Question

Compute and express each answer with the proper number of significant figures, rounding as necessary. a) \(56.7 \times 66.99=?\) b) \(1.000 \div 77=?\) c) \(1.000 \div 77.0=?\) d) \(6.022 \times 1.89=?\)

Short answer

a) 3800 b) 0.013 c) 0.0130 d) 11.4

Step-by-step solution

Perform Multiplication for Part (a)

First, multiply the numbers 56.7 and 66.99: \[ 56.7 \times 66.99 = 3796.233 \]. The result is 3796.233.

Determine Significant Figures for Part (a)

The numbers in the calculation 56.7 and 66.99 have 3 and 4 significant figures respectively. The result should be reported with the least number of significant figures, which is 3.

Round the Answer for Part (a)

Round 3796.233 to 3 significant figures: \[ 3800 \].

Perform Division for Part (b)

Divide 1.000 by 77: \[ \frac{1.000}{77} = 0.0129870 \].

Determine Significant Figures for Part (b)

1.000 has 4 significant figures, and 77 has 2 significant figures. The result should be reported with the least number, which is 2 significant figures.

Round the Answer for Part (b)

Round 0.0129870 to 2 significant figures: \[ 0.013 \].

Perform Division for Part (c)

Divide 1.000 by 77.0: \[ \frac{1.000}{77.0} = 0.0129870 \].

Determine Significant Figures for Part (c)

1.000 has 4 significant figures, and 77.0 has 3. The result should be reported with the least number of significant figures, which is 3.

Round the Answer for Part (c)

Round 0.0129870 to 3 significant figures: \[ 0.0130 \].

Perform Multiplication for Part (d)

Multiply 6.022 by 1.89: \[ 6.022 \times 1.89 = 11.38658 \].

Determine Significant Figures for Part (d)

6.022 has 4 significant figures and 1.89 has 3 significant figures. The result should be reported with the least number, which is 3.

Round the Answer for Part (d)

Round 11.38658 to 3 significant figures: \[ 11.4 \].

Key concepts

Rounding Numbers

Rounding numbers is a fundamental concept in dealing with measurements and calculations in science and engineering. It's about simplifying numbers to make them easier to work with, while still keeping them as accurate as possible. When you round a number, you decide which digits to keep and which ones to drop. Here's how you do it:

• Identify the digit where rounding is needed, based on the significant figures required.
• Look at the digit immediately after it. If it's 5 or greater, increase the identified digit by 1; if it's less than 5, keep the identified digit unchanged.

For example, in the exercise from part a: 3796.233 rounded to 3 significant figures becomes 3800. This is because the fourth digit after the decimal is 6, which is greater than 5, prompting an increase in the third digit (7) by one. Rounding helps maintain precision in calculations without complicating them with too many digits.

Multiplication and Division

In math and science, multiplication and division are operations that can change the number of significant figures in a result. It's important to remember that the number of decimal places isn't what matters most here; what's important is the number of significant digits.

When doing either multiplication or division:

• Multiply or divide the numbers as you normally would, ignoring significant figures for the time being.
• Review the inputs to your calculation: the number of significant digits in your answer will be the same as the number of significant figures in the least precise input.

In our exercise: - For multiplication, like in part d, 6.022 (4 sig figs) is multiplied by 1.89 (3 sig figs), and the answer should have the least number of significant figures, which is 3. - For division, like in part b, 1.000 (4 sig figs) divided by 77 (2 sig figs) should result in an answer with 2 significant figures. This ensures consistency and precision are maintained in scientific calculations.

Significant Figures Rules

Understanding significant figures rules is essential when handling scientific data because they tell us the precision of a number. These rules guide you in counting how many figures in a number are meaningful. Here's a simplified guide:

• All non-zero numbers (1 through 9) are always significant.
• Any zeros between significant digits are significant.
• Leading zeros, which appear before all non-zero digits, are never significant.
• Trailing zeros in a number with a decimal point are significant as they indicate precision.

In scientific calculations: - You make use of these rules to determine how many significant figures to keep in your results. For instance, in part c of the exercise: 1.000 has 4 significant figures (despite the zeros) because it's a precise measurement, while 77.0 has 3 (due to the decimal point). Being mindful of significant figures helps communicate the level of certainty in your measurements, which is crucial in both experimental and theoretical work.