Math Rounding on the GRE

You have probably encountered a question from an online quiz or exam that specifies, “round to the nearest integer as needed,” or perhaps worse, “round to the nearest hundredth.” At that moment, you might have frozen with anxiety, wishing you had paid more attention in math class in the sixth grade!

Rounding in mathematics is an important skill, and it’s quite important for getting a great score on the GRE Quantitative Reasoning section. For example, you may encounter a GRE Quantitative Comparison question or a Numeric Entry question asking you to round your answer to the nearest integer. 

So, in this article, we will answer the burning question of how to round to the nearest integer, and explore math rounding in general.

Here are the topics we’ll cover:

• Rounding and Integers
  • What Is an Integer?
  • Decimal Numbers and Place Value
  • Rounding to the Nearest Integer
    • Practice with Rounding to the Nearest Integer
  • If We Must Round Up and the Number in the Ones Place Ends in 9
  • Rounding When the Only Digit to the Left of the Decimal Point Is 0
  • Example 1: Rounding to the Nearest Integer
  • Example 2: Rounding to the Nearest Integer
• Rounding Decimal Numbers
  • Place Value Rounding
    • Rounding the Digit 9
  • Example 3: Place Value Rounding
  • Example 4: Place Value Rounding
• Summary
• What’s Next?

Let’s begin by discussing the procedure for rounding to the nearest integer.

Rounding and Integers

The most basic type of rounding question is one that asks you to round to the nearest integer. In order to understand what this means and to determine the correct answer, you must first know exactly what an integer is.

What Is an Integer?

The GRE definition of an integer is a number that can be written without a decimal point or a fractional component. For example, -82, -11, -2, -1, 0, 3, 91, and 388 are integers.

Numbers that are not integers include 1.67, -2/3, 0.6, and π.

KEY FACT:

Integers are numbers without a decimal or fractional component. They can be positive, negative, or 0.

There are some important terms that you might encounter concerning integers. For example, the positive integers include only those integers greater than or equal to 1. They are 1, 2, 3 …

The negative integers include only those integers less than or equal to -1. They are -1, -2, -3 …

Note that 0 is neither negative nor positive. Thus, the phrase “nonnegative integer” means both the positive integers and zero: 0, 1, 2, 3 …

KEY FACT:

0 is an integer but is neither positive nor negative.

Decimal Numbers and Place Value

To learn how to round to the nearest integer, we must first review what a decimal number is. Simply put, a decimal number is a number that contains a decimal point. For example, 3.10, 5.992, and 4.0 are decimal numbers.

The place values of a decimal number are each a power of 10. A place value is always 10 times the value of a place value to its right and one-tenth the value of a place value to its left. The names of the place values are important. To the left of the decimal point, they are ones (or units), tens, hundreds, etc. To the right of the decimal point, they are the tenths, hundredths, thousandths, etc. The following place value table illustrates this:

TTP PRO TIP:

Learn the names and positions on the place value table.

Dealing with and recognizing place values easily is very important when rounding numbers in math and in decimal rounding in mathematics. After all, how can you round a decimal to the nearest hundredth if you can’t locate the hundredth place of a decimal?

Rounding to the Nearest Integer

Rounding off mathematical values to the nearest integer involves looking at the digit whose place value is to its right. When talking about rounding to the nearest integer, we round to the ones (or units) place. Thus, we will use the digit in the tenths place for rounding to the nearest integer because the tenths place is the place to the immediate right of the ones place. For example, if we’re rounding the number 24.8 to the nearest integer, we look at the digit in the tenths place.

Let’s review the math round-up and round-down rules.

Round-Up: Round up the ones digit if the digit in the tenths place is 5, 6, 7, 8, or 9.

Don’t Round Up: Do not round up the ones digit if the digit in the tenths place is 0, 1, 2, 3, or 4.

What we have just learned are techniques for rounding in mathematics in general as well as for the GRE.

KEY FACT:

The digits 5, 6, 7, 8, and 9 indicate that you should round up. The digits 0, 1, 2, 3, and 4 indicate that you should not round up.

Practice with Rounding to the Nearest Integer

Let’s now use the number-rounding methods we just learned for the following examples:

• To round 24.8 to the nearest integer, we note that the digit in the tenths place is 8. So, we should round up the digit in the ones place, from 4 to 5. Thus, 24.8 rounded to the nearest integer is 25.
• To round 16.1 to the nearest integer, we note that the digit in the tenths place is 1. So, we don’t round up. Thus, 16.1 rounded to the nearest integer is 16.

