In which of the following pairs do both numbers contain the sa...

Question

In which of the following pairs do both numbers contain the same number of significant figures?

a. 0.005 75 g and 5.75 × 10^-3 g

b. 405 K and 405.0 K

c. 150 000 s and 1.50 × 10⁴ s

d. 3.8 × 10^-2 L and 3.0 × 10⁵ L

Accepted Answer

Welcome back, everyone. We need to choose the pairs that consist of measurements with different numbers of significant figures. In set one, we have 4.52 times 10 to the fourth power kilometers and 6.83 times 10 to the six power kilometers. In set two, we have 2.60 times 10 to the fifth power liters and 260,000 liters. Then in set three, we have 740.0 degrees Celsius and 704 degrees Celsius. And in set four, we have 0.84 g and 8.4 times 10 to the fourth power grams. So choice A states that pair one and two have different numbers of sig fixed choice B states that pair two and three have different numbers of sigfig choice C states that pairs three and four have different numbers of sick figs. Choice D states that pairs one and four have a different number of sig figs and choice E states that pair two and four have a different number of sick. Now, looking at our set one, we're given again 4.52 times 10 to the fourth power kilometers and 6.83 times 10 to the six power kilometers. So in this set, we have both measurements of scientific notation. And let's recall that when we're given scientific notation, we're only going to consider the base of each number. And so beginning with our first value, our base is going to be 4.52. And for our second value, our base is going to be 6.83. And we can see that both contain three significant figures in this set because our significant figures are going to be our non-zero numbers and we have 4.52 and 6.83. And so that is our analysis for set one. Now, according to choice, a, we're considering set one and set two as having different numbers of sigfig. So let's look at set two in set two. We have 2.60 times 10 to the fifth power liters, which we can see is a scientific notation number. So we're going to consider the base as 2.60 and our second unit is 260,000 liters. Now, let's recall that trailing zeros in a whole number that does not contain a decimal place are going to be nonsignificant. And so in 260,000 liters, our trailing zeros after the two and the six are all going to be nonsignificant. And so therefore, we would have just two sig figs as the two and the six in 260,000 liters. And so, based on our base for our first measurement being 2. that contained exactly three significant figures. We'll use the color blue actually, since in this unit, from our base 2.60, we have a trailing zero after a decimal plate. So it will be considered significant. So in summary, for set two, we can consider that our two measurements given 2.60 times 10 to the fifth power liters and 260 100,000 liters have overall different number of sick figs. So, so far, we're going to rule out any of our answer choices based on set one which both contained measurements with three significant figures. So we would go ahead and rule out choice A as well as choice D because choice A again stated that set one and two have different numbers of sigfig where as in choice, it states that set one and four have different number of sig figs. However, we saw that in set one, we had two scientific notation value measurements which both contain three sigfig. So now let's move on to set three which contains our measurement 740.0 degrees Celsius and degrees Celsius. So looking at 740.0 degrees Celsius, we're going to recall our rule that non-zero integers are going to be significant as well as trailing zeros in a whole number with a decimal point. So in this case, in our measurement 740.0 degrees Celsius, we would say we have a total of four sigfig and looking at our second measurement given 704°C. We're going to recall that zeros between significant nonzero numbers are going to be significant. And because we have a zero between our seven and four, we would have a total of three significant figures in our measurement 704°C. And overall, we can say that we have between our two measurements, different numbers of sex fix. So we can so far put a check by set two and set three because they both gave us measurements with different numbers of sigfig. And now let's look at set four in which we're given the measurement 0.84 g and then 8.4 times 10 to the fourth power grams. So beginning with our first measurement, a total of two significant figures. And this is because we should recall our rule where leading zeros in an integer will never be significant. So the 0.000 in the beginning of our measurement are all insignificant. So nonsignificant, meaning the only significant integers are going to be the eight and the four, which is why again, we have just two significant figures in our measurement then in set four, our second measurement given is 8.4 times 10th of the fourth power grams. So again, this is a scientific notation number and we're only going to consider the base where our base is going to be 8.4. And as we stated before, according to our rules, non zero, integers are going to be significant. So from this space, we would count a total of just two significant figures. And so based on our two measurements in set 40.84 g and 8.4 times 10 to the fourth power grams. We can say that overall, we have the same number of sig figs. So we would rule out choice four, just like we ruled out choice one and looking at our answer choices that have set four within them, we would be able to rule out choice C which stated that set three and four have different numbers of sig figs. And we would rule out set or choice E which states that set two and four have different number of sig figs. And that leaves us with our only correct answer as choice B which states that set two and three have a different number of sig figs where as we saw in set two, we had the measurement 2.60 times 10 to the fifth power liters. We considering the base, we saw a total of three significant figures in 2.60. And then our second measurement in set two was 260 100,000 liters. And so we considered that in this measurement, we only have a total of two significant figures being just a two and a six. So three and two sig figs are different values for set two. And that is true as we saw in choice B and then in choice B, it states that set three has different numbers of sigfig where in set three, we were given the measurement 740.0 degrees Celsius, which we considered having a total of four sig figs. And the second measurement given was seven 104 degrees Celsius, which we considered having three sig figs. And because four and three is a different number of sig figs. Again, choice B is our only correct final answer saying that set two and three have different numbers of sig figs as a pair. So I hope that this made sense and let us know if you have any questions.

In which of the following pairs do both numbers contain the same number of significant figures?
a. 0.005 75 g and 5.75 × 10^-3 g
b. 405 K and 405.0 K
c. 150 000 s and 1.50 × 10⁴ s
d. 3.8 × 10^-2 L and 3.0 × 10⁵ L

Key Concepts

Significant Figures

Scientific Notation

Measurement Precision

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