0.025 \times G \). To solve for \( G \), divide both sides by 0.025: \[ G = \frac{45}{0.025} = 1800 \]. This calculates the total number of games played. Additionally, to find out the percentage of games won by Team X, we use another algebraic manipulation: \[ \text{Percentage} = \frac{40}{1800} \times 100 \]. Breaking down the steps of algebraic manipulation ensures you understand each transformation, leading to the correct final answer.