Problem 14 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

Among 600 bolts, \(20 \%\) are very large \(10 \%\) are very small and the remaining are useful. One bolt is chosen at random. The probability that it is a useful bolt is (a) \(1 / 10\) (b) \(3 / 10\) (c) \(7 / 10\) (d) \(8 / 10\)

Step-by-Step Solution

Answer

The probability of picking a useful bolt is \( \frac{7}{10} \), option (c).

Step 1: Understand the Problem

You are given that there are 600 bolts in total with 20% being very large and 10% being very small. We want to find the probability of randomly picking a useful bolt.

Step 2: Calculate the Number of Non-Useful Bolts

First, calculate how many bolts are very large. Since 20% of 600 bolts are very large, we have: \[ 0.20 \times 600 = 120 \text{ bolts} \]Similarly, calculate how many bolts are very small. Since 10% of 600 bolts are very small, we have: \[ 0.10 \times 600 = 60 \text{ bolts} \]

Step 3: Find the Number of Useful Bolts

Subtract the number of very large and very small bolts from the total. The useful bolts are the remaining ones. So, \[ 600 - 120 - 60 = 420 \text{ useful bolts} \]

Step 4: Calculate the Probability of Picking a Useful Bolt

The probability is calculated as the ratio of useful bolts to the total number of bolts.\[ P(\text{useful bolt}) = \frac{\text{Number of useful bolts}}{\text{Total number of bolts}} = \frac{420}{600} \]This simplifies to:\[ P(\text{useful bolt}) = \frac{7}{10} \]

Step 5: Choose the Correct Answer Option

Compare the probability \( \frac{7}{10} \) with the given options. The correct answer that matches this probability is option (c).

Key Concepts

Percentage Problems

Percentage problems are a common element of many mathematical calculations. Understanding percentages means being able to translate a portion of a whole into a numeric form. This is useful in various applications such as finance, statistics, and probability. For example, in our problem about bolts, knowing that 20% of the bolts are very large allows us to easily calculate their exact number. To do this, we multiply the percentage (in decimal form) by the total number of items. In this case:

• Convert 20% to 0.20.
• Multiply 0.20 by 600 to find the number of very large bolts: 120 bolts.

The same method applies to other percentages, like finding the 10% of very small bolts by multiplying 0.10 by 600, resulting in 60 bolts. This simple yet powerful method helps in breaking down complex percentage problems into manageable calculations.

Random Selection

Random selection is a fundamental concept in probability. It refers to choosing an item from a set in a manner where every item has an equal chance of being selected. In our bolt example, when we pick a bolt at random, it means any of the 600 bolts have an equal chance of being chosen.
This highlights the concept of probability in action. If we want to calculate the likelihood of picking a specific item type, such as a useful bolt, we need to understand how many of those items exist compared to the total.
Random selection ensures that the process is fair and unbiased, which is a key aspect of many real-world applications, including quality control tests and surveys.

Useful Bolts

In mathematical problems involving classification, identifying the useful elements among a group can often be the goal. Here, useful bolts are defined as those that are neither very large nor very small. By determining the number of non-useful bolts first, we can easily find the count of useful bolts.
After calculating that 120 bolts are very large and 60 bolts are very small, we realize the remaining bolts (i.e., 420 out of 600) are useful. This type of categorization helps in organizing and solving problems by focusing on desired elements.

• Subtract the sum of very large and very small bolts from the total.
• The remaining count is the number of useful bolts.

This strategic approach allows for efficient computation of probabilities and informed decision-making.

Mathematical Reasoning

Mathematical reasoning involves critical thinking and logic to solve problems step-by-step. It’s about interpreting data, using appropriate methods, and confirming results. In the bolts problem, each step follows logically from the last:

• We calculate the non-useful bolt counts using percentages.
• We determine the useful bolt count by subtraction.
• We find the probability of picking a useful bolt by dividing the number of useful bolts by the total count.

These steps demonstrate a structured approach to problem-solving that can be applied universally. When tackling similar exercises, always break down the problem, calculate methodically, and verify against your possible answers. By developing these skills, mathematical reasoning becomes a powerful tool for both academic and practical challenges.