Year 11 Extended · Unit 4 Rational expressions and functions · Lesson 3 of 7

How do you add two fractions that have letters in the bottom?

Adding fractions with letters in the bottom can look like adding numeric fractions. This lesson tests a tempting shortcut, then builds a common denominator from the factorized bottoms that works for adding and subtracting.

FunctionsNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Year 7: work out 1⁄4 + 1⁄6.
    Answer5⁄12. The common denominator is 12.
  2. From Year 9: expand 3(x − 4) − (x + 2).
    Answer2x − 14. The minus sign multiplies both terms of the bracket.
  3. From lesson 1: factorize x2 − 16.
    Answer(x − 4)(x + 4).
Words used in this lesson
common denominator
a bottom that both fractions can be written over. For example: 12 for 1/4 and 1/6.
combine
write two fractions as one. For example: p/3 and q/4 together.
collect like terms
add the terms with the same letter part. For example: 7x + 2x = 9x.
rate
how much is done in one unit of time. For example: 30 pages printed each minute.
bracket
a pair of curved lines that group terms. For example: (x − 2).

Why can't you just add the tops and the bottoms?

Kwame wants the total of 3 ÷ (x + 1) and 2 ÷ (x + 2). He adds the tops and adds the bottoms and writes 5 ÷ (2x + 3).

Predict first: at x = 2, will Kwame's fraction match the sum of the two original fractions? Work out both by hand, then use the figure to see the correct single fraction and the check at x = 2.

3
2
5x + 8combined numerator
18numerator at x = 2, over 12

p = 3, q = 2: the sum is (5x + 8) over (x + 1)(x + 2). At x = 2 it is 18/12.

Show me the steps
  1. p = 3, q = 2: 3(x + 2) + 2(x + 1) = 5x + 8. At x = 2 the fractions are 3/3 and 2/4, and together they are 18/12, the same as (5 × 2 + 8)/12.
  2. p = 1, q = 4: 1(x + 2) + 4(x + 1) = 5x + 6. The x coefficient is again 5, but the constant is now 6. At x = 2 the sum is 16/12.
  3. p = 5, q = 5: 5(x + 2) + 5(x + 1) = 10x + 15. Both fractions are bigger, and at x = 2 the sum is 35/12.

Common denominator, then combine the tops. Factorize every bottom. The common denominator contains each factor once. Write each fraction over it by multiplying its top by the factors it was missing, then add or subtract the tops.

For a subtraction, put brackets round the second top so that the minus sign multiplies every term. Then expand, collect and factorize the result to see if anything cancels.

Adding tops and bottoms is wrong. With numbers, 1⁄2 + 1⁄3 = 5⁄6, but adding tops and bottoms gives 2⁄5, which is smaller than 1⁄2. The same thing goes wrong with letters.

Worked examples

1. Write 2 ÷ (x + 3) + 5 ÷ (x − 1) as one fraction.

  1. Common denominator (x + 3)(x − 1). The top is 2(x − 1) + 5(x + 3).
  2. Expand and collect: 2x − 2 + 5x + 15 = (7x + 13) ÷ ((x + 3)(x − 1)).

2. Write 3 ÷ (x − 2) − 2 ÷ (x + 4) as one fraction.

  1. The top is 3(x + 4) − 2(x − 2). The brackets matter: −2 multiplies both x and −2.
  2. 3x + 12 − 2x + 4 = (x + 16) ÷ ((x − 2)(x + 4)).

3. Write 4 ÷ (x2 − 9) + 1 ÷ (x − 3) as one fraction.

  1. Factorize first: x2 − 9 = (x − 3)(x + 3), so the common denominator is just (x − 3)(x + 3).
  2. The top is 4 + 1(x + 3) = (x + 7) ÷ ((x − 3)(x + 3)).

Your go. The first step is given. Write 5 ÷ (x + 2) − 3 ÷ (x2 − 4) as one fraction. The common denominator is (x + 2)(x − 2).

  1. Put the first fraction over the common denominator: 5(x − 2) on the top.
  2. Subtract 3, with brackets, and collect.
Check the answer

5(x − 2) − 3 = 5x − 13, so the fraction is (5x − 13) ÷ ((x + 2)(x − 2)).

Check before you go on

1. Which is x ÷ (x + 2) − (x − 3) ÷ (x − 1) as a single fraction?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. 3 ÷ (x − 1) + 4 ÷ (x + 2) = (ax + b) ÷ ((x − 1)(x + 2)). Find a + b.

3. 5 ÷ (x − 2) − 1 ÷ (x + 3) = (4x + b) ÷ ((x − 2)(x + 3)). Find b.

Level 2 · Building

4. 2 ÷ (x − 3) + 6 ÷ (x2 − 9) = (2x + c) ÷ (x2 − 9). Find c.

5. 4 ÷ (x − 1) − 4 ÷ (x + 1) = k ÷ (x2 − 1). Find k.

Level 3 · Stretch

6. 3 ÷ (x − 1) + a ÷ (x + 2) is written as one fraction whose top is 9x, with no constant term. Find a.

7. Tap A fills a tank alone in x hours and tap B alone in 2x − 2 hours. Write the fraction of the tank they fill together in one hour as a single fraction in x. Use it to find how many hours both taps take together when x = 6. Give your answer as a decimal.

Something to think about

In the figure p/(x + 1) + q/(x + 2) always has a numerator of the form (p + q)x + something. When is the combined numerator a multiple of x, with no constant? Try to find a rule for p and q.

One way to think about it

The constant is 2p + q, which is positive whenever p and q are positive. It is zero only if q = −2p, for example p = 1 and q = −2. The sliders cannot show it, but the algebra can.

Exit ticket

8. What is the simplest common denominator for 5 ÷ (x2 − 49) and 2 ÷ (x2 + 7x)?