Steply

Complex fractions

Algebra 2 · 21 problems, easy to harder

▶ Watch the first one solved

How we do it

  1. Find the common bottom
  2. Times 6 on top and bottom
  3. Multiply every part
  4. Simplify
  5. Check

The steps for Simplify (1/2 + 1/3)/(5/6). Every solution names its steps, so you always know what comes next.

Worked examples

Simplify (1/2 + 1/3)/(5/6) Easy

The question

Question: simplify this fraction.

On top: 1/2 + 1/3.

At the bottom: 5/6.

A fraction with fractions inside is called a complex fraction.

12 + 1356

1 Find the common bottom

The small fractions have bottoms 2, 3 and 6.

The smallest thing they all go into is 6.

6 is in the 2 and 3 times tables.

small bottoms: 2, 3, 6
common bottom: 6

2 Times 6 on top and bottom

6/6 is 1.

Times 1 does not change the value.

66 = 1

We multiply the top and the bottom by 6.

Then the small fractions go away.

= 6 × (12 + 13)6 × (56)

3 Multiply every part

Every part times 6.

6 times 1/2: the bottom 2 goes into 6, so we get 3.

6 times 1/3: the bottom 3 goes into 6, so we get 2.

6 × 12 = 3
6 × 13 = 2
6 × 56 = 5

So we get (3 + 2)/5.

No more small fractions!

= 3 + 25

Like terms go together: 5/5.

= 55

4 Simplify

Every part, on the top and the bottom, can be divided by 5.

So we divide the top and the bottom by it.

÷ 5: 5 → 1
÷ 5: 5 → 1

So the answer is 1.

= 1

5 Check

Let's check the long way: work out the top and the bottom first.

The top is 5/6, the bottom is 5/6.

top: 36 + 26 = 56, bottom: 56

Then top divided by bottom: times the flip.

We get 1 again.

It works!

56 ÷ 56 = 56 × 65 = 1

Let's look back.

First: find the common bottom.

Then: times 6 on top and bottom.

Then: multiply every part.

Then: simplify.

Last: check.

Answer
= 1

▶ Watch it solved

Simplify (2 + 1/4)/(3/8) Medium

The question

Question: simplify this fraction.

On top: 2 + 1/4.

At the bottom: 3/8.

A fraction with fractions inside is called a complex fraction.

2 + 1438

1 Find the common bottom

The small fractions have bottoms 4 and 8.

The smallest thing they all go into is 8.

8 is in the 4 times table.

small bottoms: 4, 8
common bottom: 8

Careful!

We multiply EVERY part by 8, also the part with no small bottom.

Leaving it out is the usual mistake.

Careful: every part times 8, the whole number too

2 Times 8 on top and bottom

8/8 is 1.

Times 1 does not change the value.

88 = 1

We multiply the top and the bottom by 8.

Then the small fractions go away.

= 8 × (2 + 14)8 × (38)

3 Multiply every part

Every part times 8.

8 times 2 is 16.

8 times 1/4: the bottom 4 goes into 8, so we get 2.

8 × 2 = 16
8 × 14 = 2
8 × 38 = 3

So we get (16 + 2)/3.

No more small fractions!

= 16 + 23

Like terms go together: 18/3.

= 183

4 Simplify

Every part, on the top and the bottom, can be divided by 3.

So we divide the top and the bottom by it.

÷ 3: 18 → 6
÷ 3: 3 → 1

So the answer is 6.

= 6

5 Check

Let's check the long way: work out the top and the bottom first.

The top is 9/4, the bottom is 3/8.

top: 84 + 14 = 94, bottom: 38

Then top divided by bottom: times the flip.

We get 6 again.

It works!

94 ÷ 38 = 94 × 83 = 6

Let's look back.

First: find the common bottom.

Then: times 8 on top and bottom.

Then: multiply every part.

Then: simplify.

Last: check.

Answer
= 6

▶ Watch it solved

Simplify (1/x + 1/3)/(1/x - 1/3) Harder

The question

Question: simplify this fraction.

On top: 1/x + 1/3.

At the bottom: 1/x − 1/3.

A fraction with fractions inside is called a complex fraction.

1x + 131x − 13

1 Find the common bottom

The small fractions have bottoms x and 3.

The smallest thing they all go into is 3x.

small bottoms: x, 3
common bottom: 3x

2 Times 3x on top and bottom

3x/3x is 1.

Times 1 does not change the value.

3x3x = 1

We multiply the top and the bottom by 3x.

Then the small fractions go away.

= 3x × (1x + 13)3x × (1x − 13)

3 Multiply every part

Every part times 3x.

3x times 1/x: the bottom x goes into 3x, so we get 3.

3x times 1/3: the bottom 3 goes into 3x, so we get x.

3x × 1x = 3
3x × 13 = x
3x × 1x = 3
3x × (−13) = −x

So we get (3 + x)/(3 − x).

No more small fractions!

= 3 + x3 − x

4 Simplify

Is there a number or an x that goes into every part, top and bottom?

No.

So it is already simple.

So the answer is (3 + x)/(3 − x).

nothing goes into every part

One more thing: no bottom can be 0.

So x can't be 0 or 3.

x ≠ 0, 3

5 Check

Let's check with a number: x = 2.

In the question, the long way: the top is 5/6, the bottom is 1/6.

x = 2: top 12 + 13 = 56, bottom 12 − 13 = 16

Top divided by bottom: 5.

question: 56 ÷ 16 = 5

Our answer with x = 2 gives 5 too.

It works!

answer: 3 + 23 − 2 = 5

Let's look back.

First: find the common bottom.

Then: times 3x on top and bottom.

Then: multiply every part.

Then: simplify.

Last: check.

Answer
nothing goes into every part

▶ Watch it solved

Practice

Try each one, then open the answer. Stuck? Open the problem to watch every step.

Easy

  1. (1/2 + 1/3)/(5/6)
    Answer
    = 1
  2. (1/2 + 1/4)/(3/8)
    Answer
    = 2
  3. (2 + 1/2)/(5/4)
    Answer
    = 2
  4. (3/4 - 1/2)/(1/8)
    Answer
    = 2
  5. (1 + 1/3)/(2/9)
    Answer
    = 6
  6. (1/3 + 1/6)/(1/2 - 1/4)
    Answer
    = 2
  7. (3/x)/(6/x^2)
    Answer
    = x2
  8. (2/x)/(4/x)
    Answer
    = 12

Medium

  1. (2 + 1/4)/(3/8)
    Answer
    = 6
  2. (1 + 1/x)/(1 - 1/x)
    Answer
    nothing goes into every part
  3. (1/x + 1/2)/(1/x - 1/2)
    Answer
    nothing goes into every part
  4. (2/x + 1)/(4/x)
    Answer
    nothing goes into every part
  5. (1 - 1/x)/(3/x)
    Answer
    nothing goes into every part
  6. (5/6 - 1/3)/(1/4 + 1/2)
    Answer
    = 23
  7. (4 - 1/2)/(1 + 3/4)
    Answer
    = 2
  8. (x + 1/x)/(2/x)
    Answer
    nothing goes into every part

Harder

  1. (1/x + 1/3)/(1/x - 1/3)
    Answer
    nothing goes into every part
  2. (2 + 3/x)/(4 - 1/x)
    Answer
    nothing goes into every part
  3. (1/2 - 1/x)/(x/2)
    Answer
    nothing goes into every part
  4. (3/x^2)/(9/x)
    Answer
    = 13x
  5. (1 + 2/x)/(1/x)
    Answer
    = x + 2