Question

Without actually performing the calculations indicated, tell to how many significant digits the answer to the calculation should be expressed. a. \((0.196)(0.08215)(295) /(1.1)\) b. \((4.215+3.991+2.442) /(0.22)\) c. \((7.881)(4.224)(0.00033) /(2.997)\) d. \((6.219+2.03) /(3.1159)\)

Short answer

a. 2 significant digits b. 2 significant digits c. 2 significant digits d. 2 decimal places

Step-by-step solution

Identify the operation and number of significant digits in each factor

In this calculation, we have a product of three factors, followed by a division. The individual factors have the following numbers of significant digits: - 0.196 → 3 significant digits - 0.08215 → 5 significant digits - 295 → 3 significant digits - 1.1 → 2 significant digits

Determine the number of significant digits in the final answer

According to the rules for significant digits, the final answer should have the least significant digits among the factors involved. In this case, the least number of significant digits in the given factors is 2. Therefore, the answer must be expressed to 2 significant digits. #b. \((4.215+3.991+2.442) /(0.22)\)#

Identify the operation and number of significant digits in each factor

In this calculation, we have a sum of three numbers, followed by a division. The numbers involved have the following number of significant digits: - 4.215 → 4 significant digits - 3.991 → 4 significant digits - 2.442 → 4 significant digits - 0.22 → 2 significant digits

Determine the number of significant digits in the final answer

According to the rules for significant digits, the result of the addition should have the same number of decimal places as the least precise number being added, and the result of the division should have the least significant digits among the factors involved. In this case, all three numbers being added have 3 decimal places, so their sum should have 3 decimal places. Then, since division by 0.22 has 2 significant digits, the final answer should be expressed to 2 significant digits. #c. \((7.881)(4.224)(0.00033) /(2.997)\)#

Identify the operation and number of significant digits in each factor

In this calculation, we have a product of three factors, followed by a division. The individual factors have the following numbers of significant digits: - 7.881 → 4 significant digits - 4.224 → 4 significant digits - 0.00033 → 2 significant digits - 2.997 → 4 significant digits

Determine the number of significant digits in the final answer

According to the rules for significant digits, the final answer should have the least significant digits among the factors involved. In this case, the least number of significant digits in the given factors is 2. Therefore, the answer must be expressed to 2 significant digits. #d. \((6.219+2.03) /(3.1159)\)#

Identify the operation and number of significant digits in each factor

In this calculation, we have a sum of two numbers, followed by a division. The numbers involved have the following number of significant digits: - 6.219 → 4 significant digits - 2.03 → 3 significant digits - 3.1159 → 5 significant digits

Determine the number of significant digits in the final answer

According to the rules for significant digits, the result of the addition should have the same number of decimal places as the least precise number being added (2.03 has 2 decimal places), and the result of the division should have the least significant digits among the factors involved. In this case, the sum should have 2 decimal places, and since 3.1159 has 5 significant digits, the final answer should be expressed to the minimum of these values, which is 2 decimal places.

Key concepts

Significant Figures Rules

When performing mathematical operations, it's essential to pay attention to significant figures or significant digits. These refer to all measurable digits in a number, which include all exact numbers or those you can confidently assess plus the first estimated digit. Understanding the rules for significant figures helps you report numbers with the appropriate precision.

Here are some important guidelines to consider:

• Counting Significant Figures: Non-zero digits are always significant. Any zeros located between significant digits, known as captive zeros, also count. Leading zeros (zeros in front of all non-zero numbers) are not significant. Trailing zeros can be tricky; they are significant only if a decimal point is present (such as in 2.500).
• Operations with Significant Figures: For multiplication and division, the number of significant digits in the result should be the same as in the input number with the least significant digits. For example, in multiplying 0.196 (3 digits) and 1.1 (2 digits), your answer should be limited to 2 significant digits.

Knowing and applying significant figures rules ensure calculations reflect reliable precision, preventing overstatement or misstatement of data precision.

Rounding in Calculations

Rounding is an essential part of working with significant figures because it helps to present numbers in a more practical and readable form. Rounding allows you to limit the number of details or digits you display, focusing only on the most significant ones.

Here is how you round numbers in calculations:

• Rounding Basics: When your first digit to be removed is less than 5, the preceding number stays the same. If it is 5 or higher, you increase the preceding number by 1. For example, rounding 2.256 to two decimal places results in 2.26 as the digit after the hundredths place (5) suggests an increment.
• Rounding in Operations: After addition or subtraction, you round the final result to the least number of decimal places from the numbers involved in the operation. For multiplication or division, you round based on the number with the fewest significant figures.

Proper rounding keeps your figures neat and maintains accuracy without exaggerating the precision beyond the capability of the measured numbers.

Precision in Measurement

Precision in measurement is a cornerstone of scientific practices. Precision refers to the consistency and repeatability of several measurements but not necessarily to their accuracy. It's about how closely the measurements match each other.

Fundamentally, significant figures underpin our understanding of precision by dictating how we express our data:

• Measurement and Error: Every measurement inherently carries a degree of uncertainty, which is why precision is so crucial. When noting measurements, you should indicate the degree of precision by writing numbers with appropriate significant figures.
• Use of Instruments: The more precise the measuring instrument, the more significant digits you can report. A ruler that measures to the millimeter will show more precision than one that only measures to the nearest centimeter.

Precision doesn't imply that data is correct; rather, it shows when repeat measurements consistently return similar results. Using significant figures to reflect precision helps ensure that the measurement and its recorded figures are properly aligned with the tools and methods used to obtain them.