Problem 40 - Algebra 1: Concepts and Skills

Question

Use \(a=\frac{p}{100} b\). Complete the sentence: When the percent p is a number greater than 100, the value of a is ____ ? than the value of the base number b.

Step-by-Step Solution

Answer

When the percent p is a number greater than 100, the value of a is greater than the value of the base number b.

Step 1: Understand the formula

The formula provided in the exercise \(a=\frac{p}{100} b\), indicates that 'a' is 'p' percent of 'b'. 'p' is divided by 100 to convert it into a decimal, which is then multiplied with 'b' to produce 'a, the resultant value.

Step 2: Evaluate for 'p' value greater than 100

We need to comprehend what happens when 'p' is greater than 100. In such cases, 'p' divided by 100 is a value greater than 1. Any number 'b' multiplied by a factor greater than 1 will result in a larger number.

Step 3: Final realization

So, when 'p' is more than 100, 'a' becomes greater than 'b' because we are scaling 'b' by a factor greater than 1. In other words, 'a' is a larger version of 'b'.

Key Concepts

Algebraic Expressions

Algebraic expressions are a crucial aspect of understanding mathematical relationships. In the provided formula, \( a = \frac{p}{100} b \), 'a', 'p', and 'b' are all part of an algebraic expression where:

• 'a' represents the resultant quantity.
• 'p' is the percentage value that modifies 'b'.
• 'b' is the base value being adjusted.

Here, 'p' needs to be converted from a percentage to a decimal by dividing by 100. This conversion is essential in carrying out computations involving algebraic expressions. Essentially, algebraic expressions allow us to model situations mathematically, providing a way to explore how changes in percentage affect various quantities. Understanding how each component interacts helps in deciphering more complex mathematical relationships.

Percentages

Percentages are a way to express proportions in relation to a whole, represented as 100. The exercise highlights the importance of converting percentages into decimals for calculations. For example, if the percentage \( p \) is 150%, it is represented in decimal form as \( \frac{150}{100} = 1.5 \).
Thus, percentages greater than 100 mean the value is more than the whole it refers to. In the context of the exercise, if \( p > 100 \), then \( a = \frac{p}{100}b \) produces a number greater than 'b'.

• A percentage over 100% implies an increase or augmentation in the number 'b'.
• Converting percentages accurately is key to problem-solving in these contexts.

Understanding percentages and how they are manipulated, such as conversion and multiplication, is fundamental for solving these types of algebra problems.

Mathematical Reasoning

Mathematical reasoning involves the ability to think logically about numbers and their relationships. In this exercise, reasoning helps us understand why \( a \) becomes greater than \( b \) when \( p \) is greater than 100. We follow these logical steps:

• Recognize that a percentage greater than 100 indicates more than the whole.
• Understand that multiplying 'b' by a factor greater than 1 scales it up.
• Draw the conclusion that \( a \) must therefore be larger than \( b \).

Mathematical reasoning is about identifying patterns, applying logical deductions, and validating the final conclusion with consistent principles. This exercise encourages students to apply these skills to figure out the impact of percent values on algebraic patterns. Good reasoning leads to deeper comprehension and more robust problem-solving capability in math.