Problem 50 - A Complete Resource Book in Chemistry for JEE Main

Question

The number of significant figures in \(306.45\) and 40440 are respectively (a) 4,5 (b) 5,5 (c) 5,4 (d) 4,6

Step-by-Step Solution

Answer

Option (c) 5,4 is correct.

Step 1: Identify Significant Figures for 306.45

The number 306.45 is a decimal number. To identify the significant figures, first consider that all non-zero digits are significant. Additionally, any zeros between non-zero digits or after the decimal point are also significant. Therefore, in 306.45, all the digits are significant: 3, 0, 6, 4, and 5. Thus, there are 5 significant figures in 306.45.

Step 2: Identify Significant Figures for 40440

In a whole number like 40440, the rules for identifying significant figures differ slightly compared to decimals. All non-zero digits are significant. Zeros between non-zero digits are significant, but trailing zeros in a whole number without a decimal point are not considered significant. For 40440, the digits 4, 0, 4, 4 are counted as significant, and the trailing 0 is not significant. Therefore, there are 4 significant figures in 40440.

Step 3: Determine the Correct Option

From the calculations, we have determined that 306.45 has 5 significant figures and 40440 has 4 significant figures. The option that matches these numbers of significant figures is (c) 5,4.

Key Concepts

Understanding Decimal Numbers and Significant Figures

Decimal numbers are numerical values that contain a decimal point, representing fractions of a whole number. When dealing with decimal numbers, identifying significant figures involves understanding which digits contribute to the precision of the number. In the case of the number 306.45, all digits are considered significant. This is because:

• All non-zero digits are significant.
• Any zeros placed between two non-zero digits are also significant.
• Zeros that come after the decimal point and after a non-zero digit are significant, as they contribute to the number's precision.

Therefore, in 306.45, each number (3, 0, 6, 4, and 5) is part of the significant figures, resulting in five significant figures altogether. The presence of a decimal point solidifies the significance of zeros within the number.

The Concept of Whole Numbers in Significant Figures

Whole numbers are simple integers without fractional or decimal components. Counting significant figures in whole numbers can be straightforward but also a bit tricky due to certain rules about zeros. For the number 40440:

• All non-zero digits are always significant.
• Zeros that are sandwiched between non-zero digits are significant, as they indicate precise measurements.
• Trailing zeros (zeros at the end) in whole numbers without a decimal point are not considered significant. They are often merely placeholders.

Thus, in the number 40440, the significant digits are 4, 0, 4, 4, making for only four significant figures. The trailing zero does not add to the precision of the measurement because there's no decimal point indicating its necessity for precision.

Identifying Trailing Zeros and Their Significance

Trailing zeros are zeros located at the end of a number. Their significance in contributing to the number's precision depends on whether the number is a decimal or a whole number:

• In decimal numbers, trailing zeros are significant when they come after the decimal point. This is because they indicate a specific level of precision. For instance, in 306.450, the trailing zero suggests precision to the thousandths place, hence it is counted as significant.
• In whole numbers, without a decimal point, trailing zeros are not usually considered significant. They serve as placeholders indicating the scale of the number rather than the precision. So, in 40440, the trailing zero is not counted as significant.

This distinction is critical when calculating significant figures, as it affects the accuracy and precision of numerical data in scientific and mathematical contexts.