Problem 23 - Chemistry for Allied Health

Question

Solve each of the following? a. 42 is \(34 \%\) of what number? b. 73 is \(82 \%\) of what number? c. 13 is \(57 \%\) of what number? d. 75 is \(25 \%\) of what number? e. 25 is \(15 \%\) of what number? f. 98 is \(76 \%\) of what number?

Step-by-Step Solution

Answer

a. 123.53, b. 89.02, c. 22.81, d. 300, e. 166.67, f. 128.95

Step 1: Understanding the Problem

To find out what number a given percentage represents, use the formula: \[\text{Part} = \frac{\text{Percent}}{100} \times \text{Whole}\]. We need to rearrange it to find the whole number: \[\text{Whole} = \frac{\text{Part}}{\frac{\text{Percent}}{100}}\].

Step 2: Solving for 42 is 34% of what number?

Use the formula with Part = 42 and Percent = 34: \[\text{Whole} = \frac{42}{0.34} = 123.5294\] or approximately 123.53.

Step 3: Solving for 73 is 82% of what number?

Use the formula with Part = 73 and Percent = 82: \[\text{Whole} = \frac{73}{0.82} = 89.0244\] or approximately 89.02.

Step 4: Solving for 13 is 57% of what number?

Use the formula with Part = 13 and Percent = 57: \[\text{Whole} = \frac{13}{0.57} = 22.8070\] or approximately 22.81.

Step 5: Solving for 75 is 25% of what number?

Use the formula with Part = 75 and Percent = 25: \[\text{Whole} = \frac{75}{0.25} = 300\].

Step 6: Solving for 25 is 15% of what number?

Use the formula with Part = 25 and Percent = 15: \[\text{Whole} = \frac{25}{0.15} = 166.6667\] or approximately 166.67.

Step 7: Solving for 98 is 76% of what number?

Use the formula with Part = 98 and Percent = 76: \[\text{Whole} = \frac{98}{0.76} = 128.9474\] or approximately 128.95.

Key Concepts

Understanding Percentage Calculations

Percentage calculations are vital mathematical concepts used in everyday situations. At its core, a percentage represents a fraction of 100. For instance, 34% means 34 out of every 100. When you're asked to find what number a given percentage represents, you are essentially looking for the 'whole' in relation to a part.
For example, "42 is 34% of what number?" means you need a total number that, when 34% is taken, equals 42. This is a typical percentage problem, which requires you to convert the percentage to a decimal first.
In this case, 34% becomes 0.34. Calculating percentages helps us understand proportions, relationships, and distributions in a wide range of contexts. Whether dealing with everyday financial calculations or scientific data analysis, understanding percentages can greatly enhance problem-solving efficiency.

The Role of Mathematical Formulas

Mathematical formulas are essential tools that help translate complex ideas into manageable calculations. In percentage problems, the formula that connects the part, percent, and whole is central.
The formula generally used is:

• \( \text{Part} = \frac{\text{Percent}}{100} \times \text{Whole} \)
• This enables us to find the part when the percentage and whole are known.

However, to solve for an unknown 'whole,' the formula is rearranged to:

• \( \text{Whole} = \frac{\text{Part}}{\frac{\text{Percent}}{100}} \)

In problems where you're given a part and a percentage, plug in these values to solve for the whole number. This re-arranged formula allows a quick and precise calculation for determining the total or full amount represented by a given percentage.

Problem-Solving Steps

Solving percentage problems can be straightforward if approached step-by-step. Begin by clearly understanding the problem and identifying the 'part' and 'percent' given.
Here's a guide to the problem-solving process:

• Identify the given 'part' and 'percent'
• Convert the percentage to a decimal by dividing it by 100
• Rearrange the formula to solve for 'whole': \( \text{Whole} = \frac{\text{Part}}{\text{Decimal}} \)
• Perform the division to find the 'whole'

By systematically breaking down each step, you ensure clarity and accuracy in your calculations. This approach is replicable, making it useful for a variety of problems beyond just percentage calculations.

Utilizing Arithmetic Operations

Basic arithmetic operations are the building blocks for solving percentage problems. These include addition, subtraction, multiplication, and particularly division. In our formula \( \text{Whole} = \frac{\text{Part}}{\text{Decimal}} \), division plays a key role.
You must first convert percentages to decimals, which is a form of division (e.g., 34% becomes 0.34). Then, you use division to find the whole. Precision in division ensures the accuracy of your result.
For example, finding the whole number for "42 is 34%" involves calculating \( \frac{42}{0.34} \). Understanding how to manipulate numbers through these operations strengthens overall mathematical fluency and problem-solving capability. By honing these arithmetic skills, solving percentage problems becomes intuitive and less cumbersome. Mastery of these operations allows for more complex calculations and understanding of mathematical relationships, developing deeper insight.