Question

For each of the following problems, 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. \(\frac{(0.2465)(25)}{1.78}\)

Short answer

a: 1.6, b: 0.01, c: 27.6, d: 3.5

Step-by-step solution

Identify Significant Figures in Each Number (Problem a)

For the multiplication problem a: \ \(45.7 \times 0.034\). \Number 45.7 has 3 significant figures, and 0.034 has 2 significant figures.

Perform Multiplication (Problem a)

Perform the multiplication: \ \(45.7 \times 0.034 = 1.5538\).

Apply Significant Figures Rule (Problem a)

The result should be rounded to the least number of significant figures in the given numbers, which is 2. Therefore, the answer is 1.6.

Identify Significant Figures in Each Number (Problem b)

For the multiplication problem b: \ \(0.00278 \times 5\). \Number 0.00278 has 3 significant figures, and 5 has 1 significant figure.

Perform Multiplication (Problem b)

Perform the multiplication: \ \(0.00278 \times 5 = 0.0139\).

Apply Significant Figures Rule (Problem b)

The result should be rounded to the least number of significant figures in the given numbers, which is 1. Therefore, the answer is 0.01.

Identify Significant Figures in Each Number (Problem c)

For the division problem c: \ \(\frac{34.56}{1.25}\). \Number 34.56 has 4 significant figures and 1.25 has 3 significant figures.

Perform Division (Problem c)

Perform the division: \ \(\frac{34.56}{1.25} = 27.648\).

Apply Significant Figures Rule (Problem c)

The result should be rounded to the least number of significant figures in the given numbers, which is 3. Therefore, the answer is 27.6.

Identify Significant Figures in Each Number (Problem d)

For the division and multiplication problem d: \ \(\frac{(0.2465)(25)}{1.78}\). \Number 0.2465 has 4 significant figures, 25 has 2 significant figures, and 1.78 has 3 significant figures.

Perform Multiplication and Division (Problem d)

Perform the calculation: \ \(\frac{(0.2465)(25)}{1.78} = \frac{6.1625}{1.78} = 3.462\).

Apply Significant Figures Rule (Problem d)

The combined operations should be rounded to the least number of significant figures from the given numbers, which is 2. Therefore, the answer is 3.5.

Key concepts

multiplication with significant figures

When multiplying numbers, it is important to pay attention to the significant figures. A significant figure (or digit) is any digit in a number that contributes to its precision. The rule for multiplication states that the result should be rounded to the same number of significant figures as the number with the fewest significant figures.

For example, consider the multiplication problem: \(45.7 \times 0.034\). The number 45.7 has 3 significant figures, while 0.034 has 2 significant figures. The product of these numbers is 1.5538. According to the rule, the final answer should be rounded to 2 significant figures, which results in 1.6.

division with significant figures

When dividing numbers, the same attention to significant figures must be maintained. The rule for division is similar to multiplication: the result should have the same number of significant figures as the number with the fewest significant figures in the problem.

Let's look at an example: \(\frac{34.56}{1.25}\). The numerator, 34.56, has 4 significant figures, and the denominator, 1.25, has 3 significant figures. Performing the division gives 27.648. Following the rule, the final answer should be rounded to 3 significant figures, resulting in 27.6.

significant figures rules

Understanding the rules for significant figures is essential for accurate scientific calculations. Here are the key rules to remember:

• All non-zero digits are significant. For instance, 123 has 3 significant figures.
• Zeros between non-zero digits are also significant. Example: 1002 has 4 significant figures.
• Leading zeros (zeros before the first non-zero digit) are not significant. For example, 0.0023 has 2 significant figures.
• Trailing zeros in a number with a decimal point are significant. For example, 45.00 has 4 significant figures.

Applying these rules ensures that calculations with significant figures are handled accurately.

rounding in significant figures

Rounding numbers to the correct number of significant figures is a vital skill in scientific calculations. Here's how to round numbers correctly:

• Identify the digit at the cutoff point (the last significant figure needed).
• If the digit immediately after the cutoff point is 5 or higher, round up the last significant figure. If it's less than 5, keep it unchanged.
• Drop all digits following the last significant figure after rounding.

For example, if you need to round 3.462 to 2 significant figures, look at the third digit (2). Since it's less than 5, the rounded number becomes 3.4.

Rounding helps maintain precision while ensuring the calculated results are not overly detailed beyond what's justified by the significant digits.