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On the Executive Assessment, terminating and repeating decimals appear in quant questions under number properties and problem solving. A terminating decimal has a finite number of digits after the decimal point (e.g., 0.25), while a repeating decimal has a recurring digit or group of digits (e.g., 0.333…). You can quickly determine whether a fraction converts to a terminating or repeating decimal by checking its denominator: if it only has factors of 2 and/or 5, it will terminate; otherwise, it will repeat. Recognizing these patterns can help you solve EA quant problems faster and avoid unnecessary calculations.
While the subject “decimals” may seem bland and mundane, there are some pretty interesting ways the GMAT can test you on your knowledge of decimal numbers. Specifically, you may be tested on how to recognize whether a decimal (when converted from a fraction) will be either a terminating decimal or a repeating decimal. In fact, your ability to recognize the difference between the two could very likely help you on test day. Let’s discuss terminating decimals first.
KEY FACT:
When you perform a fraction to decimal conversion, the fraction will convert to either a terminating decimal or a non-terminating, repeating decimal.
Here Are the Topics We’ll Cover:
- Terminating Decimals
- Non-Terminating, Repeating Decimals
- Example 1
- Example 2
- What About a Non-terminating, Non-Repeating Decimal?
- Summary
- What’s Next?
Terminating Decimals
A terminating decimal is one that has a limited number of digits after the decimal point. For instance, 3.782 and 0.25 both stop after a certain place value. While it’s straightforward to recognize a terminating decimal when written out, figuring out whether a fraction will produce one takes a bit more analysis.
Two terminating decimals rules must be met:
- The fraction must first be simplified into the form x/y, where both x and y are integers and y ≠ 0.
- The prime factors of the denominator can include only 2s and/or 5s. If any other prime factors appear, the decimal expansion will continue indefinitely and will not terminate.
Here are a few terminating decimals examples.
| Fraction | Decimal Equivalent | Terminates Because … |
|---|---|---|
| 1/2 | 0.5 | The denominator’s only prime factor is 2. |
| 7/32 | 0.21875 | The denominator’s only prime factor is 2 because 32 = 2⁵. |
| 11/50 | 0.22 | The denominator’s only prime factors are 2 and 5 because 50 = 2 ⨉ 5². |
Notice that in each case, the denominator’s prime factors are limited to 2, 5, or a combination of the two. The numerator plays no role in deciding whether the decimal terminates.
We see that each fraction’s denominator has prime factors that are only 2 or 5, or both. Note that the value of the numerator has no bearing on whether a fraction yields a terminating decimal.
KEY FACT:
If x and y are integers and y does not equal zero, x/y will produce a terminating decimal if the denominator of the reduced fraction breaks down into prime factors of only 2 and/or 5.
Now, let’s discuss how to recognize a decimal that does not terminate.
Non-Terminating, Repeating Decimals
Now that we can identify terminating decimals, let’s turn to the non-terminating, repeating decimals rule. A fraction will produce a repeating decimal if two conditions are met:
- The fraction must be simplified to lowest terms and expressed as x/y, where x and y are integers and y ≠ 0.
- The denominator, once reduced, contains at least 1 prime factor other than 2 or 5.
KEY FACT:
A non-terminating, repeating decimal is one that extends infinitely, with a repeating pattern of digits continuing after the decimal point.
Let’s look at a few examples of fractions that convert to non-terminating, repeating decimals.
| Fraction | Decimal Equivalent | Repeats Because … |
|---|---|---|
| 1/3 | 0.333333… | The denominator has a prime factor of 3 |
| 7/42 | 0.166666… | The denominator has at least 1 prime factor that is neither 2 nor 5 because 42 = 2 ⨉ 3 ⨉ 7 |
| 11/60 | 0.183333… | The denominator has at least 1 prime factor that is neither 2 nor 5 because 60 = 2² ⨉ 3 ⨉ 5 |
KEY FACT:
If x and y are integers and y ≠ 0, then x/y will produce a repeating decimal whenever the denominator of the reduced fraction contains at least 1 prime factor other than 2 or 5.
With this rule about how to identify repeating decimals in mind, we can now easily determine whether a fraction produces a terminating or repeating decimal. Let’s look at a few examples to see how it works in practice.
Example 1
Which of the following fractions will convert to a terminating decimal?
- 3/14
- 8/40
- 21/63
- 25/75
- 18/54
Solution:
A fraction in lowest terms converts to a terminating decimal if its denominator has only prime factors of 2 and/or 5. Let’s check each option.
A. 3/14 does not reduce. The denominator 14 = 2 × 7. Because of the factor 7, this fraction does not yield a terminating decimal. Eliminate A.
B. 8/40 reduces to 1/5. The denominator 5 is a prime factor allowed (only 2 and 5). So, 8/40 is a terminating decimal. (Its decimal value is 0.2.)
Since we found a terminating decimal, we can stop here.
Answer: B
Let’s try another example.
Example 2
Which of the following fractions will convert to the decimal number that contains the greatest number of nonzero digits?
- 9/30
- 16/40
- 12/36
- 7/28
- 25/50
Solution:
A fraction will produce an infinite number of nonzero digits if it is non-terminating and repeating. This happens when the fraction, in lowest terms, has a denominator with a prime factor other than 2 or 5. Let’s check each choice.
A. 9/30 reduces to 3/10. Since 10 = 2 × 5, the denominator has only 2s and 5s as prime factors. This fraction produces a terminating decimal. Eliminate A.
B. 16/40 reduces to 2/5. The denominator 5 contains only a prime factor of 5, so this is also a terminating decimal. Eliminate B.
C. 12/36 reduces to 1/3. The denominator 3 is a prime number other than 2 or 5, so this fraction produces a non-terminating, repeating decimal. Keep C.
D. 7/28 reduces to 1/4. The denominator 4 contains only a prime factor of 2, so this is also a terminating decimal. Eliminate D.
E. 25/50 reduces to 1/2. The denominator 2 has only the prime factor 2, so this is a terminating decimal. Eliminate E.
Answer: C
What About a Non-terminating, Non-Repeating Decimal?
Up to this point, we’ve focused on fractions that convert to either terminating decimals or non-terminating, repeating decimals. But what about fractions that produce non-terminating, non-repeating decimals?
Here’s the key: no fraction can ever create a non-terminating, non-repeating decimal. These decimals only come from irrational numbers, such as π or e (the base of natural logarithms). Another way to construct one is to deliberately build a sequence, for example:
4.121221222122221222221222222 …
This decimal continues forever without terminating, yet it never falls into a repeating cycle.
The EA will not test you on non-terminating, non-repeating decimals. We cover them here only to complete the picture.
Summary
We’ve explored an important arithmetic concept, terminating vs repeating decimals. Along the way, we learned 2 key rules that let us determine which type of decimal a fraction will produce:
- The fraction must first be reduced to lowest terms and written as x/y, where x and y are integers and y ≠ 0.
- If the denominator’s prime factorization contains only 2s and/or 5s, the decimal will terminate. If the denominator includes any other prime factor, the decimal will be non-terminating and repeating.
What’s Next?
The difference between terminating and repeating decimals is definitely important for the EA. If you’d like some broader guidance on boosting your EA Quant score, take a look at our article with helpful EA Quant tips. Another resource to assist with your EA Quant prep is to read the top 19 tips for increasing your EA Quant score.
You can learn the many topics tested on the GMAT by using the Target Test Prep EA course. Get a no-cost, full-access trial for 7 days.
