Fractions and decimals are a way to express parts of a whole. Once you learn the basic moves, you’ll use the same handful of strategies again and again. Let’s walk through them together.
1. Changing decimals into fractions
If a number is rational, the decimal can be converted into a fraction. A rational number means that the decimal can either terminate (end) or repeat. An irrational number, such as pi, has non-terminating and non-repeating decimals. Here is a video to help with understanding different types of numbers.
If a number has terminating decimals, here is how to convert it into a fraction:
Steps:
Split off the numbers behind the 0.
Look at the decimal part. What is this, as a number?
Example: 0.375 is “375 thousandths.” or as a fraction: 375/1000.
If the fraction can be simplified, or reduced, then express the fraction in lowest terms by dividing out the greatest common factor.
Example: 375/1000, both numerator and denominator can be divided by 125. Therefore, it can be simplified to 3/8
Example: Convert 2.375 to a mixed fraction.
Break it: 2 + 0.375
Write fraction: 0.375 = 375/1000
Simplify: divide by 125 → 3/8
Answer: 2 3⁄8
Teacher tip: Saying the decimal out loud (“thousandths,” “hundredths”) is the quickest way to know what goes on the bottom.
This video depicts another approach:
2. Fractions that repeat forever (repeating decimals)
If you see something like 0.3333… or 0.121212…, that’s a repeating decimal. To turn it into a fraction, we use a quick trick. Here is a video tutorial:
In math we use a bar on top of the number to indicate repetition. For example 0.33333… is written as 0.3̅.
Example 1: Convert 0.3̅ to a fraction
If it were 0.3, not repeating, the solution would be three tenths, or 3/10
Since it is repeating, we take the repeating number and put it in the numerator, and the denominator is the same number of digits but 9s. In this case, 3/9. 3/9 simplifies to 1/3
Take home message, the fraction is over 9 instead of 10
Example 2: Convert 0.17̅ to a fraction
This can be written as 17/99 instead of 17/100. 17/99 cannot be simplified further, so that is the solution.
Example 3: Convert 0.198̅ to a fraction
This can be written as 198/999 instead of 198/1000. 198/999 can be simplified to 22/111.
Shortcut: memorize the most common ones.
0.333… = 1/3
0.666… = 2/3
0.111… = 1/9
3. Adding and subtracting fractions
Adding and subtracting fractions is like combining parts of a whole. Imagine adding two pizzas together that are sliced into different numbers of slices. One pizza has 8 slices, the other has 10 slices. If we had 4/8 of one and 5/10 the other, we would end up with one whole pizza (half of each). However, since they are sliced differently, if we were to add different numbers of slices, we would have to consider what the equivalence of the slices would be. Hence, we find a common denominator or common total.
Example: 1 3⁄8+ 2/5
Steps:
Change from mixed fraction to improper fraction if applicable: 1 3⁄8 = 11/8
Find a common denominator (ideally the lowest common multiple). In this case the common denominator between 8 and 5 is 40.
Rewrite each fraction with that denominator. Multiply the numerator by the same factor as the denominator to ensure that they are equivalent factors. 11/8 becomes 55/40 and 2/5 becomes 16/40
Add (or subtract) the numerators. 55/40+ 16/40 = 71/40
Simplify, and if it’s improper, change it back to a mixed number. 71/40 cannot be simplified but it can be converted to a mixed number 1 31⁄40
Example: Simplify 7⅛ – 4⅔
Steps:
Change to improper: 7⅛ = 57/8, 4⅔ = 14/3
Common denominator of 8 and 3 = 24. The common denominator makes them both out of the same “total”
Rewrite by multiplying numerator and denominator by the same number: 57/8 = 171/24, 14/3 = 112/24
Subtract: 171 – 112 = 59 → 59/24
Turn back into mixed: 2 11/24
4. Multiplying fractions
This is straightforward, simply multiply the numbers horizontally.
Steps:
Convert to improper fractions.
Multiply across: top × top, bottom × bottom.
Simplify and cancel common factors early if you can.
