AT A Hardware Store The Price Of A Rake, R Is One Third The Price Of A Shovel, S. Which Equation Represents
Understanding the relationship between different tools and their prices in a hardware store is essential, especially for those interested in mathematics, shopping smarter, or managing budgets effectively. In this article, we will explore how to represent the given relationship between the price of a rake and a shovel using algebraic equations. We will analyze the problem step by step, providing clarity on how to formulate the correct equation that models this scenario.
Introduction to the Problem
Suppose you are shopping at a hardware store and need to purchase a rake and a shovel. You are told that:
- The price of a rake is R.
- The price of a shovel is S.
- The price of a rake R is one third the price of a shovel S.
The question posed is: Which equation represents the relationship between R and S?
To answer this, we need to translate the verbal statement into an algebraic expression. This process involves understanding the key phrase "R is one third the price of S" and how to express that relationship mathematically.
Breaking Down the Verbal Relationship
The statement "R is one third the price of S" can be interpreted as:
- R is equal to one third of S.
- Or, R is S divided by 3.
Mathematically, this is expressed as:
\[ R = \frac{1}{3} S \]
This equation states that the price of the rake R is equal to one third of the price of the shovel S.
Formulating the Equation
Based on the understanding, the primary algebraic equation representing the relationship is:
Equation 1:
\[ R = \frac{1}{3} S \]
This is the most straightforward and correct way to express the given relationship.
Alternative Forms
Depending on the context or the way the problem is presented, you might see this relationship manipulated into different forms:
- Multiplying both sides by 3:
\[ 3 R = S \]
- Expressing S in terms of R:
\[ S = 3 R \]
These forms are equivalent and can be useful depending on what you are trying to find or solve.
Applying the Equation in Real-World Contexts
Understanding the relationship between R and S allows consumers and store managers to:
- Calculate the expected price of one tool if the other is known.
- Create price comparisons or discounts.
- Formulate word problems for educational purposes.
Let's look at some example scenarios:
Example 1: Calculating the Price of a Rake
If the shovel costs $15, what is the price of a rake?
Using the equation:
\[ R = \frac{1}{3} S \]
Substitute S = 15:
\[ R = \frac{1}{3} \times 15 = 5 \]
So, the rake costs $5.
Example 2: Calculating the Price of a Shovel
If the rake costs $8, what is the price of the shovel?
Using the equivalent form:
\[ S = 3 R \]
Substitute R = 8:
\[ S = 3 \times 8 = 24 \]
Thus, the shovel costs $24.
Understanding the Significance of the Equation
Formulating the correct algebraic equation from a word problem is a fundamental skill in mathematics. It allows for:
- Clear representation of relationships.
- Simplification of complex problems.
- Accurate calculations and predictions.
In this scenario, the key is recognizing the phrase "one third the price," which indicates a fractional relationship.
Common Mistakes and How to Avoid Them
When translating verbal statements into equations, students often make errors. Here are some common pitfalls:
- Confusing "is" with "equals": Always interpret "is" as the equals sign (=).
- Misinterpreting "one third": Remember that "one third" indicates division by 3, not multiplication.
- Reversing the relationship: Ensure that R is expressed in terms of S, or vice versa, according to the statement.
By carefully parsing the language and translating each part into algebraic expressions, you can avoid these mistakes.
Expanding the Concept to Other Word Problems
This type of problem is common in algebra and helps develop skills such as:
- Writing equations from word problems.
- Interpreting fractional relationships.
- Solving for unknown variables.
Sample problem:
The price of a lawnmower is twice the price of a trimmer. If the trimmer costs $50, what is the lawnmower's price?
Solution:
Let T be the trimmer's price, and L be the lawnmower's price.
Given: \( L = 2 T \)
Substitute T = 50:
\[ L = 2 \times 50 = 100 \]
The lawnmower costs $100.
Summary and Key Takeaways
- The phrase "R is one third the price of S" translates to the algebraic equation:
- Alternative equivalent forms include:
- Understanding how to convert verbal relationships into equations is essential in solving real-world mathematical problems.
- Always carefully analyze the language, especially fractions and comparative phrases, to determine the correct algebraic representation.
Conclusion
Mathematics provides a powerful language to model real-world relationships, such as pricing in a hardware store. Recognizing and correctly translating phrases like "one third" into algebraic equations enables accurate problem-solving and logical reasoning. Whether you're a student honing your algebra skills or a shopper making informed choices, understanding these relationships enhances your mathematical literacy and decision-making abilities.
By mastering the translation of verbal descriptions into equations, you build a strong foundation for tackling more complex problems involving proportions, ratios, and linear relationships. Remember, the key to solving such problems lies in carefully interpreting the language and applying fundamental algebraic principles.
