Adding and subtracting algebraic fractions

The Maths Habit

Jo says 12 + 12 = 24.
Is she right?

No: two halves make 1, and 24 is only a half

Work out

  1. 13 + 14

    712

  2. 25 + 13

    1115

  3. 34 − 25

    720

The same length x, cut into 3 and into 5

xx ÷ 3xx ÷ 5

Which piece is bigger, x3 or x5? x3

Cut both bars into 15 equal pieces

x5 piecesx3 pieces

x3 = 5x15   x5 = 3x15

x3 + x5: put the pieces together

x8 pieces of 15

x3 + x5 = 5x15 + 3x15 = 8x15

The quick way: make the bottoms the same.

Simplify x3 + x5

make the bottoms the same
x × 53 × 5 + x × 35 × 3
now add or subtract the tops
= 5x + 3x15
Answer8x15

Simplify 2x3 + x4

make the bottoms the same
2x × 43 × 4 + x × 34 × 3
now add or subtract the tops
= 8x + 3x12
Answer11x12

Simplify x3 + x5

make the bottoms the same
x × 53 × 5 + x × 35 × 3
now add or subtract the tops
= 5x + 3x15
Answer8x15

Simplify 4x5 − x3

make the bottoms the same
4x × 35 × 3 − x × 53 × 5
now add or subtract the tops
= 12x − 5x15
Answer7x15

What is the common denominator?

2x + 1 + 3x + 2 (x + 1)(x + 2)

Simplify 3x + 1 + 2x + 5

make the bottoms the same
3 × (x + 5)(x + 1) × (x + 5) + 2 × (x + 1)(x + 5) × (x + 1)
now add or subtract the tops
= 3(x + 5) + 2(x + 1)(x + 1)(x + 5)
expand and collect the top
= 5x + 17(x + 1)(x + 5)
Answer5x + 17(x + 1)(x + 5)

Simplify 4x + 2 − 3x + 4

make the bottoms the same
4 × (x + 4)(x + 2) × (x + 4) − 3 × (x + 2)(x + 4) × (x + 2)
now add or subtract the tops
= 4(x + 4) − 3(x + 2)(x + 2)(x + 4)
expand and collect the top
= x + 10(x + 2)(x + 4)
Answerx + 10(x + 2)(x + 4)

Simplify 3x + 1 + 2x + 5

make the bottoms the same
3 × (x + 5)(x + 1) × (x + 5) + 2 × (x + 1)(x + 5) × (x + 1)
now add or subtract the tops
= 3(x + 5) + 2(x + 1)(x + 1)(x + 5)
expand and collect the top
= 5x + 17(x + 1)(x + 5)
Answer5x + 17(x + 1)(x + 5)

Simplify 2x + 3 + 3x + 1

make the bottoms the same
2 × (x + 1)(x + 3) × (x + 1) + 3 × (x + 3)(x + 1) × (x + 3)
now add or subtract the tops
= 2(x + 1) + 3(x + 3)(x + 3)(x + 1)
expand and collect the top
= 5x + 11(x + 3)(x + 1)
Answer5x + 11(x + 3)(x + 1)

Adding and subtracting algebraic fractions worksheet, page 1

Answer sheet

  1. Simplify 3x + 2x + 1

    5x + 3x(x + 1)

  2. Solve x2 + x3 = 10

    x = 12

  3. Simplify 6x + 4 − 2x + 3

    6(x + 3) − 2(x + 4) = 4x + 10, so 4x + 10(x + 4)(x + 3)

  1. Write x4 + x6 as a single fraction.

    5x12

  2. Write 2x + 3 + 1x − 1 as a single fraction.

    3x + 1(x + 3)(x − 1)

  3. Write 5x + 2 − 3x + 1 as a single fraction.

    2x − 1(x + 2)(x + 1)
Exit tickets to print