Adding & Subtracting Fractions with Unlike Denominators — 25 Practice Questions with Theory & Hidden Answers (Kid Einsteins Class 19)

You can only add pieces that are the same size

Think of a pizza cut into halves and another pizza cut into thirds. If someone has 1/2 of one pizza and 1/3 of the other, how much pizza do they have in total? You can’t just say “1/2 + 1/3 = 2/5.” That would mean 2 out of 5 pieces — but the halves and the thirds are not the same size, so 2 out of 5 isn’t a real amount.

The fix is to cut both pizzas into the same-sized pieces. If we cut both into sixths, the half-pizza becomes 3 pieces (1/2 = 3/6) and the third-pizza becomes 2 pieces (1/3 = 2/6). Now we can add: 3/6 + 2/6 = 5/6.

The rule: before you can add or subtract fractions, both denominators must be the same number. That shared number is called a common denominator. The smallest one that works is called the least common denominator, or LCD.

The LCD is the LCM of the two denominators

The least common denominator of two fractions is the smallest number that both denominators divide into evenly. It is the same as the least common multiple (LCM) of the two denominators.

Three ways to find the LCD:

1. List the multiples of each denominator until you find one they share. Fastest for small numbers.
2. Check if one is a multiple of the other. If it is, the larger one is the LCD. Example: for 3 and 6, the LCD is 6.
3. Multiply the denominators. This always gives a common denominator (just not always the smallest). Great as a backup: 3 × 4 = 12, so 12 is a common denominator of 3 and 4.

Common LCDs to know cold:

Add 1/3 + 1/2 step by step

Why step 2 works: to rewrite 1/3 as sixths, multiply the top and the bottom by the same number. 3 × 2 = 6, so 1 × 2 = 2, giving 2/6. Multiplying top and bottom by the same number doesn’t change the value of a fraction — it just re-labels it in smaller pieces. This is the equivalent-fractions idea from Class 10.

Exactly the same four steps — just subtract in step 3

Worked example: 2/3 − 1/2.

1. LCD of 3 and 2 is 6.
2. Rewrite: 2/3 = 4/6 and 1/2 = 3/6.
3. Subtract numerators: 4/6 − 3/6 = 1/6.
4. Simplify: 1/6 is already in lowest terms.

That’s the second circle model from the hero image at the top of this page: two-thirds of a pie minus one-half of a pie leaves one-sixth of a pie.

Adding: a/b + c/d = (a·d + b·c) / (b·d)

Subtracting: a/b − c/d = (a·d − b·c) / (b·d)

Cross-multiply each numerator with the other denominator, then the denominator is just b · d. It always works, but the denominator bd may not be the LCD, so you may need to simplify. Two examples:

• 1/3 + 1/2 = (1·2 + 3·1) / (3·2) = 5/6 — already simplified.
• 1/4 + 1/6 = (1·6 + 4·1) / (4·6) = 10/24 = 5/12 — simplify by dividing top and bottom by 2.

Use the LCD method to keep numbers small and avoid simplifying at the end. Use the cross-multiply shortcut when the LCD is awkward or you’re doing quick mental math.

1. Adding the denominators. 1/2 + 1/3 is not 2/5. The denominator names the piece size; once pieces are the same size, only the number of pieces (the numerator) changes.
2. Multiplying the numerator by one number and the denominator by another. To rewrite 1/3 as sixths, multiply both top and bottom by 2. Not one by 2 and the other by 3.
3. Forgetting to simplify. 6/8 is a correct answer, but 3/4 is the finished answer.
4. Using the wrong LCD. Every common denominator works, but the LCD keeps numbers small. 1/3 + 1/6: LCD is 6, not 18.
5. Ignoring signs in subtraction. 1/4 − 1/3 is negative (it’s 3/12 − 4/12 = −1/12). For grade 3–4 problems, always make sure the first fraction is larger before you subtract, or use a number line.