What does it mean to round a number? Rounding means to make a number shorter or simpler, but keeping it as close in value as possible to the original number. Let’s take a closer look in this edition of GRE Math Tips.
Rounding to the Nearest Integer
The most common type of rounding is to round to the nearest integer. The rule for rounding is simple: look at the digits in the tenth’s place (the first digit to the right of the decimal point). If the digit in the tenths place is less than 5, then round down, which means the units digit remains the same; if the digit in the tenths place is 5 or greater, then round up, which means you should increase the unit digit by one.
Here are a couple of other things to know:
Observation #1: under most circumstances, rounding changes the decimal to whatever integer is closer. For example, 4.3 is rounded to 4, and 4.9 is rounded to 5. The exception is when the decimal is smack dab between two integers: 4.5 is exactly equidistant to both 4 and 5, but because of the “tie-breaker” rule of rounding, anything with a 5 in the tenths digit is rounded up. This is the only case in which the “go to the closer integer” interpretation will fail.
Observation #2: Do NOT double-round. Some people look at a number like, say, 7.49, and they erroneously think — well, that 9 would round the 4 up to 5, and then a 5 gets round up, to this number would round to 8. WRONG! Never round a number “in stages.” Rounding is a one-shot deal, a one-step process. When the number we need to round is 7.49, we only need notice that the tenth’s digit is a 4, which means the number is rounded down to 7. One step, case closed. In fact, all of the following numbers get rounded to 7:
7.49
7.499999
7.4999999999999999999
—> All get rounded to 7
Here’s the truly mind-boggling part:
How many numbers would there be larger than this last number, but still lower than 7.5? INFINITY! No matter how many additional 9’s we slap on to the end of that number, there’s still a continuous infinity of decimals larger than that number and below 7.5 No matter how finely we chop up the real number line, each tiny fragment of the line, no matter how small, still contains a continuous infinite of numbers.
Observation #3: the “tie-breaker” rule can be tricky with negative values. For example, +2.5 gets rounded up to 3, but –2.5 gets rounded down … to –3. As with positive numbers, the negative number ending in .5 is rounded to the higher absolute value integer, but with negatives, that’s rounding down. (This is not the only way to formulate this rule, but this is the convention that ETS follows.)
Rounding to any other decimal place
Rounding to the nearest integer is really rounding to the nearest units place. Sometimes, you will be asked to round to the nearest hundreds, or to the nearest hundredths — to some decimal place other than the units place. The rule is just a more generalized version of the previous rounding rule.
Suppose we are asked to round to some specific decimal place — call this the “target place.” You always look at only one digit, the digit immediately to the right of the target place. If this digit immediately to the right is {0, 1, 2, 3, 4}, then you “round down”, and the digit in the target place remains unchanged. If this digit immediately to the right is {5, 6, 7, 8, 9}, then you “round up”, and the digit in the target place increases by 1.
What do you have to know about how to round to the nearest integer for the GRE?
Not many GRE questions will say, “Here’s a number: round it to the nearest such-and-such.” By contrast, many questions, in the course of asking something else, could ask you to round your answer to the nearest such-and-such. In this way, rounding is one math skill you need to know for the GRE. There are a few tricky issues, which I will address here.
Other cases of rounding
Very occasionally, a GRE math question may ask you not to round to a particular decimal place, but rather to the nearest multiple of something. For example, suppose you are asked to round, say, to the nearest 0.05 — how do you do that?
Well, let’s think about the results first. The result of rounding to the nearest 0.05 would be something divisible by 0.05 — that is to say, a decimal with either a 0 or a 5 in the hundredth place, no digits to the right of that, and any digits to the left of that. The following are examples of numbers which could be the result of rounding to the nearest 0.05:
- 0.35
- 1.40
- 3.15
- 5.2
- 8
Notice: the second, (b) is the square root of 2 (sqrt{2} = 1.414213562….) rounded to the nearest 0.05, and the third, (c), is pi rounded to the nearest 0.05.
Let’s demonstrate the rounding by means of an example. What numbers, when rounded to the nearest 0.05, would be rounded to 2.35? Well, for starters, 2.35 and other “tenths” around it would be rounded to 2.35
2.32 — rounded down to 2.30
2.33 — rounded up to 2.35
2.34 — rounded up to 2.35
2.35 — stays at value
2.36 — rounded down to 2.35
2.37 — rounded down to 2.35
2.38 — rounded up to 2.40
Now, the tricky regions are those between the values that are rounded in different directions. For example, 2.32 is rounded down and 2.33 is rounded up, so something fishy is happening between those two. Let’s think about the hundredths between 2.32 and 2.33 — exactly between them is 2.325, the midpoint between 2.30 and 2.35, and like all midpoints, according to the “tiebreaker” rule, it gets rounded up. Thus:
2.320 —- rounded down to 2.30
2.321 —- rounded down to 2.30
2.322 —- rounded down to 2.30
2.323 —- rounded down to 2.30
2.324 —- rounded down to 2.30
2.325 —- rounded up to 2.35 (the “tie-breaker” rule)
2.326 —- rounded up to 2.35
2.327 —- rounded up to 2.35
2.328 —- rounded up to 2.35
2.329 —- rounded up to 2.35
2.330 —- rounded up to 2.35
This is all probably far more detail than you will need to know for GRE math, but this does demonstrate the steps you would take to round any decimal to the nearest 0.05. By analogy, you could round any decimal to any specified multiple.
Got rounding to the nearest integer mastered? Test your skills on Magoosh’s GRE math practice questions with answers and explanations, or check out the rounding question below!
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