Rounding to a certain number of significant digits, or to a named "place", is fairly straightforward. The real question comes in how to round answers to the "appropriate" number of significant digits. If you've been given some values (lengths, say, or masses) and told to work with them, arriving at an answer which is rounded to the "appropriate" number of sig-digs, what does this mean?
The idea is this: Suppose you measure a block of wood. The length is 5.6 inches, the width is 4.4 inches, and the thickness is 1.7 inches, at least as best you can tell from your tape measure. To find the volume, you would multiply these three dimensions, to get 41.888 cubic inches.
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But can you really, with a straight face, claim to have measured the volume of that block of wood to the nearest thousandth of a cubic inch?!? Not hardly! Each of your measurements was accurate (as far as you can tell) to two significant digits: your tape measure was marked off in tenths of inches, and you wrote down the closest tenth of an inch that you could see. So you cannot claim five decimal places of accuracy, because none of your measurements exceeded two digits of accuracy.
As a result, you can only claim two significant digits in your answer. In other words, the "appropriate" number of significant digits is two, and you would report (in your physics lab report, for instance) that the volume of the block is 42 cubic inches, approximately.
How do you round when they give you a bunch of numbers to add? You would add (or subtract) the numbers as usual, but then you would round the answer to the same decimal place as the least-accurate number.
Looking at the numbers, I see that the second number, 234.6, is only accurate to the tenths place; all the other numbers are accurate to a greater number of decimal places. So my answer will have to be rounded to the tenths place:
13.214 + 234.6 + 7.0350 + 6.38 = 261.2290
The digit in the tenths place is a 2, and it's followed by another 2, so I won't be rounding up. Rounding to the tenths place, I get:
13.214 + 234.6 + 7.0350 + 6.38 = 261.2
The requirement to round your answer, when adding values, to the same "place" as the largest (that is, furthest to the left, with respect to the decimal point) last accurate place of the input values, might make a little more sense if you view the addition vertically:
13.214 234.6 7.0350 + 6.38 -------- 261.2290
When you look at the columns of digits, it kinda makes sense that, yeah, you can't claim any accuracy past the tenths place, because that's the last column that all the input numbers share.
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Here's another example:
Looking at each of the numbers they've given me, I see that I will have to round the final answer to the nearest tens place, because 450 is only accurate to the tens place. (The other numbers are accurate to the ones, tenths, and ones places, respectively.)
First, I add in the usual way:
1247 + 134.4 + 450 + 78 = 1909.4
...and then I round my result to the tens place, rounding the 0 up to 1 because of the 9 in the ones place:
1247 + 134.4 + 450 + 78 = 1910
How do you round, when they give you numbers to multiply (or divide)? You would multiply (or divide) the numbers as usual, but then you would round the answer to the same number of significant digits as the least-accurate number.
16.235 × 0.217 × 5
First, I note that 5 has only one significant digit, so I will have to round my final answer to one significant digit. (The other numbers have five and three significant digits, respectively.)
The mathematical product is:
16.235 × 0.217 × 5 = 17.614975
...but since I can only claim one accurate significant digit, I will need to round 17.614975 to one digit. I'll start with the 1 in the tens place; immediately to its right is a 7, which is greater than 5, so I'll be rounding the 1 up to 2, replacing the 7 with a zero, and dropping the decimal point and everything after it. Then I'll get 20, which is accurate to one significant digit.
16.235 × 0.217 × 5 = 20
First I multiply:
0.00435 × 4.6 = 0.02001
Looking at the original numbers, I see that 4.6 has only two significant digits, so I will have to round 0.02001 to two significant digits. The 2 is the first significant digit, so the 0 following it will have to be the second significant digit. In other words, I must report the answer as being:
0.00435 × 4.6 = 0.020
The answer should not be 0.02, because 0.02 has only one significant digit; namely, the "2". The trailing zero in 0.020 indicates that "this is accurate to the thousandths place, or two significant digits", and that trailing zero is therefore a necessary part of the answer.
Just remember the difference:
For adding, use "least accurate place".
For multiplying, use "least number of significant digits".