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Chapter 1: Problem 26

Suppose you have one quart of water/juice mix that is \(50 \%\) juice, and you add 2 quarts of juice. What percent juice is the final mix?

Short Answer

Expert verified
The final mix is 83.33% juice.

Step by step solution

01

Calculate the amount of juice in original mix

The original mix contains 1 quart of a water/juice blend that is 50% juice. Therefore, the amount of pure juice in the original mix is \( 0.5 \times 1 = 0.5 \) quarts of juice.
02

Calculate the total amount of juice added

You add 2 quarts of pure juice to the original blend. This makes the total amount of juice added \( 2 \) quarts.
03

Find the total juice in the final mix

Add the juice from the original mix to the juice added: \( 0.5 + 2 = 2.5 \) quarts of juice.
04

Determine the total volume of the final mix

The initial volume was 1 quart, and you added 2 quarts of juice, making the total volume \( 1 + 2 = 3 \) quarts.
05

Calculate the percentage of juice in the final mix

Divide the total amount of juice by the total volume and multiply by 100 to get the percentage: \( \frac{2.5}{3} \times 100 = 83.33\% \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mixture Problems
Mixture problems are a common type of problem in mathematics that involve combining two or more substances to form a new substance with particular properties. These problems often require you to determine the concentration or proportion of one component in the mixture.
Let's take the example provided: we have a water/juice mix and then add more juice to it.
  • The initial mixture is 50% juice, which means out of 1 quart, half of that quart is just juice.
  • By adding more juice, the task is to figure out what percentage the juice makes of the new total mixture.
Understanding how the properties of each component interact is essential for solving these problems. Always make sure to maintain an organized approach to solving mixture problems by identifying and isolating information about each component. This will make it easier to track the changes step by step.
Fraction to Percentage Conversion
Converting a fraction to a percentage is a crucial mathematical skill, especially when solving problems that deal with proportions.
To make this conversion, you essentially express a fraction as a pattern out of 100, since a percentage is a fraction with a denominator of 100.
  • First, you take your fraction, such as in the final example where you have a fraction of juice to the total mixture, which is \( \frac{2.5}{3} \).
  • Then, you multiply it by 100 to convert this fraction into a percentage.
The calculation would look like this: \( \frac{2.5}{3} \times 100 \). This conversion simplifies understanding proportions in everyday context, like how much of the final mix is actually juice, as in our exercise where the final mix results in approximately 83.33% juice.
Problem Solving Steps
Problem solving is a systematic process. It often involves breaking down a complex problem into smaller, more manageable parts.
With percentage and mixture problems, following a structured approach is key.
  • Start by identifying all relevant data. Know the initial volumes and concentrations.
  • Apply logical mathematical operations to transform initial data into usable information. This includes calculating initial quantities of specific components such as juice in the original blend.
  • Add any additional components, meanwhile noting the changes in total volume.
  • Execute the calculation step to find the new concentration, typically converting a fraction into a percentage.
Following these steps, not only clarifies each part of the problem but also ensures you aren't misled by complex wording or details. This approach avoids errors and makes solving similar problems in the future easier and faster.

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Most popular questions from this chapter

Estimate the values by making reasonable approximations for unknown values, or by doing some research to find reasonable values. How much does the water in a 6 -person hot tub weigh?

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The store is selling lemons at 2 for \(\$ 1 .\) Each yields about 2 tablespoons of juice. How much will it cost to buy enough lemons to make a 9 -inch lemon pie requiring \(1 / 2\) cup of lemon juice?

Solve: \(\frac{2}{5}=\frac{6}{x}\)
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