Note that when dealing with precision and rounding in math, we pay no attention to the value of other digits to the right of the decimal point when we are rounding to the nearest integer.

For example, to round 83.79935 to the nearest integer, we consider only the digit 7 in the tenths place to make our rounding decision. Our rounding rule specifies that we will round up, so the correct answer is 84.

If We Must Round Up and the Number in the Ones Place Ends in 9

Hopefully, you are thinking that the mathematical rounding techniques you’ve learned so far for rounding to the nearest integer are pretty straightforward. Don’t get too comfortable just yet! There are still a couple of slightly more involved math roundoff strategies you must learn. Consider the following:

Let’s round 19.51 to the nearest integer. We see that the digit in the tenths place is 5, so we know to round up. So far, so good. But we have to round the digit 9 (in the ones place) up, meaning that we have to round 9 up to 0. But technically, we are rounding that 9 up to 10, and so we round the ones place to 0, and we also round up the 1 to 2 in the tens place. Thus, the correctly rounded integer answer is 20.

KEY FACT:

If you must round up a place value containing the digit 9, round the 9 up to 0 and increase the digit to its left by 1.

Rounding When the Only Digit to the Left of the Decimal Point Is 0

Another possibly confusing rounding situation is if you are asked to round a number to the nearest integer with only a 0 in the ones place, such as 0.8 or 0.4.

For 0.8, we see that because of the 8 in the tenths place, we round up, and the rounded answer is 1.

But for 0.4, with the 4 in the tenths place, we do not round up, and so the rounded answer is 0. This might feel uncomfortable, to say that “something” (like 0.4) is equal to “nothing,” but note that 0.4 is closer to 0 than it is to 1. Therefore, rounding to 0 is not only correct but also logical.

KEY FACT:

Sometimes a decimal number will round to 0.

Let’s practice rounding to the nearest integer with an example.

Rounding Decimal Numbers

Some very good news is that rounding decimal numbers is essentially the same procedure as rounding to the nearest integer. You just have to keep track of which decimal place you want to round to.

Place Value Rounding

Consider the decimal number 45.3094. You want to round this number to the nearest hundredth. The digit in the hundredth position is 0. Now, the digit to the right of 0 is 9, which is in the thousandths place. Because its value is 9, we must round the 0 up to become 1. Thus, when we round 45.3094 to the nearest hundredth, the answer is 45.31.

In this example, note that after the rounding decision is made, all extraneous digits are dropped. So, any digits past the hundredth place will no longer be part of the rounded number.

Let’s round 34.6199 to the nearest tenth. The digit in the tenths place is 6. The digit to its right is 1, meaning that we do not round up the 6. Thus, rounded to the nearest tenth, the number is 34.6. Note that we dropped all digits past the tenths place after the rounding was done.

TTP PRO TIP:

Use the same rounding rules for rounding a decimal number to any place as you used when rounding to the nearest integer.

Rounding the Digit 9

Earlier, we learned that a number like 29.7, rounded to the nearest integer, requires that we round up the 9 to 10. So, the digit in the ones place becomes 0, and the number in the tens place must be increased by 1. Thus, the rounded number is 30.

We use identical logic for rounding a decimal number, no matter where the digit 9 is located. For example, to round the number 12.496 to the nearest hundredth, we see that the digit to the right of the 9 is 6. So, we will round up. Thus, the 9 in the (current) hundredths place must be rounded up to 10, so the hundredths place will now contain a 0, and the tenths place will increase from 4 to 5. Therefore, we round 12.496 to 12.50.

In the above example, note that our rounded answer of 12.50 could be rewritten as 12.5. But because we were told to round to the nearest hundredth, it is appropriate to leave the answer as 12.50.

Let’s try another example.

Summary

• An integer is written without a decimal point or fractional component.
• An integer can be positive, negative, or 0.
• It is important to know the place values of decimal numbers, such as ones, tens, hundreds, tenths, hundredths, and thousandths.
• To round to a particular place value, look at the digit of the number directly to the right of that position. If the number is 5, 6, 7, 8, or 9, round up. If the number is 0, 1, 2, 3, or 4, do not round up.
• If we round up an integer that has a 9 in the ones position, we change the 9 to 0 and increase the digit in the tens position by 1.
• Place value rounding for any decimal number follows the same rules as those for rounding integers.

What’s Next?

Rounding numbers is a useful skill to learn for your GRE preparation. However, there are many other tools and techniques you’ll need in order to reach your target score. One of these is increasing your speed at answering GRE questions without sacrificing accuracy. 

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