Decimals

A decimal represents a fraction. For example:

• $0.1$ is equal to $1/10$
• $0.25$ is equal to $1/4$
• $0.5$ is equal to $1/2$

You can put these fractions into the calculator to verify for yourself that they come out to the equivalent decimal value.

The names of the slots to the right of the decimal point are the tenths, hundredths, and thousandths places.

• $0.100$ has a $1$ in the tenths place
• $0.010$ has a $1$ in the hundredths place
• $0.001$ has a $1$ in the thousandths place

It’s important to memorize these because most problems with decimals will ask you to round your answer to a specific decimal place. Remember, we round up if the digit to the right is 5 or above, and we round down if the digit to the right is 4 or below.

• $0.15492$ rounded to the nearest tenth is $0.2$
• $0.15492$ rounded to the nearest hundredth is $0.15$
• $0.15492$ rounded to the nearest thousandth is $0.155$

Only round your final answer!

Many questions will ask you to round your answer to a specific precision. Do your calculations at the highest precision possible, since rounding during your calculations could give you a different result.

Let’s quiz your rounding skills:

What is $0.13579$ rounded to the nearest hundredth?

If a problem asks for the value of a number far to the right of the decimal, look for a pattern in the digits to deduce its value. The GRE calculator will not allow you to see 30 decimal places down the line. However, if the decimal representation is $.2424242424$ continuously, you can see that all even slots to the right of the decimal (the 2nd, 4th, 6th, and so on) are $4$. So, the value of the 30th decimal place to the right must also be $4$.

Sidenote

Are decimals even or odd?

Neither! Because decimals are not whole numbers, they are not even or odd. They’re just decimals.

Translating simple decimals to fractions

GRE questions might ask you to directly translate decimal values to fractions, and vice versa. You probably already know that $0.1$ is equivalent to $1/10$, but what if a question asked what fraction is equivalent to $0.124$?

The simple way to translate a decimal to a fraction is to divide the digits by 10 raised to some exponent. For instance, the decimal $0.124$ has digits reaching out to the thousandths place, so you would divide the digits by 1,000, and then simplify.

$$0.124=\frac{124}{1000}=\frac{31}{250}$$

These same steps can be applied to any decimal value. Here are a few more examples:

$$\begin{array}{rlrl}.98& =98/100& =& 49/50\\ .345& =345/1000& =& 69/200\\ .6278& =6278/10000& =& 3139/5000\end{array}$$

There’s another way to translate a decimal to a fraction. It’s a more complicated process, but it helps you think in a different way that can be helpful to solve more complicated problems.

When we turned $0.124$ into $124/1000$ in the previous example, we treated all three digits as one number. Instead, we could break it down by digits individually, expressing it like this:

$$\begin{array}{rl}0.124& =0.1+0.02+0.004\\ & =1/10+2/100+4/1000\\ & =100/1000+20/1000+4/1000\\ & =(100+20+4)/1000\\ & =124/1000\\ & =31/250\end{array}$$

The result will be the same, but using this piecewise approach gives us more flexibility when we have decimals that repeat infinitely.

Translating repeated decimals to fractions

Not all decimals can be expressed with a fixed number of digits. For instance, consider $1/3$. How would you write it as a decimal? Using $0.3$ isn’t really accurate… $0.33$ gets closer… maybe $0.333$, $0.3333$, or even $0.333333333333$? No matter how many $3$s we add, we’ll never precisely represent $1/3$. Instead, to represent a repeating decimal, we need to use a “bar” on top of the number.

$$1/3=0.\overline{3}\approx \mathrm{0.3333333...}$$

If we want to translate $0.\overline{3}$ into a fraction, these infinitely repeating digits pose an issue with the method we described above: we’ll never be able to precisely represent the fraction using $10^n$ in the denominator, since there are always more digits further out.

Fortunately, we can memorize the fractions for some common repeating decimal numbers, and combine them with other parts using the approach above. Dividing a single digit by $9$ gives you a repeating decimal with just that digit. We used this earlier, since $1/3=3/9=0.\overline{3}$.

Fraction	“Bar” notation	Approximate value
$1/9$	$0.\overline{1}$	0.111111
$2/9$	$0.\overline{2}$	0.222222
$3/9$	$0.\overline{3}$	0.333333
$4/9$	$0.\overline{4}$	0.444444
$5/9$	$0.\overline{5}$	0.555555
$6/9$	$0.\overline{6}$	0.666666
$7/9$	$0.\overline{7}$	0.777777
$8/9$	$0.\overline{8}$	0.888888

When a bar goes over multiple numbers, those numbers are repeated together as a group.

Fraction	“Bar” notation	Approximate value
$1/11$	$0.\overline{09}$	0.090909
$2/11$	$0.\overline{18}$	0.181818
$3/11$	$0.\overline{27}$	0.272727
$4/11$	$0.\overline{36}$	0.363636
$5/11$	$0.\overline{45}$	0.454545
$6/11$	$0.\overline{54}$	0.545454
$7/11$	$0.\overline{63}$	0.636363
$8/11$	$0.\overline{72}$	0.727272
$9/11$	$0.\overline{81}$	0.818181
$10/11$	$0.\overline{90}$	0.909090

Did you notice the pattern with these ones? The two digits under the bar always sum up to 9, and the first digit is always one less than the numerator.

Let’s try using these techniques on an example question.

Quantity A: $2(1/10+2/100+3/900)$
Quantity B: $0.246$

Try to solve it using what we’ve just learned! Don’t use a calculator.

The approach we’ll use is to start by rewriting the fractions in QA after they are multiplied by 2, then to translate the fractions individually into decimals, and finally to add them together.

$$\begin{array}{rl}\text{QA}& =2(1/10+2/100+3/900)\\ & =2/10+4/100+6/900\\ & =0.2+0.04+6/900\end{array}$$

It was going well, but we’ve hit a roadblock. What do we do with $6/900$?

The trick is to break it down further into pieces we can deal with:

$$6/900=6/9\ast 1/100$$

We know $6/9=0.\overline{6}$, and multiplying by $1/100$ just shifts over the decimal place two spaces, so…

$$6/900=0.00\overline{6}$$

And now we can put it all together.

$$\begin{array}{rl}\text{QA}& =0.2+0.04+6/900\\ & =0.2+0.04+0.00\overline{6}\\ & =0.24\overline{6}\end{array}$$

We’re left with the following:

• QA: $0.24\overline{6}\approx 0.246666$
• QB: $0.246$

And now the quantities are easy to compare!

Bringing it all together: question walkthrough video

Here’s a video going through one of our practice questions to demonstrate these ideas in action: