Problem 63 - A Complete Resource Book in Chemistry for JEE Main

Question

In the final answer of the expression \(\frac{(29.2-20.2)\left(1.79 \times 10^{5}\right)}{1.37}\) The number of significant figures is (a) 2 (b) 4 (c) 6 (d) 7

Step-by-Step Solution

Answer

(a) 2

Step 1: Understand the Problem

We are given an expression that involves subtraction, multiplication, and division. We need to determine the number of significant figures in the final answer.

Step 2: Identify Significant Figures in Each Component

- In the subtraction part \((29.2 - 20.2)\), each number has 3 significant figures (because of one decimal place). - In the scientific notation \(1.79 \times 10^5\), there are 3 significant figures (\(1.79\)). - In the division by \(1.37\), there are 3 significant figures.

Step 3: Perform Arithmetic Operations with Attention to Significant Figures

- Compute the subtraction: \(29.2 - 20.2 = 9.0\) (resulting in 2 significant figures due to the decimal).- Multiply: \(9.0 \times 1.79 \times 10^5\) results in \(1.611 \times 10^6\) with 2 significant figures because the limiting factor is \(9.0\) (2 significant figures).- Divide: \(1.611 \times 10^6 / 1.37\) results in \(1.176 \times 10^6\) rounded to 2 significant figures, as 2 is the smallest number of significant figures from our input values.

Step 4: Determine the Number of Significant Figures

The limiting term throughout the arithmetic operations had 2 significant figures (from the subtraction result \(9.0\)), therefore, the final answer should also be expressed with 2 significant figures.

Key Concepts

Scientific Notation

Scientific notation is a method of expressing very large or very small numbers in a compact form that is easy to work with, especially in calculations. It is often used in science and engineering to ensure that calculations remain precise and manageable.

In scientific notation, a number is written as the product of a number (between 1 and 10) and a power of ten. For instance, the number 179,000 can be written as \( 1.79 \times 10^5 \). Here, \( 1.79 \) is the significant part, while \( 10^5 \) indicates that the decimal point is moved five places to the right.

• This format makes it simple to compare and perform operations, particularly when dealing with different magnitudes or units.
• It reduces large computations into more manageable steps by focusing on the significant digits.

By using scientific notation, you can ensure that all calculations respect the precision of the numbers involved, preserving the number of necessary significant figures.

Arithmetic Operations

Arithmetic operations are basic calculations that include addition, subtraction, multiplication, and division. Each of these operations can have an impact on the number of significant figures that should be used in the final result.

When performing operations like multiplication and division, the rule is to express the final answer using the smallest number of significant figures from the involved numbers.

• When multiplying \(9.0\) by \(1.79 \times 10^5\), the number of significant figures to keep in mind is the smaller count from \(9.0\) which is 2.
• For division, such as \(\frac{1.611 \times 10^6}{1.37}\), the significant figures of \(1.37\) (which is 3) would typically dictate the outcome, but since 2 significant figures arise from earlier calculations, that limits the final answer.

Following these guidelines helps maintain the integrity and precision of your results in scientific work by preventing errors that can arise from inappropriate rounding or excessive approximation.

Subtraction in Calculations

Subtraction is one of the basic arithmetic operations and plays a critical role in determining significant figures for a subsequent calculation. In subtraction, the number of decimal places in the result should match the number with the least decimal places from the numbers being subtracted.

For instance, in the calculation \( 29.2 - 20.2 \), both numbers have one decimal place. Therefore, the result \(9.0\) is expressed with one decimal place as well—this results in it having 2 significant figures.

• This initial step limits the number of significant figures throughout the entire chain of operations that follows.
• It is crucial to carry these figures correctly as they form the basis of subsequent calculations, affecting outcomes in multiplication and division.

Knowing the rules for subtraction ensures that any inaccuracies are minimized and that scientific data remains reliable and precise across all forms of calculation.