Problem 34 - Prealgebra

Question

A number \(n\) is \(25 \%\) of some number \(a\) and \(35 \%\) of a number \(b\). Tell the relationship between \(a\) and \(b\). Is \(ab,\) or is it impossible to determine the relationship? Explain.

Step-by-Step Solution

Answer

The relationship is \( a > b \).

Step 1: Identify Mathematical Expressions

We know that a number \( n \) is \( 25\% \) of some number \( a \). This can be expressed as \( n = 0.25a \). Also, \( n \) is \( 35\% \) of a number \( b \), which can be expressed as \( n = 0.35b \). Our goal is to find the relationship between \( a \) and \( b \).

Step 2: Set Expressions Equal

Since both expressions equal \( n \), we can set the expressions equal to each other: \( 0.25a = 0.35b \). This allows us to create a single equation that relates \( a \) and \( b \) directly.

Step 3: Solve for One Variable

Divide both sides of the equation \( 0.25a = 0.35b \) by \( 0.25 \) to express \( a \) in terms of \( b \): \( a = \frac{0.35}{0.25}b \). Simplifying gives \( a = 1.4b \).

Step 4: Determine the Relationship

From \( a = 1.4b \), we can see that \( a \) is \( 1.4 \) times \( b \), meaning \( a > b \). Therefore, \( a \) is greater than \( b \).

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are a fundamental concept in solving mathematical problems. These expressions are combinations of numbers, variables, and arithmetic operations, such as addition, subtraction, multiplication, and division. In our exercise, we encountered expressions like \( n = 0.25a \) and \( n = 0.35b \). Here,

• \( n \) is a variable representing a number we do not yet know.
• \( a \) and \( b \) are the variables related to \( n \) through percentages.
• The coefficients 0.25 and 0.35 represent the percentage relationship of \( n \) with \( a \) and \( b \) respectively.

These expressions are essential because they translate real-world percentage problems into a language that can be manipulated mathematically. In this way, algebraic expressions enable us to establish relationships between different quantities.

The Art of Solving Equations

Solving equations involves finding the values of unknown variables that satisfy given relationships. In this exercise, both expressions \( n = 0.25a \) and \( n = 0.35b \) are set equal because they both describe the same variable, \( n \). Thus, we equate them to solve for one variable in terms of the other:

• \( 0.25a = 0.35b \)
• Our goal is to isolate one variable. By dividing by the coefficient (0.25), we find:
• \( a = \frac{0.35}{0.25}b \)
• Simplifying the fraction gives \( a = 1.4b \)

This manipulative step allows us to express \( a \) in terms of \( b \), revealing how these two quantities relate. Equipped with this interpretation, we gain the insight needed to compare \( a \) and \( b \).

Analyzing Inequalities

Inequalities deal with the comparison of values to determine which is greater, lesser, or if they are equal. In our problem, once we derive the equation \( a = 1.4b \), we can deduce that an inequality exists between \( a \) and \( b \). Since \( a = 1.4b \), it indicates that \( a \) is 1.4 times the size of \( b \). Thus, \( a \) is greater than \( b \), which can also be written as:

• \( a > b \)
• The factor 1.4 emphasizes that for every value \( b \) has, \( a \) has that value plus 0.4 times more.

By knowing how to manipulate and interpret equations and inequalities, we can draw significant conclusions about their relationships, which is a valuable skill in both academics and real-world scenarios. Understanding these comparisons helps us make informed decisions based on the mathematical relationships present.