Calculating with algebraic fractions

The same rules as for fractions

Good news: you calculate with algebraic fractions exactly like with ordinary fractions. No new rules. You just write x instead of numbers sometimes.

Adding? You need a common denominator — like with 1/2 + 1/3. Multiplying? Numerator times numerator, denominator times denominator. Dividing? You multiply by the reciprocal fraction.

Who can do fractions can do this too. Who has forgotten fractions now has a great chance to wake them up — there is training waiting below.

The only extra is the conditions from the last lesson. For every expression, note right at the start what must not be zero in the denominator.

Today we will go through all three operations on simple expressions. No long bracket monsters — those do not usually show up in entrance exams. What shows up is exactly this: a short expression, two operations, a condition.

Filip and Sofie compare homework. Filip has 3/x for 1/x + 2/x, Sofie has 3/2x. “I added the numerators and the denominators,” Sofie defends herself. Filip pulls out ordinary fractions: “If that worked, 1/4 + 1/4 would be 2/8, which is 1/4 again. Two quarters of pizza would be the same as one. I would like that in a shop, but not on a plate.” Sofie laughs: “Right. You do not add denominators. The denominator says how big the pieces are — and that does not change.”

💡

If you are not sure about a rule, try it on ordinary fractions with numbers. What holds for 1/2 and 1/3 also holds for 1/x and 1/y.

Three operations, three methods

Adding with the same denominator: add the numerators, copy the denominator. 2/x + 3/x = 5/x. Condition x ≠ 0.

Adding with different denominators: find a common denominator. For 1/(2x) + 1/(3x) the common denominator is 6x. You expand the first fraction by three: 3/(6x). The second by two: 2/(6x). Sum: 5/(6x).

Multiplying: top times top, bottom times bottom. (2/x) · (3/5) = 6/(5x). Before multiplying it pays to cancel across — you save yourself big numbers.

Dividing: flip the second fraction and multiply. (4/x) : (2/3) = (4/x) · (3/2) = 12/(2x) = 6/x.

With division an extra condition appears: you must not divide by zero, so the whole second fraction must not be zero either. Its numerator therefore must not be zero either.

Method for every question: conditions first, then the operation, finally cancel the result if you can. Three lines, no rush. Most mistakes come from skipping steps in your head — on paper a mistake has nowhere to hide.

Example 1: 3/x + 4/x

Question: Add 3/x + 4/x.

Method: Condition: x ≠ 0. The denominators are the same, so add only the numerators: 3 + 4 = 7. コピー the denominator.

Result: 7/x, condition x ≠ 0.

Check by substituting: For x = 2 the question is 3/2 + 4/2 = 1.5 + 2 = 3.5. The result 7/2 = 3.5. It matches.

Example 2: 1/(2x) + 1/(3x)

Question: Add 1/(2x) + 1/(3x).

Method: Condition: x ≠ 0. Denominators 2x and 3x — the common denominator is 6x.

Expand the first fraction by three: 1/(2x) = 3/(6x). Expand the second by two: 1/(3x) = 2/(6x).

Add the numerators: 3 + 2 = 5.

Result: 5/(6x), condition x ≠ 0.

Check: For x = 1 the question is 1/2 + 1/3 = 5/6. The result 5/6. It matches exactly.

Example 3: (6/x) : (3/2)

Question: Divide (6/x) : (3/2).

Method: Condition: x ≠ 0. Dividing by a fraction = multiplying by the reciprocal: (6/x) · (2/3).

Top: 6 · 2 = 12. Bottom: x · 3 = 3x. You have 12/(3x).

Cancel by three: 12/(3x) = 4/x.

Result: 4/x, condition x ≠ 0.

Check: For x = 2 the question is (6/2) : (3/2) = 3 : 1.5 = 2. The result 4/2 = 2. It matches.

⚠️

You do not add denominators when you add. 1/x + 1/x is not 2/(2x), but 2/x. The denominator says the size of the pieces — that does not change when you add. You only add the number of pieces, so the numerators. Who mixes this up should remember pizza: a quarter plus a quarter is a half, not a quarter.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. スタート with ten. When it goes well, add more.

Practise adding ordinary fractions — it is the same method as for algebraic fractions. Write fractions as a/b.

Practise the questions

On paper

Three operations in a row. For each question the first line is the condition, the last line is the cancelled result.

Add: 2/x + 5/x and then 1/(2x) + 1/(4x). For the second one find the common denominator 4x.
Multiply: (3/x) · (2/5) and then (x/2) · (4/x). For the second one cancel across before multiplying.
Divide: (8/x) : (2/3). Flip, multiply, cancel.
Check any one result by substituting x = 2. The question and the result must give the same number.

Now you 💪

You added fractions with different denominators through a common denominator, not by adding the denominators.
For division you flipped the second fraction and then multiplied.
Every question has the condition x ≠ 0 on the first line.

✅

Done when: You have solved adding, multiplying and dividing algebraic fractions, and one check by substituting came out right.

What to take from this lesson

You calculate algebraic fractions the same way as ordinary fractions.
Adding wants a common denominator. You never add denominators.
Multiplying: top times top, bottom times bottom. Cancel across beforehand.
Dividing: flip the second fraction and multiply.
Write conditions on the first line — in entrance exams they are worth marks.