Problem 102 - Introductory Algebra for College Students

Question

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. Because there are four quarters in a dollar, I can use the formula \(q=4 d\) to determine the number of quarters, \(q,\) in \(d\) dollars.

Step-by-Step Solution

Answer

The statement 'Because there are four quarters in a dollar, I can use the formula \(q=4d\) to determine the number of quarters, \(q\), in \(d\) dollars' does make sense. The formula accurately converts dollars into quarters.

Step 1: Understanding the Statement and Variables

First, we have to understand what the statement is saying and what are the defined variables. In this case, \(q\) represents the 'number of quarters' and \(d\) represents the 'number of dollars'. The statement suggests that if you multiply the amount of dollars (d) by 4, you will get the equivalent number of quarters (q). This conversion is straightforward as we know that one dollar is equivalent to four quarters.

Step 2: Validity of the Statement

Next, to see if the statement 'makes sense' or 'does not make sense', let's test this statement with a simple example: Assume you have 1 dollar, then according to the proposed formula \(q=4d\), the number of quarters would be \(q=4*1=4\), which is true as we know 1 dollar equals 4 quarters. Thus, the given formula is correct.

Step 3: Confirming the Statement with a Different Example

To assure the correctness of the statement, let's test with a different number of dollars: Assume 10 dollars is given to be converted into quarters, then according to the formula \(q=4d\), the number of quarters would be \(q=4*10=40\), which is indeed the correct conversion. Hence, the statement does make sense and the provided formula is accurate for converting dollars to quarters.