How to generate two random irreducible fractions

When to use this: Use this method when you want students to compute the sum of two fractions and ensure that each generated fraction is already irreducible (i.e., simplified before the operation).

What you'll achieve: You will generate two random fractions whose numerators and denominators are coprime, guaranteeing that both fractions are irreducible before students calculate their sum.

See it in action: Watch how this logic is implemented inside Learning Lemur:

Before you begin

Requirements

• Basic knowledge of how to create a Learning Lemur question
• Familiarity with adding an algorithm to generate random variables

Steps

Generate the first numerator — Select a random value

a = random(1,9)

This defines the numerator of the first fraction.

Generate a compatible denominator — Ensure coprimality

B = {b with b in 2..9 where gcd(a,b) = 1}
b = random(B)

This creates a set of denominators coprime to a, ensuring the fraction is irreducible.

Generate the second numerator — Select a new random value

c = random(1,9)

Generate the second denominator — Ensure coprimality and avoid duplication

D = {d with d in 2..9 where gcd(c,d) = 1 && d != b}
d = random(D)

This ensures:

• The second fraction is irreducible
• The denominators are not identical

Define the solution — Compute the sum

sol = a/b + c/d

This stores the correct answer for grading.

Verify it worked

• Preview the question multiple times
• Confirm that each generated fraction is already simplified
• Check that no denominator equals 0
• Ensure the sum is computed correctly

Full algorithm (copy-paste version)

Use the complete version below if you want to copy the logic directly into your question algorithm:

# Generate the first fraction
a = random(1,9)    

B = {b with b in 2..9 where gcd(a,b) = 1}  
b = random(B)               

# Generate the second fraction
c = random(1,9)   

D = {d with d in 2..9 where gcd(c,d) = 1 && d != b} 
d = random(D)                  

# Compute the solution
sol = a/b + c/d

Options and variations

• If you want to allow identical denominators: Remove the condition d != b from the second denominator set
• If you want larger fractions: Increase the interval (e.g., 1..15 instead of 1..9), but test performance to avoid excessive regeneration
• If you want to enforce simplified student answers: Combine this pattern with answer validation constraints in grading logic.

Common errors

• Fractions are not irreducible: Check that gcd(a,b)=1 is correctly written in the set definition
• Infinite regeneration loop: Ensure the denominator interval contains enough values to satisfy the coprimality condition
• Unexpected duplicate fractions: Add additional constraints if you need both fractions to differ entirely

Related

• Reference: [feature/settings]
• Explanation: [concept]