Problem 3 - Prealgebra

Question

Solve each of the following problems by first restating it as one of the three basic percent problems of Section 7.2 . In each case, be sure to show the equation. A basketball player made 63 out of 75 free throws. What percent is this?

Step-by-Step Solution

Answer

The player made 84% of his free throws.

Step 1: Identify the Total Quantity

The total number of free throws attempted is 75. In percent problems, this is considered as the base or the whole.

Step 2: Identify the Part of the Total

The part, or the actual quantity made, is 63 free throws. This is the portion we want to express as a percent of the whole.

Step 3: Set Up the Percentage Equation

To find what percent 63 is of 75, use the percentage equation: \[ \frac{\text{part}}{\text{whole}} = \frac{63}{75} = x\% \] where \( x \) is the percent value we want to find.

Step 4: Solve for the Percent

Solve the equation for \( x \) by converting the fraction to a percentage:\[ x = \left( \frac{63}{75} \right) \times 100 \].Calculate the fraction: \( \frac{63}{75} = 0.84 \).Then multiply by 100: \( 0.84 \times 100 = 84\% \).

Step 5: Interpret the Result

The basketball player made 63 out of 75 free throws, which is 84% of the total attempts.

Key Concepts

Percentage Calculation

Percentage calculation is often used to determine how one quantity compares to another, as a part of a whole expressed in hundredths. When we need to find out what percent one number is of another, we use the formula:

• \( \frac{\text{part}}{\text{whole}} \times 100 = \text{percent} \)

The 'part' is the specific quantity we are analyzing, while the 'whole' is the total or complete quantity. By dividing the part by the whole, we convert that fraction to a decimal. Multiplying by 100 shifts this decimal to a percentage, which represents a way of expressing numbers as proportions of 100. For example, if a player makes 63 out of 75 shots, we calculate:

• \( \frac{63}{75} \approx 0.84 \) and \( 0.84 \times 100 = 84\% \)

These calculations show us that the player succeeded with 84% of their attempts.

Fraction to Percent Conversion

Converting a fraction to a percentage involves a simple process that helps make comparisons more relatable and understandable. To convert a fraction to a percent, you follow these steps:

• Divide the numerator (the top number) by the denominator (the bottom number).
• Multiply the result by 100 to convert the decimal to a percent.

This process allows us to see what percent the fraction represents. For instance, with the fraction \( \frac{63}{75} \), you divide and get approximately 0.84. By multiplying 0.84 by 100, you convert it to 84%. This shows us clearly how much 63 is of the whole 75, making it easier to interpret the significance of the fraction in practical terms, like performance percentages in sports.

Interpreting Mathematical Results

Interpreting mathematical results means understanding what the numbers actually represent in real-world terms. When you calculate a percent, like 84%, it's not just a number; it tells a story. In the context of our basketball problem, a success rate of 84% indicates how often the player successfully made free throws during their attempts.
This interpretation is crucial because it provides insights:

• Is the player's performance above or below average?
• What does this percentage tell us about consistency?
• How does this compare to other players or previous games?

Understanding these results helps us make informed decisions, like identifying areas for improvement or recognizing strengths. Interpreting results transforms numerical data into actionable knowledge essential in many fields.