A fraction is a value in mathematics that defines a portion of a whole. In other words, a fraction is a two-number ratio. The decimal, on the other hand, is a number in which the entire number and fractional parts are separated by a decimal point. Different varieties of decimals exist, including terminating and non-terminating decimals, as well as repeating and non-repeating decimals. If a number has a fixed number of digits (e.g. 0.532) then this is known as a terminating decimal as it has an endpoint.
The conversion of decimal to fractional value is favored when solving different mathematical problems because fractional values are easier to simplify. In this post, we look at how to convert recurring decimals to fractions.
What is a repeating and terminating decimal?
Non-terminating decimals are divided into two types of decimals: repeating and terminating decimals. The term repeating decimals refers to non-terminating decimals that repeat. If the digits after the decimal point end, the number has a terminating decimal expansion.
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Repeating Decimal
A repeating decimal has an endless number of digits, yet all of the digits are known. For the decimal to be considered repeating, the digits after the decimal point cannot all be zero. Not all of the digits are known for non-terminating decimals that do not repeat. There will always be a digit after it that needs to be determined, no matter how many digits are known.
Ex:
| Non-terminating Decimal | 0.3333333… |
| Repeating Decimal | 0.090909090 |
It’s worth noting that 1/3 is both a non-terminating and a repeating decimal. When attempting to discern between rational and irrational numbers, it is critical to understand the variations between both sorts of decimals. All decimals that end in a digit are rational numbers. A fraction can be used to express both terminating and recurring decimals.
Irrational numbers, the most famous of which is pi (?), are both non-terminating and non-repeating. Pi (3.14159…) and the square root of 2 (1.4142135…) are two examples. Neither will ever finish or repeat, no matter how many digits we compute.
Terminating Decimal
If the digits after the decimal point end, the number has a terminating decimal expansion. Because digits after the decimal point terminate after one digit, the fraction 5/10 has the decimal expansion 0.5, which is a terminating decimal expansion. A terminating decimal expansion or a non-terminating recurring decimal expansion are both possible for a rational number.
What are terminating and recurring decimals in a fraction?
Terminating decimals are fairly straightforward. The decimal number is written as the numerator, and the place value is written as the denominator, to convert a decimal to a fraction. For example:
0.03 is 3 hundredths
Therefore, we write 0.03 as 3/100
But what happens when we convert repeating decimals into fractions?
Repeating Decimals to Fractions
This is where it gets a little more complex. Take the example of 0.3333333…
We shall call this decimal x.
x = 0.333333333…
We then set up another calculation whereby we multiply the above equation by 10.
x = 0.33333333…
10x = 3.333333…
We then subtract the second equation from the first:
10x – x = 9x
3.3333333… – 0.333333… = 3
9x = 3
You then need to get x on its own by dividing the attached number as such:
9x/9 = 3/9
X = 3/9
This can be simplified to 1/3 which is a third.
How to write repeating decimals
Recurring and repeating decimals are never-ending. Obviously, you can’t simply keep writing never-ending pages and pages of decimals so you need a notation to express the difference between terminating, repeating, and recurring decimals.
The most common way to express a recurring decimal is with a dot above the recurring number:
0.3333333… is thus 0.3̇
A repeating decimal is slightly more difficult. How do you express 0.191919…? If we were to write 0.19̇ that would translate as 0.19999999. As such we write 0.1̇9̇ and insert dots above both numbers which repeat.
How to find decimals from a fraction
We divide the numerator by the denominator to find decimals from a fraction.
For example:
2/8
Firstly, we divide the numerator by the denominator using long division:
0.25
8)2.00
40
0
Therefore, 2/8 as a decimal is 0.25
So how does it work when we have a recurring or repeating decimal?
Let us look at 4/9:
0 .444…
9)4.000
36
40
36
40
As you can see, this will just go on and on and on. As such, you can conclude that this decimal is repeating and stick a dot above the decimal number to show that it’s recurring.
Terminating and Repeating Decimal Example
The following are some examples of fractions that have terminating and repeating decimals:
Convert 0.191919 to a fraction
x = 0.191919191
As the decimal recurs in the hundredths rather than just the tenths, we should use 100x rather than 10x.
100x = 19.1919191919
The reason for this is because if we used 10x then we would be subtracting
1.91919 – 0.191919 which would be 1.72728 which then messes up the equation.
If we subtract the true second equation by the first we get:
100x – x = 99x
19.191919 – 0.191919 = 19
Thus, we get 99x = 19
We then divide each side by 99
99x / 99 = 19 / 99
Thus x = 19/99
As such, 0.191919 as a fraction is 19/99
Convert 7/12 into a decimal
0.5833…
12 ) 7.0000
60
100
96
40
36
40
36
4
As such, 7/12 = 0.58333… or 0.583̇
Which of the fractions below will result in a terminating decimal?
- 1/7
- 9/60
- 12/45
Solution:
Remember that if the denominator of a fraction reduced to the lowest terms has only prime factors of 2 and/or 5, it will convert to a terminating decimal. Let’s take a look at each option.
A. 1/7 has already been reduced to its simplest form. 1/7 is not a terminating decimal since its denominator is a prime number other than 2 or 5. Option A should be eliminated.
B. 9/60 can be reduced to 3/20. We can see that 9/60 is a terminating decimal since all of the prime elements in the reduced denominator are either 2 or 5. In reality, 0.15 is the decimal equivalent. We can now stop because we found an answer with a concluding decimal form. B is the correct answer.
Which of the following are irrational numbers?
Q: 1/3
When you divide 1 by 3, you get the decimal 0.3. The bar implies that the number 3 will continue to repeat endlessly. Because it repeats, 0.3 or 0.333… is a rational number. It’s also a decimal that doesn’t end.
Q: 3/11
The decimal equivalent of 3 divided by 11 is 0.27. The number 27 appears twice in this number. Because it repeats, 0.27, or 0.2727…, it is a rational number. It’s also a decimal that doesn’t end.
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