Calculating Broth Concentration After Adding Water To Soup A Step-by-Step Guide

Introduction

Hey guys! Today, we're diving into a classic mixture problem that involves calculating the concentration of broth in a soup after adding water. This type of problem often appears in math classes and even in everyday cooking scenarios. Understanding how to solve these problems can be super useful. We'll break down the problem step by step, making sure everyone gets the hang of it. So, let's get started and figure out how adding water affects the broth concentration in our soup!

Problem Statement

We're starting with a 50-liter pot of soup that is 50% broth. Then, we add 10 liters of pure water to the pot. Our goal is to figure out the new ratio of broth to the total solution. We'll use a table to organize our information and set up an equation to solve for the new concentration. This is a pretty common scenario, and understanding how to tackle it will help in many similar situations. So, let's jump into the details and work through the solution together!

Setting up the Table

To solve this problem effectively, we'll use a table to organize the information. This helps us visualize the different components and how they change when we add water. The table will have columns for the amount of solution, the percentage of broth, and the amount of broth. We'll have rows for the initial state, the amount of water added, and the final state. This structured approach makes it easier to track the changes and set up our equation. Let's create the table and fill in the known values.

Initial State

Initially, we have 50 liters of soup, and 50% of it is broth. To find the amount of broth, we multiply the total volume by the percentage of broth. So, 50 liters * 50% = 25 liters of broth. We can fill this information into our table. Having a clear starting point is crucial for solving mixture problems. It gives us a solid foundation to build on as we add more ingredients. Let's make sure this initial state is crystal clear before moving on.

Water Added

Next, we add 10 liters of pure water. This means the percentage of broth in the added water is 0%, and the amount of broth added is 0 liters. Adding pure water dilutes the soup, which will change the overall concentration of the broth. It's important to recognize that adding water increases the total volume but doesn't change the amount of broth. This is a key concept in mixture problems. We'll update our table with this information, noting the increase in total volume and the constant amount of broth.

Final State

After adding the water, the total volume of the solution is 50 liters (initial soup) + 10 liters (water) = 60 liters. The amount of broth remains the same at 25 liters because we only added water. Now, we need to find the new percentage of broth in the final solution. This is where our equation will come in handy. By organizing our information in the table, we've made it much easier to see the relationships between the volumes and concentrations. Let's move on to setting up the equation to calculate the final broth percentage.

Setting up the Equation

To find the new ratio of broth to solution, we need to divide the amount of broth by the total volume of the solution. In our case, we have 25 liters of broth and 60 liters of total solution. So, the new ratio is 25 liters / 60 liters. This fraction represents the proportion of broth in the final mixture. We can express this ratio as a percentage to better understand the concentration of broth. This step is crucial in solving mixture problems, as it directly gives us the final concentration we're looking for. Let's simplify this fraction and convert it to a percentage.

Calculating the New Ratio

So, the new ratio of broth to solution is 25 liters / 60 liters. To simplify this fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 5. This gives us 5 / 12. This simplified fraction represents the proportion of broth in the final solution. Now, let's convert this fraction to a percentage to make it easier to understand the concentration. Converting to a percentage involves multiplying the fraction by 100. This will give us a clear picture of how diluted the soup has become.

Converting to Percentage

To convert the fraction 5/12 to a percentage, we multiply it by 100. (5/12) * 100 = 41.67%. So, the final solution is approximately 41.67% broth. This means that after adding 10 liters of water, the concentration of broth in the soup decreased from 50% to 41.67%. This is a significant change and highlights how adding water can dilute a mixture. Understanding this percentage helps us in real-world scenarios, like adjusting recipes or understanding chemical concentrations. Let's summarize our findings and conclude the problem.

Final Answer

After adding 10 liters of water to the 50-liter pot of soup, the new concentration of broth is approximately 41.67%. We found this by organizing the information in a table, calculating the total volume and the amount of broth, and then finding the new ratio. This problem demonstrates a classic mixture problem, and the steps we followed can be applied to similar scenarios. Whether you're cooking in the kitchen or solving math problems, understanding these concepts is super helpful. Great job working through this problem with me, guys! You've now got another tool in your math arsenal.

Table Summary

Keywords Explanation

Repair Input Keyword

The repair input keyword focuses on understanding the equation that relates the new ratio of broth to the solution after adding water. It emphasizes the process of calculating the final concentration of the broth. This involves setting up the equation correctly, which is crucial for solving mixture problems accurately. The explanation covers how to determine the new concentration by dividing the amount of broth by the total volume of the solution, and then converting it to a percentage. This concept is fundamental in understanding how dilutions affect concentrations.

Mathematics

The mathematics keyword highlights the mathematical principles involved in solving the problem. This includes understanding percentages, ratios, and proportions. Mixture problems like this one are a common application of these mathematical concepts. The solution demonstrates how to use these principles to calculate the new concentration of a solution after adding a diluent. This is applicable in various fields, including chemistry, cooking, and even everyday situations where you need to adjust mixtures. By understanding the math behind it, you can confidently solve similar problems.