Problem 61 - College Algebra and Calculus: An Applied Approach

Question

The percent of patients in a clinical trial of a cancer treatment showing a decrease in tumor size is 49.2\%. There are 3445 patients in the trial. How many patients show a decrease in tumor size?

Step-by-Step Solution

Answer

1695 patients show a decrease in tumor size.

Step 1: Understand the problem

We need to find out how many patients show a decrease in tumor size. We have been given that 49.2% of patients show a decrease in tumor size and the total number of patients is 3445.

Step 2: Convert the percentage to a decimal

To convert a percentage to a decimal, divide the percent by 100. So, 49.2% becomes \(49.2 \div 100 = 0.492\).

Step 3: Calculate the number of patients

To find the number of patients with decreased tumor size, multiply the total number of patients by the decimal equivalent of the percentage. So, \(0.492 * 3445 = 1695.24\). We cannot have a fraction of a patient, so we round this number to the nearest whole number, ending up with 1695 patients.

Key Concepts

Percentage Calculations

When you're dealing with percentages, it's essentially a way to express numbers as parts of a whole divided into 100 equal parts. Understanding this can simplify many calculations in everyday and academic contexts.

To calculate a percentage of a given number, you multiply the total by the percentage expressed in decimal form. For instance, if 49.2% of a group shows a characteristic feature, we find how many there are by first converting this percentage to its decimal form, then multiplying by the total number in the group.

• The formula becomes: \( \text{Width in percentage form} \times \text{Total Number} = \text{Amount} \).
• Real-world application: Suppose in a classroom of 200 students, 25% have blue eyes. You'd calculate \( 0.25 \times 200 = 50 \) students with blue eyes.

Remember, understanding how to calculate percentages is invaluable as it appears frequently in scenarios ranging from financial planning to academic problem-solving.

Decimal Conversion

Decimals are an essential part of both basic and advanced math. Converting percentages to decimals is a straightforward process that forms the backbone of percentage calculations.

Here's how to convert percentages to decimals:

• Simply divide the percentage value by 100.
• This process shifts the decimal point two places to the left.

For example, to convert 49.2% to a decimal, perform the operation:

• \( 49.2 \div 100 = 0.492 \)

Understanding decimal conversion is critical because it translates percentages into a format that is directly usable in most mathematical calculations.

When dealing with complex problems, like determining a statistic across a large number of subjects, decimal conversion simplifies calculations by transforming percentages into a form suitable for multiplication with whole numbers.

Rounding Numbers

Rounding numbers is a valuable skill, especially when dealing with real-world data that doesn't always fit neatly into whole numbers. This is particularly important when you're calculating tangible things, like the number of people in a group.

**Why Round and How to Round: **

• Rounding provides a simplification for numbers with decimals, converting them into a form that is easy to interpret.
• To round a number, look at the digit directly after the decimal point of the place you're rounding to. If it's 5 or higher, round up. Otherwise, round down.

In the context of our example, after converting the percentage of patients to a decimal and multiplying by the number of total patients, we calculated 1695.24 as our answer. Since it's unrealistic to have a fraction of a patient, you round 1695.24 to 1695.

Rounding numbers makes final results more practical and applicable while maintaining an acceptable level of accuracy in most scenarios.