Worksheet for practicing adding and subtracting algebraic fractions with progressive difficulty levels and a literacy challenge.

Let’s solve each problem step by step. We’ll go section by section, starting with Skill 1 and moving through all the skills and stretches.

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Skill 1: ROK - Warm up

These are simple fraction additions/subtractions with same or different denominators.

1)
3/7 + 2/7 = (3+2)/7 = 5/7

2)
1/2 + 3/4 → common denominator is 4
= 2/4 + 3/4 = 5/4 or 1¼

3)
1/4 + 1/2 → common denominator is 4
= 1/4 + 2/4 = 3/4

4)
3/5 – 2/7 → common denominator is 35
= (3×7)/(5×7) – (2×5)/(7×5) = 21/35 – 10/35 = 11/35

✔ Skill 1 Answers:
1) 5/7
2) 5/4 or 1¼
3) 3/4
4) 11/35

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Skill 2

Algebraic fractions — combine like terms after getting common denominators.

1)
x/2 + x/3 → LCD = 6
= (3x)/6 + (2x)/6 = 5x/6

2)
2x/8 + 3x/8 = (2x + 3x)/8 = 5x/8

3)
7x/9 – x/8 → LCD = 72
= (7x×8)/72 – (x×9)/72 = 56x/72 – 9x/72 = 47x/72

4)
3y/9 + 8y/5 → simplify first: 3y/9 = y/3
Now: y/3 + 8y/5 → LCD = 15
= (5y)/15 + (24y)/15 = 29y/15

5)
x/4 + y/4 = (x + y)/4

✔ Skill 2 Answers:
1) 5x/6
2) 5x/8
3) 47x/72
4) 29y/15
5) (x + y)/4

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Skill 3

More algebraic fractions — watch signs and variables.

1)
x/4 + x/2 → LCD = 4
= x/4 + 2x/4 = 3x/4

2)
5x/4 – 3x/14 → LCD = 28
= (5x×7)/28 – (3x×2)/28 = 35x/28 – 6x/28 = 29x/28

3)
3x/5 + x/4 → LCD = 20
= (12x)/20 + (5x)/20 = 17x/20

4)
5y/7 + 2y/3 → LCD = 21
= (15y)/21 + (14y)/21 = 29y/21

5)
7x/7 – x/10 → simplify 7x/7 = x
So: x – x/10 = 10x/10 – x/10 = 9x/10

✔ Skill 3 Answers:
1) 3x/4
2) 29x/28
3) 17x/20
4) 29y/21
5) 9x/10

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Skill 4

Fractions with variables in denominator — find LCD carefully.

1)
3/x + 2/(2x) → simplify 2/(2x) = 1/x
So: 3/x + 1/x = 4/x

2)
2/(5x) – 7/(10x) → LCD = 10x
= (4)/(10x) – 7/(10x) = -3/(10x)

3)
2/(7x) + 3/7 → LCD = 7x
= 2/(7x) + (3x)/(7x) = (2 + 3x)/(7x)

4)
5/y + 3x/(xy) → LCD = xy
= (5x)/(xy) + 3x/(xy) = (5x + 3x)/(xy) = 8x/(xy) = 8/y *(simplify by canceling x)*

Wait — let’s check that again:

5/y = (5x)/(xy)
3x/(xy) stays as is
Sum: (5x + 3x)/(xy) = 8x/(xy) = 8/y ✔

5)
2/(3x) – 5/(2y) → LCD = 6xy
= (2×2y)/(6xy) – (5×3x)/(6xy) = 4y/(6xy) – 15x/(6xy) = (4y – 15x)/(6xy)

✔ Skill 4 Answers:
1) 4/x
2) -3/(10x)
3) (2 + 3x)/(7x)
4) 8/y
5) (4y – 15x)/(6xy)

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Stretch 1

Three-term fractions — add them step by step.

