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Chapter 20: Problem 2

In jar \(\mathrm{A}, 60 \%\) of the marbles are red and the rest are green. \(40 \%\) of the red marbles are moved to an empty jar B. \(60 \%\) of the green marbles are moved to an empty jar C. The marbles in both B and C are now moved to another empty jar D. What fraction of the marbles in jar A were moved to jar D? (A) 0.12 (B) 0.24 (C) 0.36 (D) 0.48 (E) 0.6

Short Answer

Expert verified
(D) 0.48

Step by step solution

01

Calculate the Red Marbles in Jar A

Initially, let's consider that Jar A has a total of 100 marbles. Since 60% of them are red, the number of red marbles is calculated as \(0.6 \times 100 = 60\).
02

Calculate the Green Marbles in Jar A

The rest of the marbles in Jar A are green. Since 60% are red, the remaining 40% must be green. Therefore, the number of green marbles is \(0.4 \times 100 = 40\).
03

Move Red Marbles to Jar B

From the red marbles, 40% are moved to Jar B. The number of red marbles moved is \(0.4 \times 60 = 24\).
04

Move Green Marbles to Jar C

From the green marbles, 60% are moved to Jar C. The number of green marbles moved is \(0.6 \times 40 = 24\).
05

Total Marbles Moved to Jar D

Jar D will contain all the marbles from Jars B and C. This means the total number of marbles in Jar D is the sum of those moved from Jar B and Jar C, which is \(24 + 24 = 48\).
06

Compute Fraction of Marbles Moved to Jar D

Since Jar A initially had 100 marbles, the fraction that was moved to Jar D is calculated as \(\frac{48}{100} = 0.48\).
07

Conclude with the Correct Fraction

Therefore, the fraction of the marbles in Jar A moved to Jar D is \(0.48\), which corresponds to the answer choice D.

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Key Concepts

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

Probability
Probability is a fascinating concept that often appears in GMAT math problems. It involves predicting how likely an event is to happen. When solving probability questions, you often work out percentages and fractions to express how possible an outcome is. In the marble problem, if we think of the movement from one jar to another as an event, we use percentages to express these movements. For example:
  • 40% of the red marbles are moved to a different jar, describing the chance any particular red marble ends up in a new jar.
  • 60% of the green marbles are transferred, highlighting the probability for each green marble.
Understanding probability in questions like these help in predicting outcomes based on given percentages. It's about calculating and knowing how changes affect the whole system in terms of likelihood and distribution. Remember, percentage is just another way to express probability.
Word Problems
Word problems are all about translating words into mathematical expressions or equations. They involve understanding a scenario described by words and then breaking it down into numbers and operations. This exercise demonstrates how essential it is to carefully translate written information (like percentages) into mathematical operations:
  • Recognize the initial conditions (60% of the marbles are red, 40% are green).
  • Identify the actions described (40% of red moved, 60% of green moved).
  • Extract necessary mathematical components from sentences.
Word problems test your ability to interpret and process information logically, turning it into a numerical format ready for calculation. It鈥檚 about connecting the dots from a textual problem to actionable equations.
Fractions
Fractions are a crucial part of mathematics, especially in solving GMAT problems. A fraction tells us how much of a whole we have or need. In the context of this exercise, fractions were used to express:
  • The parts of marbles moved (e.g., \(\frac{40}{100}\) represents 40% of marbles moved to jar C).
  • The solution requires understanding how these fractions combine and represent parts of a whole system (like jar A).
  • Finally, the fraction \(\frac{48}{100}\) shows the total marbles that ended up in jar D from the original jar.
Fractions make sense of parts of a whole. They are essential tools in breaking down complex problems into manageable parts. Always remember to simplify fractions obtained in calculations for accuracy and clarity.
Problem Solving
Problem-solving in mathematics involves more than just calculations; it鈥檚 also about strategy. As you can see from the step-by-step solution of the exercise, problem-solving consists of:
  • Breaking the problem into smaller, manageable steps (like calculating number of red and green marbles first).
  • Following logical progression (moving from one step to another smoothly to get accurate results).
  • Verifying and interpreting your results, ensuring they make sense based on the context.
By approaching problems methodically, you prevent errors and streamline thought processes. Every problem is a challenge that requires critical thinking and strategic planning for an effective solution.

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