Problem 33 - Algebra 2

Question

A newspaper reported that the average American laughs 15 times per day. Write an equation to represent the average number of laughs produced by \(m\) household members during a period of \(d\) days.

Step-by-Step Solution

Answer

The equation is \(15md\).

Step 1: Understand the Problem

The problem tells us that an average American laughs 15 times per day. We need to find an equation to model the total number of laughs based on the number of people and the number of days.

Step 2: Identify Variables

Let \(m\) represent the number of household members and \(d\) represent the number of days. We seek the total number of laughs in terms of \(m\) and \(d\).

Step 3: Write the Expression for One Person

If one person laughs 15 times per day, then over \(d\) days, they will laugh \(15d\) times.

Step 4: Account for Multiple Members

Since the household has \(m\) members, and each laughs \(15d\) times in \(d\) days, for \(m\) members, we multiply the previous expression by \(m\).

Step 5: Formulate the Equation

The equation representing the total number of laughs by \(m\) household members over \(d\) days is given by: \( \, \text{Total Laughs} = 15md \).

Key Concepts

Equation formulation

In mathematics, formulating an equation involves creating a mathematical statement that represents a real-world situation. When faced with a problem description, like in our example about average laughs per day, it's pivotal to extract the relevant details and relationships.

The first step is understanding the problem context. Here, the focus is to determine the total number of laughs over several days, involving multiple household members. Next, we translate the identified details into mathematical expressions.

Let's break down the formulation process:

• Recognize the constant: In the given scenario, an average American laughs 15 times per day is a key figure.
• Identify the operations: Each individual's daily laughs (15) must be multiplied by the number of days (\(d\)) to get laughs per person over those days.
• Expanding to multiple members: Multiply the per person result by the number of members (\(m\)) to cover the entire household.

This results in the equation \(15md\). Hence, equation formulation condenses detailed information into a succinct mathematical representation, facilitating straightforward evaluation and computation.

Variables in mathematics

In algebra, variables are symbols used to represent unknown values or quantities that can change. They are the building blocks for setting up equations and can represent everything from the number of household members to the number of days in a laughing scenario.

In our problem, we assign letters to these changing quantities for simplicity:

• \(m\): Represents the number of household members.
• \(d\): Denotes the period in days.

These variables allow us to generalize the equation beyond a specific instance.

Mathematicians employ variables to formulate equations because they enable a flexible analysis of situations. Instead of listing every possible scenario, variables make it easy to adjust numbers and see how outcomes differ with varying inputs. When the equation \(15md\) uses \(m\) and \(d\) effectively, it reflects that the solution applies to any number of household members over any number of days.

Word problems in algebra

Algebraic word problems present real-life scenarios that require a thoughtful translation into mathematical language. Word problems like the one about average laughs challenge students to discern the relevant numeric relationships and express them through equations.

Approaching word problems involves several steps:

• Understanding the Context: Begin by reading the problem thoroughly to grasp what's being asked. Recognize key figures, such as the average laughs, and what they represent.
• Identifying Variables: Determine what quantities in the problem change, and decide how to represent them using variables (e.g., people, days).
• Setting Up the Equation: Use the information to form an equation. Make sure the relationships between variables and constants (like the laughing frequency) are correctly represented. In our case, the formula \(15md\) shows total laughs for the given number of members and days.
• Solving the Equation: Although not required for formulation, ensure you understand how to solve the equation for specific values to find solutions if needed.

By breaking down word problems into these manageable steps, solving them becomes more intuitive, and reduces the stress of tackling complex text-based mathematical questions.