1)
x/2 + x/3 + x/4 → LCD = 12
= (6x)/12 + (4x)/12 + (3x)/12 = 13x/12

2)
7x/10 + 2x/5 – x/2 → LCD = 10
= 7x/10 + 4x/10 – 5x/10 = (7x + 4x – 5x)/10 = 6x/10 = 3x/5

3)
5x/6 – x/3 + 4x/9 → LCD = 18
= (15x)/18 – (6x)/18 + (8x)/18 = (15x – 6x + 8x)/18 = 17x/18

✔ Stretch 1 Answers:
1) 13x/12
2) 3x/5
3) 17x/18

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Stretch 2

Variables in denominators — be careful with factoring.

1)
4/(xy²) – x/y → LCD = xy²
= 4/(xy²) – (x·xy)/(y·xy) → wait, better:
x/y = (x · xy)/(y · xy)? No.

Actually:
x/y = (x · xy)/(y · xy)? That’s wrong.

Correct way:
To get denominator xy² from y, multiply numerator and denominator by xy:
x/y = (x · xy) / (y · xy) = x²y / (xy²)? No — that’s overcomplicating.

Better:
LCD of xy² and y is xy².
So:
4/(xy²) – x/y = 4/(xy²) – (x · xy)/(y · xy) → still messy.

Wait — simpler:
x/y = (x * x y) / (y * x y)? No.

Standard method:
Multiply numerator and denominator of second term by xy to get denominator xy²? Let's see:

Denominator y → to become xy², multiply by xy.
So:
x/y = (x * xy) / (y * xy) = x²y / (xy²)

But that gives us:
4/(xy²) – x²y/(xy²) = (4 – x²y)/(xy²)

That seems correct but unusual. Maybe we can leave it as is? Or did I make a mistake?

Wait — perhaps the problem expects simplification without expanding.

Alternative approach:
Write both terms with denominator xy²:

First term: 4/(xy²) — already good.

Second term: x/y = ? / (xy²)
Multiply numerator and denominator by xy:
x/y = (x * xy) / (y * xy) = x²y / (xy²)

Yes, so:
4/(xy²) – x²y/(xy²) = (4 – x²y)/(xy²)

But this looks odd. Let me double-check with numbers.

Suppose x=1, y=1:
Original: 4/(1*1) – 1/1 = 4 – 1 = 3
My answer: (4 – 1*1)/(1*1) = 3/1 = 3 → OK.

Another test: x=2, y=1
Original: 4/(2*1) – 2/1 = 2 – 2 = 0
My answer: (4 – 4*1)/(2*1) = 0/2 = 0 → OK.

So it’s correct: (4 – x²y)/(xy²)

But maybe they want it factored? Probably not necessary.

2)
3x/(x²) + 2y/(yx) → simplify first:
3x/x² = 3/x
2y/(yx) = 2/x (since y cancels)
So: 3/x + 2/x = 5/x

3)
5/(2xyz) + 6x/(x²yz) → simplify second term:
6x/(x²yz) = 6/(xyz)
Now: 5/(2xyz) + 6/(xyz) = 5/(2xyz) + 12/(2xyz) = 17/(2xyz)

✔ Stretch 2 Answers:
1) (4 – x²y)/(xy²)
2) 5/x
3) 17/(2xyz)

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Stretch 3

Subtracting algebraic fractions — watch signs.

1)
(2x+1)/2 – (3x+1)/5 → LCD = 10
= [5(2x+1) – 2(3x+1)] / 10
= [10x + 5 – 6x – 2] / 10
= (4x + 3)/10

2)
(7x+4)/2 – (2x–1)/7 → LCD = 14
= [7(7x+4) – 2(2x–1)] / 14
= [49x + 28 – 4x + 2] / 14
= (45x + 30)/14
Simplify? Factor numerator: 15(3x + 2)/14 → no common factors, so leave as (45x + 30)/14

Or factor out 3: 3(15x + 10)/14 — still not simpler. So best as (45x + 30)/14.

