Problem 3 - Prealgebra

Question

Solve each of the following word problems by translating the statement into a proportion. Be sure to show the proportion used in each case. [Examples \(1-4]\) Basketball A basketball player scores 162 points in 9 games. At this rate how many points will he score in 20 games?

Step-by-Step Solution

Answer

The player will score 360 points in 20 games.

Step 1: Identify Known and Unknown Values

First, identify the known values and the unknowns from the problem statement. The player scores 162 points in 9 games. We need to find out how many points he will score in 20 games. Let the number of points scored in 20 games be represented by the variable \( x \).

Step 2: Set Up the Proportion

Since the player's scoring rate is consistent, set up a proportion using the known values and \( x \). The proportion relates the number of points to the number of games:\[ \frac{162}{9} = \frac{x}{20} \]

Step 3: Cross-Multiply to Solve for x

Cross-multiply the terms in the proportion to find \( x \). Solve the equation: \[ 162 \times 20 = 9 \times x \]Simplifying the equation gives:\[ 3240 = 9x \]

Step 4: Solve for x

To solve for \( x \), divide both sides of the equation by 9:\[ x = \frac{3240}{9} \]\[ x = 360 \]

Step 5: Conclusion

The player is expected to score 360 points in 20 games at this rate. Therefore, the consistent scoring rate and proportion indicate this total.

Key Concepts

Understanding Word Problems

Word problems in mathematics can often feel intimidating, but they are simply questions that use real-world scenarios to involve math in decision-making. These problems require us to extract numerical information from text and turn it into mathematical equations or expressions.

In the case of this exercise, the word problem involves a basketball player’s scoring rate across games. The key to solving such problems is recognizing the relevant information:

• Identify what is known – here, the player scores 162 points in 9 games.
• Identify what is unknown – we want to find out how many points he will score in 20 games.

Once you have identified these values, you can start framing your problem using mathematical concepts such as proportions.

Mastering Cross-Multiplication

Cross-multiplication is a valuable technique to solve equations involving proportions. It transforms a proportion into a simple equation that can be easily solved. The process is quick and efficient:

Given a proportion such as \( \frac{a}{b} = \frac{c}{d} \), cross-multiplication involves multiplying across the equals sign diagonally:

• Multiply \( a \) by \( d \), resulting in \( a \times d \).
• Multiply \( b \) by \( c \), resulting in \( b \times c \).
• Set them equal to each other, \( a \times d = b \times c \).

In our example, using cross-multiplication turns \( \frac{162}{9} = \frac{x}{20} \) into an equation: \( 162 \times 20 = 9 \times x \). This simplifies solving for \( x \), reducing the problem to straightforward arithmetic.

Solving Proportion Equations

Proportion equations are equations that state two ratios are equal. Solving them is all about finding the unknown variable that makes this statement true.

The first step is always to set up a proportion that represents the given information. In this exercise, we knew the player's scoring rate: 162 points in 9 games, and needed to calculate the unknown: points in 20 games. The correct proportion is \( \frac{162}{9} = \frac{x}{20} \).

After establishing the proportion, use cross-multiplication, resulting in \( 162 \times 20 = 9 \times x \). Solving this equation involves simple division. After calculating \( x \), substituting back into the context gives us the total points scored, hence solving our word problem efficiently.