| Getting your Trinity Audio player ready... |
While the subjects of fractions and decimals may seem boring and mundane, there are some pretty interesting ways the SAT can test you on your knowledge of converting fractions to decimals. Specifically, you may be tested on how to recognize whether a decimal (when converted from a fraction) will be either a terminating or repeating decimal. Additionally, if you are filling in the box of a student-produced response with either a decimal or a fraction, you need to know the rather strict rules for correctly entering your answer.
Here are the topics we’ll cover:
- How to Make Fractions into Decimals
- Terminating Decimals
- Repeating Decimals
- What About Non-terminating, Non-repeating Decimals?
- SAT Fraction to Decimal Examples
- The Rules for Entering Fractions and Decimals into the Student-Produced Response Box
- General Guidelines
- Guidelines Specific to Fractions and Decimals
- Summary
- What’s Next?
How to Make Fractions into Decimals
The traditional technique for converting fractions into decimals is by dividing the numerator by the denominator. Without a calculator, this requires the use of long division. However, because you have access to the online calculator on the SAT for all math questions, just perform the division by using the division button. For example, to obtain the decimal equivalent of 3/4 , enter 3 ➗ 4 , and the answer will appear just below and to the right of your input on the Desmos calculator.
KEY FACT:
To convert a fraction to a decimal on the SAT, use the division button on the calculator.
When we perform a fraction-to-decimal conversion, the fraction will convert to either a terminating decimal or a non-terminating repeating decimal. Let’s discuss terminating decimals first.
Terminating Decimals
A terminating decimal is a decimal in which the digits after the decimal point terminate. For example, 7.128 and 0.45 are terminating decimals. It’s easy to see when a decimal is a terminating decimal because it has only values of 0 after the point of termination. However, determining whether a particular fraction will result in a terminating decimal may be more challenging. There are two requirements for a fraction to convert to a terminating decimal:
1) First, make sure that the fraction has been reduced to its lowest terms and is written in the form x / y, where x and y are integers and y is not equal to zero.
2) If you can factor the denominator into prime factors of only 2 and/or 5, then the fraction yields a terminating decimal. If there are any other prime numbers in the denominator’s prime factorization, then the decimal will NOT terminate.
Let’s look at this fractions to decimals chart that illustrates some terminating decimal examples.
| Fraction | Decimal Equivalent | Terminates Because … |
|---|---|---|
| 1/2 | 0.5 | The denominator’s only prime factor is 2. |
| 9/16 | 0.5625 | The denominator’s only prime factor is 2 because 16 = 2^4. |
| 17/50 | 0.34 | The denominator’s only prime factors are 2 and 5 because 50 = 2 ⨉ 5². |
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 effect on whether a fraction becomes a terminating decimal.
KEY FACT:
If x and y are integers and y does not equal zero, the fraction x / y will be a terminating decimal if the denominator of the reduced fraction has prime factors of only 2 and/or 5.
Now, let’s discuss how to recognize a decimal that repeats rather than terminates.
Repeating Decimals
Once we know the criteria for a terminating decimal, we can quickly determine how to recognize a non-terminating, repeating decimal. There are two requirements for a fraction to convert to a repeating decimal:
1) First, make sure that the fraction has been reduced to its lowest terms and is written in the form x / y, where x and y are integers and y is not equal to zero.
2) The denominator of the reduced fraction has at least one prime factor that is neither 2 nor 5.
Let’s look at the following fraction-to-decimal conversion chart that illustrates a few examples of fractions that convert to non-terminating, repeating decimals.
| Fraction | Decimal Equivalents | Repeats Because … |
|---|---|---|
| 1/11 | 0.09090909… | The denominator has a prime factor of 11 and a recurring decimal pattern of 090909 |
| 1/6 | 0.166666… | The denominator has at least one prime factor that is neither 2 nor 5 because 6 = 2 ⨉ 3 |
| 11/60 | 0.183333… | The denominator has at least one prime factor that is neither 2 nor 5 because 60 = 2² ⨉ 3 ⨉ 5 |
KEY FACT:
If x and y are integers and y is not equal to zero, then x / y converts to a repeating decimal if the denominator of the fraction has at least one prime factor that is neither 2 nor 5. (Note: the fraction must be reduced to lowest terms.)
What About Non-terminating, Non-repeating Decimals?
Interestingly, every fraction will give us a decimal that either terminates or repeats. The only type of number that yields a decimal number that is both non-terminating and non-repeating is an irrational number, such as π or √2. You may recall that the definition of an irrational number is one that cannot be expressed as a fraction. Thus, dividing numerators by denominators is the only way to obtain terminating or repeating decimals.
KEY FACT:
Irrational numbers are the source of non-terminating, non-repeating decimal values. Expressing rational numbers as decimals is the only way to obtain either a terminating or repeating decimal.
We now know how to determine if we will get terminating decimals or non-terminating, repeating decimals by first simplifying fractions into lowest terms and then analyzing the denominators. Let’s practice with a couple of repeating or terminating decimal questions for SAT.
SAT Fraction to Decimal Examples
Example 1
Of the following fractions, which one will convert to a terminating decimal?
