Solving Mixture Problems: A Step-by-Step Guide

by SLV Team 47 views

Hey guys! Let's dive into a classic mixture problem. We've got a container, some liquids, and a bit of ratio fun. The goal is to figure out the initial amount of liquid A. Don't worry, it's not as scary as it sounds. We'll break it down step by step, making sure you understand every bit of the solution. So, grab your pencils, and let's get started!

Understanding the Problem: Mixture Ratios

Mixture problems are like puzzles. You're given information about the proportions of different components in a mixture, and you have to use that information to find an unknown quantity. In this case, we know the initial ratio of liquids A and B, how much of the mixture is removed, and the final ratio after some liquid B is added. The key to solving these types of problems is to set up the correct equations and use a bit of algebra. The problem is a word problem, so first you must understand the problem itself.

Let's break down the question: A container has a mix of liquids A and B, and the ratio of these liquids is initially 5:6. This means that for every 5 parts of liquid A, there are 6 parts of liquid B. When 66 liters of the mix is removed and replaced with liquid B, the ratio changes to 5:9. We are asked to determine the original amount of liquid A in the container.

Setting up the Initial Conditions

Initially, let's represent the quantities of liquids A and B as 5x and 6x, respectively. The total initial mixture is then 5x + 6x = 11x. The problem gives us the total amount removed from the container is 66 liters. When 66 liters of the mixture is removed, the proportion of A and B in the removed portion remains the same as in the original mixture. That's because when you take some of the mixture away, you're taking a little bit of both liquids A and B, in the same ratio as they appear in the mixture.

After Removing the Mixture and Replacing

When we draw off 66 liters of the mixture, the ratio of A and B in the removed mixture is the same as the original ratio of 5:6. Since the total initial mixture is 11x, the fraction of A in the mixture is 5/11 and the fraction of B is 6/11. The amount of A removed is (5/11) * 66 and the amount of B removed is (6/11) * 66.

After removing the mixture, the remaining amount of A is 5x - (5/11) * 66 and the remaining amount of B is 6x - (6/11) * 66. Then, 66 liters of liquid B is added to the container. The amount of A remains unchanged, and the new amount of B is (6x - (6/11) * 66) + 66. The new ratio of A to B is 5:9. This ratio gives us our second equation.

Solving the Problem: Step-by-Step

Step 1: Define Variables

Let's start by defining our variables. Let:

  • A be the initial quantity of liquid A.
  • B be the initial quantity of liquid B.
  • The initial ratio of A:B is 5:6.
  • The total initial quantity of the mixture is 5x + 6x = 11x.

Step 2: Set Up Equations

Based on the information given, we can form two key equations:

  1. Initial Ratio: A / B = 5 / 6 . This tells us the initial proportions of the liquids.

  2. After Replacement: When 66 liters are removed, and replaced with liquid B, the ratio becomes 5:9. This is the crucial part that changes things. We need to account for the removal and the addition. Let's break this down further.

    • Removed Amounts: In the 66 liters removed, the amount of A is (5/11) * 66 = 30 liters, and the amount of B is (6/11) * 66 = 36 liters.
    • Amounts After Removal: The remaining amount of A is A - 30 and the remaining amount of B is B - 36.
    • Amounts After Replacement: We replace the 66 liters with liquid B, so the new amount of B becomes B - 36 + 66 = B + 30.
    • New Ratio Equation: The new ratio of A to B is 5:9, which gives us the equation: (A - 30) / (B + 30) = 5 / 9.

Step 3: Solve the Equations

Now, let's solve these equations. We have two equations:

  1. A / B = 5 / 6 , so A = (5/6) * B
  2. (A - 30) / (B + 30) = 5 / 9

Substitute A from equation 1 into equation 2:

((5/6) * B - 30) / (B + 30) = 5 / 9

Cross-multiply to get rid of the fractions:

9 * ((5/6) * B - 30) = 5 * (B + 30)

Simplify:

(15/2) * B - 270 = 5B + 150

Multiply the entire equation by 2 to remove the fraction:

15B - 540 = 10B + 300

Rearrange the equation:

15B - 10B = 300 + 540

5B = 840

B = 840 / 5

B = 168 liters

Now, substitute the value of B back into the equation A = (5/6) * B:

A = (5/6) * 168

A = 140 liters

Step 4: Find the Initial Quantity of Liquid A

We found that the initial quantity of liquid A (A) is 140 liters. This is the answer to the question.

Conclusion: Mixture Mastery!

Great job, everyone! We've successfully navigated a mixture problem. Remember, the key is to break down the problem into smaller, manageable steps. By carefully defining your variables, setting up the equations, and solving them systematically, you can tackle any mixture problem that comes your way. Keep practicing, and you'll become a mixture master in no time!

Recap

  • Initial setup: Understand the initial ratio and the total volume of the mixture.
  • Removal: Calculate the amounts of each liquid removed based on the initial ratio and the volume removed.
  • Replacement: Account for the addition of the new liquid.
  • New Ratio: Use the new ratio to set up a second equation.
  • Solve: Solve the equations to find the unknowns.

This method can be applied to many mixture problems, so keep practicing to get comfortable with the steps. You've got this, guys!