A decimal represents a fraction. For example:
You can enter these fractions into a calculator to confirm that they produce the same decimal values.
The place values to the right of the decimal point are the tenths, hundredths, and thousandths places.
It’s important to know these place names because many decimal problems ask you to round to a specific place. Remember:
Round up if the digit to the right is or above.
Round down if the digit to the right is or below.
rounded to the nearest tenth is
rounded to the nearest hundredth is
rounded to the nearest thousandth is
Let’s quiz your rounding skills:
What is rounded to the nearest hundredth?
Answer:
The hundredth place is two digits to the right of the decimal point (0.13). Look one digit further right (0.135). Because that digit is , you round the hundredth digit up, giving .
If a problem asks for the value of a digit far to the right of the decimal point, look for a repeating pattern in the digits. The GRE calculator won’t show you 30 decimal places, but patterns let you reason without seeing every digit.
For example, if the decimal is , then the digits in the even positions to the right of the decimal (2nd, 4th, 6th, and so on) are all . So, the 30th digit to the right of the decimal must also be .
GRE questions might ask you to translate decimals to fractions (or fractions to decimals). You probably already know that is equivalent to , but what fraction is equivalent to ?
A straightforward method is:
The decimal reaches the thousandths place (3 digits), so you divide by and simplify:
These same steps work for any terminating decimal. Here are a few more examples:
There’s another way to translate a decimal to a fraction. It takes more steps, but it’s useful because it treats the decimal as a sum of place values.
In the example above, we treated as a single number of digits () over . Instead, you can break it into tenths, hundredths, and thousandths:
You get the same result, but this “piecewise” approach is especially helpful when you start working with decimals that repeat.
Not all decimals end after a fixed number of digits. For example, consider . Writing it as isn’t exact, and neither is or . No matter how many s you write, you still won’t represent exactly.
To show that a decimal repeats forever, we use a bar over the repeating digit(s):
If you try to use the “divide by ” method on , you run into a problem: there is no final digit, so there’s no single power of 10 that captures the whole decimal.
A practical workaround is to memorize a few common repeating decimals and then combine them with place-value reasoning. A key fact is:
For example, .
| Fraction | “Bar” notation | Approximate value |
|---|---|---|
| 0.111111 | ||
| 0.222222 | ||
| 0.333333 | ||
| 0.444444 | ||
| 0.555555 | ||
| 0.666666 | ||
| 0.777777 | ||
| 0.888888 |
When a bar goes over multiple digits, that entire block repeats as a group.
| Fraction | “Bar” notation | Approximate value |
|---|---|---|
| 0.090909 | ||
| 0.181818 | ||
| 0.272727 | ||
| 0.363636 | ||
| 0.454545 | ||
| 0.545454 | ||
| 0.636363 | ||
| 0.727272 | ||
| 0.818181 | ||
| 0.909090 |
Did you notice the pattern here? The two digits under the bar always add to 9, and the first digit is always one less than the numerator.
Let’s use these ideas in an example.
Quantity A:
Quantity B:
Try to solve it using what you’ve just learned. Don’t use a calculator.
Answer: Quantity A is greater
Start by distributing the 2 in Quantity A. Then convert each term into a decimal.
Now focus on . Rewrite it in a form that uses a familiar repeating decimal:
You know . Multiplying by shifts the decimal point two places to the left, so:
Now add the pieces:
So you’re comparing:
Since , Quantity A is greater.
Here’s a video going through one of our practice questions to demonstrate these ideas in action:
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