- 1/7
- 3/20
- 4/15
- 1/3
- 1/30
Solution:
Rather than waste time using your calculator to determine which fraction converts to a terminating decimal, recall that a fraction reduced to the lowest terms will convert to a terminating decimal if its denominator contains prime factors of only 2 and/or 5. Let’s consider each answer choice.
A. 1/7 is already reduced to lowest terms. Its denominator of 7 is a prime number other than 2 or 5, so 1/7 does not yield a terminating decimal. Eliminate choice A.
B. 3/20 is in reduced form. We can factor the denominator as 20 = 2² ⨉ 5. Since all the prime factors of the reduced denominator are either 2 or 5, we see that 3/20 is a terminating decimal. (In fact, its decimal equivalent is 0.15.)
We have found the correct answer, so we can stop.
Answer: B
Example 2
Which of the following fractions will convert to the decimal that contains the greatest number of nonzero digits?
- 7/25
- 18/36
- 3/30
- 5/15
- 4/20
Solution:
A fraction’s decimal representation will have an infinite number of nonzero digits if it is a non-terminating decimal. Recall that a fraction reduced to lowest terms whose denominator has a prime factor other than 2 or 5 will be a non-terminating, repeating decimal. Let’s consider each answer choice.
A. 7/25 has been reduced to lowest terms already. Its denominator factors to 25 = 5². Thus, only 5 is a prime factor. This fraction converts to a terminating decimal. Eliminate choice A.
B. 18/36 must be reduced to 1/2. The denominator is the prime number 2. Thus, this fraction will convert to a terminating decimal. Eliminate choice B.
C. 3/30 can be reduced to 1/10. Its denominator factors to 10 = 2 ⨉ 5, both of which are prime, so 3/30 converts to a terminating decimal. Eliminate choice C.
D. 5/15 can be reduced to 1/3. Because the denominator contains a prime factor of 3, this fraction is non-terminating and repeating. Thus, it has an infinite number of nonzero digits.
Note that we can stop here, as we have found the correct answer.The fraction 1/3 yields a non-terminating, repeating decimal that has an infinite number of nonzero digits after the decimal point.
Also, note that it was not necessary to evaluate each fraction’s decimal equivalent. The quick factorization of each answer choice’s denominator was all that was necessary.
Answer: D
The Rules for Entering Fractions and Decimals into the Student-Produced Response Box
The SAT has specific rules for entering fraction and decimal answers into the answer box of student-produced response questions, for which there are no answer choices. Let’s first look at some general guidelines and then zero in on fractions and decimals.
General Guidelines
- The maximum number of characters that you can input into the box is 5 for positive numbers and 6 for negative numbers.
- The decimal point acts as one character. So does the “slash” symbol for a fraction.
- Numerical answers can now be negative.
- If you obtain several answers during your calculations, enter only one of them.
- Don’t enter symbols, such as a comma, a dollar sign, or a percent sign.
Guidelines Specific to Fractions and Decimals
- If a fraction exceeds the limit of 5 characters for positive or 6 characters for negative answers, reduce the fraction to fit the box or enter its decimal equivalent.
- If your answer is a mixed number (e.g., 4 1/2), convert it to an improper fraction (9/2) or its decimal equivalent (4.5).
- If a decimal answer exceeds the allowable space, truncate it or round it at the fourth digit. For example, you would enter 7.6666666 as 7.666 or 7.667.
- If a decimal number has a leading 0, then the leading 0 can count as one of the digits or not, as you choose. For example, 0.6666666 could be entered as 0.666 (truncated at the fourth digit) or as 0.667 (rounded at the fourth digit). If you choose not to use a leading 0 to express your answer, then you will enter either .6666 (truncated at the fourth digit) or .6667 (rounded at the fourth digit.
Note that if you do not follow these rules and guidelines, your answer will be marked as incorrect. For example, if you round your answer of 0.6666666 and enter 0.67 into the answer box, you will not receive credit for a correct response.
While you are taking your SAT, you will have access to these rules every time you encounter a question with a student-produced response. Thus, if you forget the rules or want to double-check that your answer is correctly entered, you can read the rules and the examples provided. However, this will waste valuable test time. A better strategy is to ensure you have mastered the rules during your SAT preparation time.
TTP PRO TIP:
Know the rules and guidelines for entering fractions and decimals into the student-produced response box!!
Summary
To convert a fract5ion to a decimal, you can use traditional long division. However, since the SAT provides a calculator for all math problems, you can use it to divide the numerator by the denominator.
A terminating decimal occurs when only 2s and/or 5s are factors of the reduced fraction’s denominator.
A nonterminating, repeating decimal occurs when at least one factor of the reduced fraction is a prime number other than 2 or 5.
It is important to learn the rules for entering fractions and decimals into the box of a student-produced response question. By improperly entering your answer, you could end up missing a question, even though your answer is correctly calculated.
What’s Next?
Fractions and terminating and repeating decimals may not be the most thrilling topics tested on the SAT, they are easy to master. The rules that guide us to distinguish between fractions that do and do not give us terminating decimals are a simple application of prime number theory, and your knowledge of those rules may well come in handy on test day!
For some more helpful SAT math tips, check out our article about the math concepts that are tested on the SAT.