✔ Stretch 3 Answers:
1) (4x + 3)/10
2) (45x + 30)/14

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Skill 5

Adding fractions with binomial numerators.

1)
x/2 + (x+1)/4 → LCD = 4
= (2x)/4 + (x+1)/4 = (2x + x + 1)/4 = (3x + 1)/4

2)
(x+2)/2 + (x+3)/4 → LCD = 4
= [2(x+2) + (x+3)] / 4 = [2x + 4 + x + 3]/4 = (3x + 7)/4

3)
(x–3)/3 + (x+4)/4 → LCD = 12
= [4(x–3) + 3(x+4)] / 12 = [4x – 12 + 3x + 12]/12 = (7x)/12

4)
(x+2)/5 + (x–3)/3 → LCD = 15
= [3(x+2) + 5(x–3)] / 15 = [3x + 6 + 5x – 15]/15 = (8x – 9)/15

5)
(2x+1)/2 + (3x+1)/5 → LCD = 10
= [5(2x+1) + 2(3x+1)] / 10 = [10x + 5 + 6x + 2]/10 = (16x + 7)/10

✔ Skill 5 Answers:
1) (3x + 1)/4
2) (3x + 7)/4
3) 7x/12
4) (8x – 9)/15
5) (16x + 7)/10

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Stretch 4

Equations — solve for x.

1)
x/5 + x/3 = 16 → LCD = 15
= (3x + 5x)/15 = 8x/15 = 16
→ 8x = 16 × 15 = 240
→ x = 240 / 8 = 30

2)
4x/3 – x/7 = 23 → LCD = 21
= (28x – 3x)/21 = 25x/21 = 23
→ 25x = 23 × 21 = 483
→ x = 483 / 25 = 19.32 or as fraction 483/25

Check: 23 × 21 = 483? 20×21=420, 3×21=63 → 483 yes.
483 ÷ 25 = 19.32 — but maybe keep as fraction: 483/25

3)
5x/7 + 2x/6 = 22 → simplify 2x/6 = x/3
So: 5x/7 + x/3 → LCD = 21
= (15x + 7x)/21 = 22x/21 = 22
→ 22x = 22 × 21 = 462
→ x = 462 / 22 = 21

4)
(x+2)/2 + (x–2)/3 = 7 → LCD = 6
= [3(x+2) + 2(x–2)] / 6 = [3x + 6 + 2x – 4]/6 = (5x + 2)/6 = 7
→ 5x + 2 = 42
→ 5x = 40
→ x = 8

5)
(3x–6)/2 + (2x+4)/3 = 7 → LCD = 6
= [3(3x–6) + 2(2x+4)] / 6 = [9x – 18 + 4x + 8]/6 = (13x – 10)/6 = 7
→ 13x – 10 = 42
→ 13x = 52
→ x = 4

✔ Stretch 4 Answers:
1) x = 30
2) x = 483/25 or 19.32
3) x = 21
4) x = 8
5) x = 4

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## Final Answer:

Skill 1:
1) 5/7
2) 5/4
3) 3/4
4) 11/35

Skill 2:
1) 5x/6
2) 5x/8
3) 47x/72
4) 29y/15
5) (x + y)/4

Skill 3:
1) 3x/4
2) 29x/28
3) 17x/20
4) 29y/21
5) 9x/10

Skill 4:
1) 4/x
2) -3/(10x)
3) (2 + 3x)/(7x)
4) 8/y
5) (4y – 15x)/(6xy)

Stretch 1:
1) 13x/12
2) 3x/5
3) 17x/18

Stretch 2:
1) (4 – x²y)/(xy²)
2) 5/x
3) 17/(2xyz)

Stretch 3:
1) (4x + 3)/10
2) (45x + 30)/14

Skill 5:
1) (3x + 1)/4
2) (3x + 7)/4
3) 7x/12
4) (8x – 9)/15
5) (16x + 7)/10

Stretch 4:
1) x = 30
2) x = 483/25
3) x = 21
4) x = 8
5) x = 4