Problem 14 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

A sum of money is rounded off to the nearest rupee. The probability that round off error is at least ten paise is (A) \(\frac{81}{100}\) (B) \(\frac{82}{101}\) (C) \(\frac{19}{100}\) (D) \(\frac{19}{101}\)

Step-by-Step Solution

Answer

The answer is (A) \(\frac{81}{100}\).

Step 1: Understand the Problem

We are asked to find the probability that the rounding error, when a sum of money is rounded off to the nearest rupee, is at least ten paise. This means we need to find cases where the rounding causes a change of at least 0.10 rupees.

Step 2: Define the Range

When rounding to the nearest rupee, the decimal portion of the money determines how rounding occurs. A sum with 0.00 to 0.49 paise will be rounded down, and 0.50 to 0.99 paise will be rounded up. Thus, the range for potential rounding errors is 0.00 paise to 0.99 paise.

Step 3: Identify Significant Error Ranges

We want the rounding error to be at least 0.10 rupees. Therefore, the rounding error is significant (at least 0.10 rupees) if the decimal is either between 0.10 to 0.49 (when rounded down) or 0.50 to 0.99 (when rounded up).

Step 4: Calculate the Probability

Count the favorable outcomes. Errors of at least 10 paise occur in two segments: 0.10 to 0.49 and 0.50 to 0.99. Both ranges have 40 outcomes each (10, 11, ..., 49 and 50, ..., 99), totaling 80 out of 100 possible outcomes (00 through 99). Thus, the probability is \(\frac{80}{100} = 0.8\).

Step 5: Match the Probability With Given Options

Compare the calculated probability \(\frac{80}{100}\) with the given options. The closest option is \(\frac{81}{100}\).

Key Concepts

Rounding Errors

Rounding errors occur when a number is adjusted to a specified precision, often to make calculations simpler and results easier to read. Generally, this happens in financial contexts, where figures are converted to the nearest whole unit, like rounding to the nearest rupee. While rounding makes numbers easier to handle, it introduces a slight error because the altered value is not identical to the original.
Errors can vary depending on the rounding method used, either rounding up or down.
In the context of probability, we examine how often and how significant these rounding discrepancies can be.

• Rounding Down: When the decimal portion is less than 0.5, the entire number moves to the nearest lower whole number.
• Rounding Up: When the decimal portion is 0.5 or greater, the entire number rises to the next whole unit.

Understanding rounding errors is crucial in financial applications to ensure accuracy and clarity in monetary transactions.

Nearest Rupee Rounding

In monetary transactions, rounding to the nearest rupee is common practice. This means adjusting the value of a sum of money to the closest whole rupee.
For example, amounts like ₹4.99 would round to ₹5, while ₹4.49 would round down to ₹4. This method helps maintain a straightforward representation of money.
The rounding process impacts calculations due to the small discrepancies introduced whenever the money is not already a whole rupee. In practical terms, this adjustment affects how cash transactions are processed or how financial records are kept.

• Range for Rounding Up: Decimal components from 0.50 to 0.99 push the number up to the next rupee.
• Range for Rounding Down: Decimal components from 0.00 to 0.49 lower the figure to the previous rupee.

Adopting consistent rounding rules provides a method to minimize discrepancies over time and ensures that all transactions or calculations abide by the same standard.

Decimal Rounding

Decimal rounding involves adjusting the decimal part of a number to enhance readability or meet specific precision requirements. Generally, decimal rounding impacts both the accuracy of figures and their subsequent use in computations. In financial arenas, it ensures figures are manageable and practical without extensive decimal places.
The rules of decimal rounding dictate whether a number's decimal component will contribute to the whole number or be omitted.
This concept is significant in estimating probability, especially when dealing with situations like rounding to the nearest rupee.

• When the decimal is below 0.5, it typically results in rounding down.
• If the decimal is 0.5 or above, it contributes to rounding up.

Focusing on decimal rounding enables us to identify likely patterns of rounding errors, calculate associated probabilities, and grasp the larger implications on calculations and data integrity.