Find Charm Cost: Which Function Is Correct?

Let's dive into a classic word problem that involves finding the right mathematical function to represent a real-world scenario. Jharna's bracelet charm purchase is a perfect example of how we can use math to understand costs and quantities. We need to figure out which function accurately calculates the total cost ($y$) based on the number of bracelet charms ($x$) she buys, given that each charm has the same price. This is a fantastic way to practice algebraic thinking and understand the relationship between variables.

Understanding the Problem: Jharna's Bracelet Charm Purchase

Jharna bought 6 bracelet charms and paid a total of **$13.92**. The key piece of information here is that **each bracelet charm cost the same amount**. This tells us we're dealing with a proportional relationship, which is fundamental to understanding linear functions. Our goal is to find a function that takes the *number of charms* ($x$) and outputs the *total cost* ($y$). This type of problem is super common in mathematics, especially when you're first learning about functions and how they model situations. We're essentially trying to find the rate at which the cost increases per charm.

Calculating the Unit Cost: The First Step to the Right Function

Before we can determine the correct function, we absolutely must know the cost of a single bracelet charm. Since Jharna paid $13.92 for 6 charms, and each cost the same, we can find the price per charm by dividing the total cost by the number of charms. This is a crucial step in solving this problem. The calculation is as follows: $13.92 / 6. Let's do the math:

$13.92

6

This division gives us $2.32. So, each individual bracelet charm costs $2.32. This unit cost is the rate of change in our function – it's how much the total cost increases for every additional charm purchased. In the context of functions, this unit cost is often referred to as the slope of the line, representing the price per item. Understanding this unit cost is the cornerstone of setting up the correct equation. Without it, we'd be guessing, but with it, we have a solid mathematical foundation to build upon.

Evaluating the Function Options

Now that we know the cost of one bracelet charm is $2.32, let's examine the given function options to see which one accurately reflects this relationship. Remember, we're looking for a function where $y$ (total cost) is equal to the cost per charm multiplied by the number of charms ($x$). The general form of such a function is $y = (cost\_per\_charm) \times x$.

Let's break down each option:

• A. $y = 13.92x$: This function suggests that each charm costs $13.92. If Jharna bought 6 charms, the total cost would be $13.92 imes 6 = $83.52. This is clearly incorrect, as we know she paid only $13.92 for 6 charms. So, option A is out.

• B. $y = 2.32x$: This function uses our calculated unit cost of $2.32 per charm. If Jharna bought 6 charms, the total cost would be $2.32 imes 6 = $13.92. This matches the information given in the problem! This option looks very promising. It directly incorporates the price per item and the quantity purchased to find the total cost, which is exactly what we need.

• C. $y = 6x$: This function implies that each charm costs $6. If Jharna bought 6 charms, the total cost would be $6 imes 6 = $36. This doesn't match the $13.92 total cost, so option C is incorrect.

• D. $y = 83.52x$: This function suggests a very high cost per charm. If Jharna bought 6 charms, the total cost would be $83.52 imes 6 = $501.12. This is wildly incorrect. It seems this number might be related to the total cost of buying 6 charms at $13.92 each, but it's not the cost per charm. This option is definitely not the one we're looking for.

The Correct Function and Why It Works

Based on our analysis, the only function that accurately represents the cost of buying bracelet charms is B. $y = 2.32x$. This function is a linear function because the relationship between the number of charms ($x$) and the total cost ($y$) is constant. For every additional charm ($x$) Jharna buys, the total cost ($y$) increases by a fixed amount, which is the price of one charm ($2.32). This is a direct and proportional relationship. The variable $x$ represents the independent variable (the number of charms you choose to buy), and $y$ represents the dependent variable (the total cost, which depends on how many charms you buy). The constant $2.32$ is the rate of proportionality or the slope of the line represented by this function. It tells us the cost escalates linearly with each additional charm.

This concept is fundamental in pre-algebra and algebra 1. Understanding how to identify the unit rate and translate it into a function is a skill that will be used repeatedly in more complex mathematical problems. For example, if you were calculating the distance traveled based on speed, or the amount of ingredients needed for a recipe based on servings, you'd use a similar approach. You'd find the rate (speed or amount per serving) and multiply it by the quantity (time or number of servings) to find the total. The structure $y = mx + b$ is the general form of a linear equation, where $m$ is the slope (our unit cost) and $b$ is the y-intercept. In this specific problem, the y-intercept ($b$) is $0$ because if Jharna buys $0$ charms ($x=0$), the total cost ($y$) is also $0$. So, the equation simplifies to $y = 2.32x$, fitting the $y = mx$ form perfectly.

Conclusion: Mastering Proportional Relationships

In summary, to find the function that correctly determines the total cost ($y$) for buying $x$ bracelet charms, we first needed to calculate the cost of a single charm. By dividing Jharna's total expense ($13.92) by the number of charms she purchased (6), we found that each charm costs $2.32. This unit cost is the crucial factor in our function. The function **$y = 2.32x$** accurately models this scenario because it multiplies the cost per charm by the number of charms to yield the total cost. This is a prime example of a proportional relationship, a core concept in mathematics that helps us understand how quantities change together at a constant rate. Recognizing these patterns allows us to predict costs, calculate quantities, and solve a wide array of real-world problems using the power of algebra. Keep practicing these types of problems, and you'll become a whiz at translating word problems into functional equations!

For further exploration into linear functions and proportional relationships, check out resources from Khan Academy.