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Chapter 2: Problem 27

Perform each of the following calculations, and give an answer with the correct number of significant figures: a. \(45.7 \times 0.034\) b. \(0.00278 \times 5\) c. \(\frac{34.56}{1.25}\) d. \(\frac{(0.2465)(25)}{1.78}\) e. \(\left(2.8 \times 10^{4}\right)\left(5.05 \times 10^{-6}\right)\) f. \(\frac{\left(3.45 \times 10^{-2}\right)\left(1.8 \times 10^{5}\right)}{\left(8 \times 10^{3}\right)}\)

Short Answer

Expert verified
a. 1.6, b. 0.01, c. 27.6, d. 3.5, e. 0.14, f. 0.8.

Step by step solution

01

Calculate 45.7 times 0.034

Multiply the numbers: \( 45.7 \times 0.034 = 1.5538 \)Identify the number of significant figures in each factor. 45.7 has 3 significant figures, and 0.034 has 2 significant figures. So, the result should have 2 significant figures. Therefore, 1.5538 rounds to 1.6.
02

Calculate 0.00278 times 5

Multiply the numbers: \( 0.00278 \times 5 = 0.0139 \)Identify the number of significant figures in each factor. 0.00278 has 3 significant figures, and 5 has 1 significant figure. So, the result should have 1 significant figure. Therefore, 0.0139 rounds to 0.01.
03

Calculate 34.56 divided by 1.25

Divide the numbers: \( \frac{34.56}{1.25} = 27.648 \)Identify the number of significant figures in each factor. 34.56 has 4 significant figures, and 1.25 has 3 significant figures. So, the result should have 3 significant figures. Therefore, 27.648 rounds to 27.6.
04

Calculate (0.2465) times 25 divided by 1.78

First multiply 0.2465 by 25: \( 0.2465 \times 25 = 6.1625 \)Then divide by 1.78: \( \frac{6.1625}{1.78} = 3.4626404494 \) Identify the number of significant figures. 0.2465 has 4 significant figures, 25 has 2 significant figures, and 1.78 has 3 significant figures. So, the result should have 2 significant figures. Therefore, 3.4626404494 rounds to 3.5.
05

Calculate (2.8 times 10^4) times (5.05 times 10^-6)

Multiply the numbers \( (2.8 \times 10^4) \times (5.05 \times 10^{-6}) = 2.8 \times 5.05 \times 10^{4+(-6)} = 14.14 \times 10^{-2} = 0.1414 \)Identify the number of significant figures in each factor. 2.8 has 2 significant figures, and 5.05 has 3 significant figures. So, the result should have 2 significant figures. Therefore, 0.1414 rounds to 0.14.
06

Calculate (3.45 times 10^-2) times (1.8 times 10^5) divided by (8 times 10^3)

First multiply (3.45 times 10^-2) by (1.8 times 10^5): \( (3.45 \times 10^{-2}) \times (1.8 \times 10^5) = 6.21 \times 10^{3} \)Then divide by (8 times 10^3): \( \frac{6.21 \times 10^3}{8 \times 10^3} = 0.77625 = 7.7625 \times 10^{-1} \)Identify the number of significant figures in each factor. 3.45 has 3 significant figures, 1.8 has 2 significant figures, and 8 has 1 significant figure. So, the result should have 1 significant figure. Therefore, 0.77625 rounds to 0.8.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Scientific Notation
Scientific notation is a way of expressing very large or very small numbers in a compact form. It consists of a number between 1 and 10 multiplied by a power of 10. For example, the number 5,000 can be written as \(5 \times 10^{3} \). This method makes it easier to handle and calculate with such numbers.It is particularly useful in science and engineering where we frequently encounter extremely large or tiny numbers. To convert a number into scientific notation:
  • Move the decimal point so that there's only one non-zero digit before it.
  • Count the number of places the decimal point has moved; this becomes the exponent of 10.
  • If you moved the decimal to the left, the exponent is positive; if you moved it to the right, the exponent is negative.
Multiplication
When multiplying numbers, we follow a few essential steps to ensure accuracy and the correct number of significant figures. For instance, multiplying 45.7 by 0.034:
  • First perform the multiplication: \(45.7 \times 0.034 = 1.5538 \)
  • Determine significant figures in each factor: 45.7 has 3 significant figures, and 0.034 has 2 significant figures.
  • The result should have the same number of significant figures as the factor with the fewest significant figures, which in this case is 2 significant figures.
  • Therefore, round 1.5538 to 1.6.
Whenever multiplying numbers in scientific notation, multiply the significant figures separately from the powers of 10, then combine them.
Division
Division follows similar rules to multiplication when it comes to maintaining significant figures. Consider dividing 34.56 by 1.25:
  • Perform the division: \( \frac{34.56}{1.25} = 27.648 \)
  • Determine the significant figures in each factor: 34.56 has 4 significant figures, and 1.25 has 3 significant figures.
  • The result should have the same number of significant figures as the divisor with the fewest significant figures, which is 3 in this case.
  • Therefore, round 27.648 to 27.6.
For numbers in scientific notation, divide the significant figures first and then deal with the exponents separately, combining them in the end.
Rounding
Rounding is crucial for reporting calculations with the right number of significant figures. When rounding a number, there are a few key rules to remember:
  • If the digit to be dropped is less than 5, the last retained digit remains unchanged.
  • If the digit to be dropped is 5 or greater, the last retained digit is increased by one.
To illustrate this, consider the example where we've calculated \( \frac{(0.2465)(25)}{1.78} \). The raw result is 3.4626404494. Since the smallest number of significant figures in the calculation is 2, we round 3.4626404494 to 3.5.Accurate rounding ensures that our results are reported precisely and meaningfully, avoiding overstatement or understatement.

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Most popular questions from this chapter

Using conversion factors, solve each of the following clinical problems: a. The physician has ordered \(1.0 \mathrm{~g}\) of tetracycline to be given every six hours to a patient. If your stock on hand is 500 -mg tablets, how many will you need for one day's treatment? b. An intramuscular medication is given at \(5.00 \mathrm{mg} / \mathrm{kg}\) of body weight. What is the dose for a 180 -lb patient? c. A physician has ordered \(0.50 \mathrm{mg}\) of atropine, intramuscularly. If atropine were available as \(0.10 \mathrm{mg} / \mathrm{mL}\) of solution, how many milliliters would you need to give? d. During surgery, a patient receives \(5.0 \mathrm{pt}\) of plasma. How many milliliters of plasma were given?

Round off or add zeros to the following calculated answers to give a final answer with three significant figures: (2.2) a. \(2.784 \mathrm{~kg}\) b. \(76.016 \mathrm{~L}\) c. \(0.006212 \mathrm{~cm}\)

Write each of the following in scientific notation with two significant figures: a. \(5100000 \mathrm{~g}\) b. \(26000 \mathrm{~s}\) c. \(40000 \mathrm{~m}\) d. \(0.000820 \mathrm{~kg}\)

Identify the numbers in each of the following statements as measured or exact: a. My chemistry book weighs 8 lb. b. There are 12 red roses in this bouquet. c. In metric system, \(1 \mathrm{~m}\) is equal to \(100 \mathrm{~cm}\). d. There are 20 types of cakes in this bakery.

Identify each of the following as measured or exact and give the number of significant figures (SFs) in each measured number: a. The mass of a neonate is \(1.607 \mathrm{~kg}\). b. The Daily Value (DV) for iodine for an infant is \(130 \mathrm{mcg}\). c. There are \(4.02 \times 10^{6}\) red blood cells in a blood sample. d. In November, 23 babies were born in a hospital.
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