Carry out the following multiplications and divisions,...

Carry out the following multiplications and divisions, expressing your answer to the correct number of significant figures. Assume that all numbers are measured numbers. a. $2.0000 \times 2.00 \times 0.0020$ b. $4.1567 \times 0.00345$ c. $0.0037 \times 3700 \times 1.001$ d. $\frac{6.00}{33.0}$ e. $\frac{530,000}{465.300}$ f. $\frac{4670 \times 3.00}{2.450}$

Carry out the following multiplications and divisions, expressing your answer to the correct number of significant figures. Assume that all numbers are measured numbers.
a. $2.0000 \times 2.00 \times 0.0020$
b. $4.1567 \times 0.00345$
c. $0.0037 \times 3700 \times 1.001$
d. $\frac{6.00}{33.0}$
e. $\frac{530,000}{465.300}$
f. $\frac{4670 \times 3.00}{2.450}$

Key Concepts

Rounding Techniques

Rounding is the process of adjusting a number to a specified degree of precision to correctly represent significant figures. After calculations, rounding guarantees that the result reflects the proper precision, eliminating excess digits that could imply false accuracy.

Measured Numbers

Measured numbers are values obtained through experimentation or observation that inherently carry an uncertainty. The number of significant figures in a measured number indicates its precision, and they are treated differently from exact numbers in calculations involving multiplication and division.

Significant Figures

Significant figures reflect the precision of a measured number by including all the digits that are known reliably, excluding leading zeros and sometimes trailing zeros when they serve as placeholders. They communicate the uncertainty in measurements and determine how precise the derived results should be.

Multiplication and Division Rules

When performing multiplication or division with measured numbers, the final result should be rounded to contain the same number of significant figures as the factor with the fewest significant figures. This rule ensures that the computed result does not imply greater accuracy than the least precise measurement.

Transcript

00:02 So for this problem, what we want to do is we want to solve these arithmetic problems, but we want to do it with the appropriate number of significant digits.

00:11 So the green line over on the right underneath that 2 .54, that's to signify that that is an exact number.

00:18 That is not a measured number.

00:19 It has no limit to its precision.

00:23 It has infinite precision.

00:25 It is an exact number.

00:26 It has infinite numbers of significant digits.

00:30 One thing to notice about this is that the entire problem involves multiplication and division.

00:36 There's no addition or subtraction.

00:38 So anytime you have multiplication or division, you just look to see, all right, well, what's the least number of sig figs i have? that will determine the amount of precision i have in my final value.

00:49 So when you do all of these operations, you should get 17 .36 in your calculator.

00:58 But we are limited with our sig -figs.

01:01 So let's count them up top.

01:03 So here we have four sig -figs.

01:06 Here we have three sig -figs.

01:08 That 10 to the third doesn't count because it just tells you how many extra places to add.

01:16 This one over here has four and then that's an exact.

01:19 So it has infinite numbers.

01:21 So we should be limited to three sig -figs.

01:25 That's our limiting value right there.

01:27 So one way you can say that is that this is approximately then equal to 17 .4 would be the number you come up with.

01:38 Second one here, again, let's notice only multiplication and divisions.