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Chapter 1: Problem 7

Miriam and Betty buy a total of 42 stamps. Miriam bought 6 more stamps than Betty did. How many stamps did Miriam buy? A. 18 B. 24 C. 30 D. 36

Short Answer

Expert verified
B. 24

Step by step solution

01

Define the variables

Let x represent the number of stamps Betty bought, and let (x + 6) represent the number of stamps Miriam bought (as she bought 6 more than Betty).
02

Create an equation based on the given information

We know that Miriam and Betty together bought 42 stamps, so we can add the number of stamps bought by both individuals (x for Betty and x + 6 for Miriam) to create an equation: \(x + (x + 6) = 42\)
03

Solve the equation for x

Combine the x terms: \(2x + 6 = 42\) Subtract 6 from both sides: \(2x = 36\) Divide both sides by 2: \(x = 18\) Now we know that Betty bought 18 stamps.
04

Find the number of stamps Miriam bought

Since Miriam bought 6 more stamps than Betty, and we know that Betty bought 18 stamps, we can find the number of stamps Miriam bought: Miriam bought: \(18 + 6 = 24\) stamps So the correct answer is B. 24.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Variable Definition
Understanding variables is crucial in solving algebra problems. In this word problem, we define a variable to represent the unknown quantity we need to find.
Let's denote Betty's stamps by the variable \(x\). This is our base variable as we don't know the exact number of stamps she purchased.
To account for the additional stamps Miriam bought, we define her stamps as \(x + 6\). This is because Miriam bought six more than Betty.
Variables act as placeholders for unknowns, which we will solve through equations. Defining them clearly at the start helps in setting up and solving the problem efficiently.
Equation Creation
Equations help us mathematically represent and solve problems. Based on the problem, we know that Miriam and Betty together bought 42 stamps.
Thus, the total number of stamps can be shown as \(x + (x + 6) = 42\), representing both Betty's and Miriam's purchases.
Creating equations effectively involves translating word problems into mathematical expressions.
Here, knowing that Miriam bought 6 more than Betty allows us to combine the expressions in an equation that illustrates the entire situation.
Equation Solving Techniques
Solving equations involves isolating the variable by performing operations that maintain equality.
First, we combine the terms on the left side of the equation: \(2x + 6 = 42\). This consolidates Betty's and Miriam's stamps.
Next, we isolate the \(2x\) term by subtracting 6 from both sides, resulting in \(2x = 36\).
Finally, divide both sides by 2 to find \(x = 18\). This tells us the number of stamps Betty purchased.
Through these systematic steps, solving any linear equation becomes manageable and straightforward.
Word Problem Analysis
Analyzing word problems involves understanding the given information and determining how to use it.
Key indicators in this problem include 'together', 'more than', and the total number of stamps.
These clues help us form a plan: define variables for each person's stamps, create an equation, and solve it step-by-step.
The solution reveals not just the number of stamps Miriam bought, but also demonstrates the power of breaking down complex situations into simpler components using algebra.

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