Solving Proportion Problems Finding The Cost Of 3.5 Pounds Of Grapes

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In this comprehensive guide, we'll dive deep into solving a common mathematical problem involving ratios and proportions: determining the cost of a specific quantity of grapes given the price per pound. We'll break down the problem step by step, explore the underlying concepts, and explain why a particular equation correctly represents the relationship between the quantities involved.

Problem Statement

The problem states: One pound of grapes costs $1.55. Which equation correctly shows a pair of equivalent ratios that can be used to find the cost of 3.5 pounds of grapes?

We are given two options:

A. $\frac{1.55}{1}=\frac{x}{3.5}$

B. $\frac{1.55}{1}=\frac{3.5}{x}$

Our goal is to identify the equation that accurately sets up the proportional relationship needed to calculate the cost of 3.5 pounds of grapes.

Understanding Ratios and Proportions

Before we delve into the solution, let's refresh our understanding of ratios and proportions.

Ratios

A ratio is a comparison of two quantities. It can be expressed in several ways, including:

  • Using a colon: a:b
  • As a fraction: a/b
  • Using the word "to": a to b

In our problem, the given information tells us the ratio of the cost of grapes to the weight of grapes: $1.55 for 1 pound. This can be written as the ratio 1.55/1.

Proportions

A proportion is a statement that two ratios are equal. It expresses the idea that two relationships are equivalent. For example, if we know that 2 apples cost $1, then we can set up a proportion to find the cost of 4 apples. The proportion might look like this: $\frac{2 \text{ apples}}{$1} = \frac{4 \text{ apples}}{$x}$, where x represents the unknown cost.

Setting Up the Proportion for the Grape Problem

The key to solving proportion problems lies in setting up the ratios correctly. We need to ensure that the corresponding quantities are in the same positions in both ratios. In our grape problem, we have the following information:

  • Cost of 1 pound of grapes: $1.55
  • Weight of grapes we want to find the cost for: 3.5 pounds
  • Unknown cost of 3.5 pounds of grapes: x (which we need to find)

We can set up a ratio comparing the cost to the weight: $\frac{\text{Cost}}{\text{Weight}}$

Using the given information, we have one complete ratio: $\frac{1.55}{1 \text{ pound}}$

We want to find the cost (x) of 3.5 pounds of grapes, so our second ratio will be: $\frac{x}{3.5 \text{ pounds}}$

Now, we can set up a proportion by equating these two ratios:

1.551=x3.5\frac{1.55}{1} = \frac{x}{3.5}

Analyzing the Options

Let's revisit the options provided in the problem:

A. $\frac{1.55}{1}=\frac{x}{3.5}$

B. $\frac{1.55}{1}=\frac{3.5}{x}$

Comparing our correctly set up proportion with the options, we can see that option A matches our equation perfectly. It correctly places the cost ($1.55 and x) in the numerators and the corresponding weights (1 pound and 3.5 pounds) in the denominators.

Option B, on the other hand, incorrectly places the weight (3.5 pounds) in the numerator of the second ratio and the unknown cost (x) in the denominator. This arrangement does not represent the proportional relationship between the cost and weight of the grapes.

Solving the Proportion

Although the problem only asks for the correct equation, let's briefly look at how we would solve the proportion to find the actual cost of 3.5 pounds of grapes.

To solve the proportion $ rac{1.55}{1} = \frac{x}{3.5}$, we can use cross-multiplication. This involves multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa, and then setting the two products equal to each other:

  1. 55 * 3.5 = 1 * x

  2. 425 = x

Therefore, x = $5.425. Since we're dealing with money, we would typically round this to two decimal places, giving us $5.43. So, 3.5 pounds of grapes would cost $5.43.

Why Option A is Correct

Option A is the correct equation because it accurately represents the proportional relationship between the cost and weight of the grapes. It sets up the ratios consistently, with the cost in the numerator and the corresponding weight in the denominator for both ratios. This arrangement allows us to correctly solve for the unknown cost (x) using the principles of proportions.

Common Mistakes to Avoid

When setting up proportions, it's easy to make mistakes if you're not careful. Here are some common pitfalls to avoid:

  1. Inconsistent Placement of Quantities: Make sure that corresponding quantities are in the same position in both ratios. For example, if you have cost in the numerator of the first ratio, you must also have cost in the numerator of the second ratio.

  2. Mixing Up Units: Ensure that the units are consistent within each ratio. For instance, if you're comparing dollars to pounds, make sure both ratios use dollars and pounds.

  3. Misinterpreting the Problem: Carefully read and understand the problem statement before setting up the proportion. Identify what quantities are known and what quantity you need to find.

Real-World Applications of Proportions

Proportions are a fundamental concept in mathematics with numerous real-world applications. They are used in:

  • Cooking: Scaling recipes up or down while maintaining the correct ratios of ingredients.
  • Map Reading: Calculating distances on a map based on the map scale.
  • Construction: Determining the correct amounts of materials needed for a project.
  • Finance: Calculating interest rates and currency exchange rates.
  • Science: Converting units of measurement and analyzing experimental data.

By mastering the concept of proportions, you gain a valuable tool for solving a wide range of problems in everyday life and various professional fields.

Conclusion

In conclusion, the correct equation to find the cost of 3.5 pounds of grapes, given that one pound costs $1.55, is option A: $\frac{1.55}{1}=\frac{x}{3.5}$. This equation accurately represents the proportional relationship between the cost and weight of the grapes. By understanding the principles of ratios and proportions, you can confidently solve similar problems and apply these concepts in various real-world scenarios. Remember to set up your ratios consistently, ensuring that corresponding quantities are in the same positions, and avoid common mistakes like mixing up units or misinterpreting the problem statement. With practice, you'll become proficient in using proportions to solve a wide range of problems.