Question

how many significant figures does the difference 218.7201 - 218.63 contain?
1
2
3
5
7

Explanation:

Step1: Calculate the difference

First, we calculate \(218.7201 - 218.63\).
\(218.7201 - 218.63 = 0.0901\)

Step2: Determine significant figures

Now, we analyze the significant figures of \(0.0901\). Leading zeros are not significant, but zeros between non - zero digits are significant. So, the significant figures are \(9\), \(0\), and \(1\)? Wait, no. Wait, \(0.0901\): the first non - zero digit is \(9\), then the zero after \(9\) is between non - zero digits (well, after \(9\) and before \(1\))? Wait, no, let's recall the rules of significant figures:
1. All non - zero digits are significant.
2. Zeros between non - zero digits are significant.
3. Leading zeros (zeros before the first non - zero digit) are not significant.
4. Trailing zeros in a decimal number are significant.

For \(0.0901\):
- The leading zeros (\(0.0\)) are not significant.
- The \(9\) is significant.
- The \(0\) between \(9\) and \(1\) is significant (because it's between two non - zero digits? Wait, no, \(9\) is non - zero, then \(0\), then \(1\) is non - zero. So the zero between \(9\) and \(1\) is significant.
- The \(1\) is significant.

Wait, but let's recalculate the subtraction:

\(218.7201-218.63 = 0.0901\)? Wait, no:

\(218.7201 - 218.63\):

We subtract the two numbers:

\(218.7201-218.63 = 0.0901\)? Wait, \(218.7201-218.63 = 0.0901\)? Let's do the subtraction properly:

\(218.7201-218.6300 = 0.0901\)

Now, for significant figures in subtraction, the result should have the same number of decimal places as the least precise measurement.

\(218.7201\) has four decimal places, \(218.63\) has two decimal places. So the result should be rounded to two decimal places? Wait, no, the rule for addition and subtraction is that the number of decimal places in the result is equal to the number of decimal places in the term with the least number of decimal places.

\(218.7201\) has 4 decimal places, \(218.63\) has 2 decimal places. So when we subtract, we look at the decimal places:

\(218.7201-218.63 = 0.0901\), but we should round to the least number of decimal places, which is 2. Wait, no, \(218.63\) has two decimal places, \(218.7201\) has four. So the difference should be reported to two decimal places? Wait, no, let's do the subtraction:

\(218.7201-218.63 = 0.0901\). Now, the number of decimal places in \(218.63\) is 2, in \(218.7201\) is 4. So the result should have 2 decimal places? Wait, no, the rule is that the result has the same number of decimal places as the measurement with the fewest decimal places. So \(218.63\) has two decimal places, so the result should be rounded to two decimal places. Wait, \(0.0901\) rounded to two decimal places is \(0.09\)? No, that's not right. Wait, maybe I made a mistake in the subtraction.

Wait, \(218.7201-218.63\):

\(218.7201-218.63 = 0.0901\). Let's check the decimal places:

\(218.7201\): the decimal part is \(0.7201\) (four decimal places)

\(218.63\): the decimal part is \(0.63\) (two decimal places)

When subtracting, we align the decimals:

\(218.7201\)

\(-218.6300\)

\(= 0.0901\)

Now, the number of decimal places in the result is determined by the least number of decimal places in the inputs. Here, \(218.63\) has two decimal places, so the result should be reported to two decimal places. Wait, but \(0.0901\) to two decimal places is \(0.09\)? But that would be two significant figures? Wait, no, \(0.09\) has one significant figure? Wait, no, \(0.09\): the non - zero digit is \(9\), so only one significant figure? But that can't be right. Wait, maybe I messed up the subtraction.

Wait, let's do the subtraction again:

\(218.7201 - 218.63\):

\(218.7201-218.63 = 0.0901\). Now, let's consider the significant figures in the result of subtraction. The rule is that the result has the same precision (number of decimal places) as the least precise measurement. \(218.63\) has a precision of two decimal places, \(218.7201\) has a precision of four decimal places. So the result should be rounded to two decimal places. So \(0.0901\) rounded to two decimal places is \(0.09\)? But \(0.09\) has one significant figure? Wait, no, \(0.09\): the first non - zero digit is \(9\), so only one significant figure? But that seems wrong. Wait, maybe the question is not about rounding, but just about the number of significant figures in the difference before rounding? Wait, the question is "How many significant figures does the difference \(218.7201 - 218.63\) contain?"

So let's calculate the difference exactly: \(218.7201-218.63 = 0.0901\)

Now, let's apply the rules of significant figures to \(0.0901\):

- Leading zeros: \(0.0\) are not significant.
- The \(9\) is significant.
- The \(0\) between \(9\) and \(1\) is significant (because it's between two non - zero digits? Wait, \(9\) is non - zero, \(0\) is between \(9\) and \(1\) (which is non - zero), so yes, the zero between \(9\) and \(1\) is significant.
- The \(1\) is significant.

Wait, so that's three significant figures? Wait, \(0.0901\): the significant figures are \(9\), \(0\), \(1\)? Wait, no, the zero after \(9\) and before \(1\) is significant. So the number of significant figures is 3? Wait, but let's check the original numbers:

\(218.7201\) has 6 significant figures (2,1,8,7,2,0,1? Wait, no: \(218.7201\): all digits are significant because it's a decimal number and there are no leading zeros. So \(2\), \(1\), \(8\), \(7\), \(2\), \(0\), \(1\) – 7 significant figures? Wait, no: \(218.7201\): the digits are \(2\), \(1\), \(8\), \(7\), \(2\), \(0\), \(1\). All non - zero digits are significant, and the zero between \(2\) and \(1\)? Wait, no, \(218.7201\): the decimal part is \(0.7201\), so the zero in \(0.7201\) is between \(2\) and \(1\), so it's significant. So \(218.7201\) has 7 significant figures.

\(218.63\): the digits are \(2\), \(1\), \(8\), \(6\), \(3\) – 5 significant figures.

When subtracting, the number of decimal places in the result is determined by the least number of decimal places in the inputs. \(218.7201\) has 4 decimal places, \(218.63\) has 2 decimal places. So the result should have 2 decimal places. But if we don't round, and just look at the difference \(0.0901\), let's count the significant figures:

- The first non - zero digit is \(9\) (position 3 after the decimal, but in terms of significant figures, leading zeros are not significant). So \(9\) is significant, the \(0\) after \(9\) is significant (because it's between \(9\) and \(1\)), and \(1\) is significant. Wait, so that's three significant figures? Wait, \(0.0901\): the significant figures are \(9\), \(0\), \(1\) – three significant figures? But let's check the options: the options are 1,2,3,5,7.

Wait, maybe I made a mistake in the subtraction. Let's recalculate \(218.7201 - 218.63\):

\(218.7201-218.63 = 0.0901\). Now, let's consider the significant figures in \(0.0901\):

- Leading zeros: not significant.
- \(9\): significant.
- \(0\): significant (because it's between two non - zero digits? Wait, \(9\) is non - zero, \(0\) is next, then \(1\) is non - zero. So the zero between \(9\) and \(1\) is significant.
- \(1\): significant.

So that's three significant figures. So the answer should be 3.

Answer:

3