Question

Translate to an equation and solve. A basketball team in one game made \(42 \%\) of the total number of attempted shots. If the players made a total of 38 baskets, how many shot attempts did they have in the game?

Short answer

Answer: 90 shot attempts

Step-by-step solution

Write the equation

Translate the problem into an equation: 0.42x = 38.

Solve for x

To solve for x, divide both sides of the equation by 0.42: x = 38 / 0.42

Calculate the result

Divide 38 by 0.42 to find the total number of shot attempts: x = 90.476

Round the result

Since we cannot have a fraction of a shot attempt, round the result to the nearest whole number: x ≈ 90 Therefore, there were approximately 90 shot attempts in the game.

Key concepts

Solving Linear Equations

Understanding how to solve linear equations is a fundamental skill in algebra, which is important for progressing through more complex mathematics. A linear equation is an algebraic equation where each term is either a constant or the product of a constant and a single variable. When we are given a word problem, the first step is to translate the words into an algebraic expression.

In the exercise, we see that the team made 42% of their shots and scored 38 baskets. To form the equation, we associate the percentage with the part of the total shots made (38) and the unknown total shots attempted as the variable (let's call it 'x'). Writing this as an equation, it looks like this: \(0.42x = 38\.\)

To isolate 'x' and solve for the total number of attempted shots, we need to perform the inverse operation. For this example, we divide both sides of the equation by 0.42, which gives us \(x = \frac{38}{0.42}\). After performing the division, we find that 'x' is approximately 90, which is the answer after rounding to the nearest whole number, as we can't attempt a fraction of a shot.

Percentage Problems

Percentage problems are ubiquitous in everyday life, from calculating tips at a restaurant to understanding statistics in the news. When translating percentage problems into equations, it's crucial to remember that 'percent' means 'per hundred.' So, if a basketball team makes 42% of its shots, that's the same as saying 42 out of every 100 shots.

To convert a percentage to a decimal, which is necessary for creating equations, divide the percentage by 100. This means that 42% becomes 0.42. In our exercise, the equation \(0.42x = 38\) represents the situation where 0.42, or 42%, of the total attempts (x) resulted in made baskets (38). Moreover, mastering percentage problems helps students not only in math but also in understanding real-world financial situations like interest rates and discounts.

Prealgebra Mathematics

Prealgebra serves as the foundation for all higher-level math studies. It introduces the basic concepts and operations needed to understand algebra. It includes arithmetic but also introduces variables, equations, and problem-solving techniques. In prealgebra, students learn how to translate word problems into mathematical expressions and equations, which is critical for their success in algebra and beyond.

For instance, in the basketball problem, students must discern the relevant information (42% success rate and 38 successful shots) and disregard any potentially superfluous details. Then, translating these into an algebraic equation is a key skill learned in prealgebra. The problem-solving process involves identifying the unknown variable, forming an equation, and then solving that equation—a fundamental framework that will be used throughout a student’s math education.