Generate Random Fractions
Generate random proper, improper or mixed-number fractions from the denominators you choose, or whole worksheets of fraction addition, subtraction, multiplication, division and comparison problems. Pick lowest terms or fractions that need simplifying, and show or hide the answers.
Answers in lowest terms, as mixed numbers and as exact decimals
Proper, Improper and Mixed Fractions
| Kind | Examples | What it means |
|---|---|---|
| Proper | 3/8, 5/6 | Numerator smaller than denominator; less than 1 |
| Improper | 7/4, 9/9 | Numerator equal to or larger than denominator; 1 or more |
| Mixed number | 1 3/4 | A whole number and a proper fraction; 1 3/4 = 7/4 |
| Unit fraction | 1/5, 1/12 | Numerator 1; the building block in early grades |
| Equivalent | 2/4 = 3/6 = 1/2 | Different numbers, same value |
For improper fractions and mixed numbers, Largest value caps the size: at 3, results stay under 3 (or equal to it if whole numbers are allowed). Whole numbers such as 6/3 are left out by default, because a "fraction" that is really 2 usually isn't what a practice question wants.
Worked Example: 3/4 + 5/6
The Grade 5 preset produced 3/4 + 5/6, and the answer column says 19/12 = 1 7/12. Here is the working a student should show.
Step 1: Lowest common denominator
Step 2: Rewrite both fractions
Step 3: Add the numerators
Step 4: Simplify and convert
Multiplying the denominators (4 × 6 = 24) also works, giving 18/24 + 20/24 = 38/24, but then the answer has to be simplified by 2. Using the LCD keeps numbers smaller. The generator's answers are always fully reduced, so either method should end at 19/12.
Which Denominators for Which Grade
The US Common Core standards limit the denominators children meet each year. Type the list into Denominators, or use the presets.
| Grade | Denominators | Typical fraction work |
|---|---|---|
| 3 | 2, 3, 4, 6, 8 | Unit fractions, fractions on a number line, comparing with the same numerator or denominator |
| 4 | 2, 3, 4, 5, 6, 8, 10, 12, 100 | Equivalent fractions, adding and subtracting like denominators, mixed numbers, fraction × whole number |
| 5 | Any | Adding and subtracting unlike denominators, multiplying fractions, dividing with unit fractions |
| 6 | Any | Dividing a fraction by a fraction |
In the UK, the National Curriculum moves through a similar order across Years 3 to 6. Outside a curriculum, 2 to 12 is a good general range: big enough to need real common denominators, small enough to work mentally.
How Many Random Fractions Need Simplifying?
Fewer than you might think. Of the 66 proper fractions with denominators 2 to 12, 45 (68%) are already in lowest terms. For large numbers the share settles at 6/π², about 61%: the chance that two random whole numbers share no common factor.
So a page of "simplify these" questions made from fully random fractions would have mostly nothing to do. Set Form to Needs simplifying and every result has a common factor; set it to Already in lowest terms when you don't want simplification to distract from the skill being practised. Prime denominators such as 7 or 11 have no proper fractions that simplify, so a list made only of primes won't work with Needs simplifying.
To simplify, divide the top and bottom by their greatest common factor: for 18/24 that is 6, giving 3/4. If you only spot a smaller common factor, divide by it and repeat until none is left.
Which Fractions Give Repeating Decimals
A fraction in lowest terms has a terminating decimal only if its denominator has no prime factors other than 2 and 5. Every other denominator repeats forever. The answers here show the repeating block in brackets, so 0.1(6) means 0.1666…
| Fraction | Decimal | Terminates? |
|---|---|---|
| 1/2 | 0.5 | Yes (2) |
| 1/3 | 0.(3) | No, repeats 3 |
| 1/4 | 0.25 | Yes (2 × 2) |
| 1/5 | 0.2 | Yes (5) |
| 1/6 | 0.1(6) | No, 3 is a factor |
| 1/7 | 0.(142857) | No, 6-digit cycle |
| 1/8 | 0.125 | Yes (2 × 2 × 2) |
| 1/9 | 0.(1) | No |
| 1/10 | 0.1 | Yes (2 × 5) |
| 1/11 | 0.(09) | No, 2-digit cycle |
| 1/12 | 0.08(3) | No, 3 is a factor |
Simplify before deciding: 3/12 looks like it should repeat, but it reduces to 1/4 = 0.25. For a worksheet that converts fractions to decimals without long repeats, use denominators 2, 4, 5, 8, 10, 20, 25 and 100.
Building a Fraction Worksheet
- Pick a preset or set Make, the fraction type and the denominators for your class.
- Untick Show answers and press Copy. Each line is numbered, ready to paste into a document.
- Tick Show answers again and copy a second time for the answer key. The problems stay the same until you press Generate new or change a setting.
Subtraction problems are ordered so the answer is never negative. Division answers can be larger than either fraction, since dividing by a proper fraction makes a number bigger: 1/2 ÷ 1/4 = 2. For more than one worksheet with the same settings, generate, copy, then generate again.
Frequently asked questions
How do I get only fractions less than 1?
Set Fraction type to Proper. Every numerator is then smaller than its denominator, so values range from 1/d up to (d−1)/d.
Can I choose specific denominators like halves, quarters and eighths?
Yes. Type them as a list, such as 2, 4, 8, or as a range such as 2-12. You can mix both: 2-6, 8, 10.
What does the number in brackets in the decimal mean?
It is the repeating part. 0.(3) is 0.333… and 0.08(3) is 0.08333… Exact terminating decimals have no brackets.
Can I make fraction problems with mixed numbers?
Yes. Set Fraction type to Mixed numbers and choose an operation. Problems and answers are written as mixed numbers, such as 1 1/2 + 2 2/3 = 4 1/6.
Sources
- Common Core State Standards: Grade 3, Number and Operations—Fractions: denominators limited to 2, 3, 4, 6 and 8.
- Common Core State Standards: Grade 4, Number and Operations—Fractions: denominators 2, 3, 4, 5, 6, 8, 10, 12 and 100.
- Common Core State Standards: Grade 5, Number and Operations—Fractions: adding and subtracting unlike denominators.
- OEIS A059956: Decimal expansion of 6/π²: the probability that two random integers are coprime.
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