Simplify complex fractions in this math practice worksheet.

Let’s solve each problem step by step. These are complex fractions — that means a fraction on top of another fraction. To simplify them, we divide the top fraction by the bottom fraction. And dividing by a fraction is the same as multiplying by its reciprocal (flip the bottom fraction).

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Problem 1:
We have:
  ½
  —
  ⁄₅ → Wait, no! It’s ½ over ⁵⁄₁? No — looking again: it’s ½ over ⁵, which is written as:

Actually, in the image, Problem 1 is:
  ½
  —
  

That means: (1/2) ÷ 5

But 5 is the same as 5/1.

So:
(1/2) ÷ (5/1) = (1/2) × (1/5) = 1/10

✔ Final Answer for #1: 1/10

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Problem 2:
  4/3
  —
  8/8 → Wait, no — it’s 4/3 over 8, meaning:

(4/3) ÷ 8

8 = 8/1

So:
(4/3) × (1/8) = 4/(3×8) = 4/24 = 1/6

✔ Final Answer for #2: 1/6

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Problem 3:
  8/9
  —
  20

Meaning: (8/9) ÷ 20

20 = 20/1

So:
(8/9) × (1/20) = 8/(9×20) = 8/180

Simplify: divide numerator and denominator by 4 → 2/45

✔ Final Answer for #3: 2/45

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Problem 4:
  5/6
  —
  12

Meaning: (5/6) ÷ 12

12 = 12/1

So:
(5/6) × (1/12) = 5/(6×12) = 5/72

Already simplified.

✔ Final Answer for #4: 5/72

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Problem 5:
  8
  —
  11/4
  —
  5

Wait — this one looks like:
Top: 8
Middle: 11/4
Bottom: 5

Actually, looking at the layout: it’s 8 divided by (11/4 divided by 5)? Or is it (8 / (11/4)) / 5?

No — standard interpretation of stacked fractions:
It’s 8 over (11/4 over 5) → but actually, in math notation, when you see:

8
----
11/4
----
5

That usually means: 8 ÷ ( (11/4) ÷ 5 )

But let’s check the structure. In many textbooks, if it's written as:

8
-----
11/4
-----
5

It might mean: (8) / ( (11/4) / 5 )? That would be unusual.

Alternatively, perhaps it’s meant to be:
8 divided by (11/4), all over 5? That doesn’t make sense.

Wait — looking back at the original image description:
Problem 5 is written as:

8
----
11/4
----
5

This is ambiguous, but in most school contexts, this is interpreted as:

(8) ÷ (11/4) ÷ 5 — but order matters.

Actually, better way: think of it as a complex fraction where the main division line is between 8 and the rest.

Standard rule: The longest horizontal line is the main division.

In this case, likely:
Numerator: 8
Denominator: (11/4) / 5 → which is (11/4) ÷ 5

So overall: 8 ÷ [ (11/4) ÷ 5 ]

First, compute inside: (11/4) ÷ 5 = (11/4) × (1/5) = 11/20

Then: 8 ÷ (11/20) = 8 × (20/11) = 160/11

Which is an improper fraction. Can leave as is or write as mixed number: 14 6/11

But since others are simple fractions, maybe they expect 160/11.

Wait — let me double-check the structure.

Alternative interpretation: Maybe it’s (8 / (11/4)) / 5

That would be: first 8 ÷ (11/4) = 8 × 4/11 = 32/11

Then divide by 5: (32/11) ÷ 5 = 32/11 × 1/5 = 32/55

Hmm — which is correct?

Looking at how it’s typeset:
In the user’s message, it’s written as:

5.
8
--
11/4
--
5

This suggests two fraction bars. Typically, the middle bar is the main one. So:

The entire expression is:
[8]
———
[ (11/4) / 5 ]

Which is 8 ÷ ( (11/4) ÷ 5 )

As above: (11/4) ÷ 5 = 11/20

Then 8 ÷ (11/20) = 8 × 20/11 = 160/11

But let’s see problem 6 for pattern.

Problem 6 is:

1/3
---
8/9

Which is clearly (1/3) ÷ (8/9)

So for consistency, problem 5 should be interpreted similarly.

But problem 5 has three levels. Perhaps it’s:

Top: 8
Middle: 11/4
Bottom: 5

And the main fraction is between top and bottom, with middle being part of denominator? I think the safest is to assume it’s:

8 divided by (11/4 divided by 5)

Which is 8 ÷ (11/20) = 160/11

But let’s calculate both ways.

Option A: ((8) / (11/4)) / 5 = (32/11)/5 = 32/55

Option B: 8 / ((11/4)/5) = 8 / (11/20) = 160/11

Now, 160/11 is about 14.5, while 32/55 is about 0.58. Which makes more sense?

Looking at other problems, answers are small fractions. 32/55 is plausible.

Perhaps the formatting is: the "11/4" is the denominator of the first fraction, and then that whole thing is over 5? No.

Another way: in some notations,

a
b/c
d

means a / (b/c) / d, but that’s not standard.

I recall that in such worksheets, sometimes it’s written as:

For example, problem 5 might be intended as: \frac{8}{\frac{11}{4}} \div 5 ? But that’s not what’s shown.

Looking back at the user’s text: “5. \frac{8}{\frac{11}{4} \over 5}” — no, in plain text it’s:

“5.
8
--
11/4
--
5”

I think the best interpretation is that it’s a complex fraction where the numerator is 8, and the denominator is the fraction (11/4) over 5, i.e., (11/4)/5.

So denominator = (11/4) * (1/5) = 11/20

Then 8 / (11/20) = 8 * 20/11 = 160/11

But let’s check online or standard practice.

Upon second thought, in many textbooks, when you have:

a
b
c

with two lines, it often means a / (b/c) if b and c are fractions, but here b is 11/4, c is 5.

Perhaps it's (a / b) / c.

Let me look at problem 6 for clue.

Problem 6 is:

1/3
---
8/9

Which is clearly (1/3) ÷ (8/9) = (1/3)*(9/8) = 9/24 = 3/8

So for problem 5, if it were similar, but it has three parts.

Another idea: perhaps the "11/4" is meant to be the denominator, and "5" is separate, but that doesn't make sense.

Let's read the user's input carefully: "5. \frac{8}{\frac{11}{4} \over 5}" — no, in the initial request, it's written as:

"5.
8
--
11/4
--
5"

I think there might be a typo or misformatting. In many such worksheets, problem 5 is likely: \frac{8}{\frac{11}{4}} divided by 5, but that's not standard.

Perhaps it's \frac{8}{\left( \frac{11}{4} \div 5 \right)}

I found a better way: let's assume that the main fraction bar is the one between 8 and the rest, and the rest is (11/4) over 5, so yes, 8 / [(11/4)/5] = 8 / (11/20) = 160/11

But 160/11 is 14 6/11, which is fine.

However, let's calculate numerically:

If we do 8 divided by (11/4) = 8 * 4/11 = 32/11 ≈ 2.909

Then divided by 5 = 2.909 / 5 = 0.5818, which is 32/55

If we do (11/4) / 5 = 2.75 / 5 = 0.55, then 8 / 0.55 = 14.545, which is 160/11

Now, which one is intended? Given that other problems have small answers, and 32/55 is small, while 160/11 is large, perhaps it's the first interpretation.

Moreover, in the list, problem 5 is after simpler ones, so likely it's (8 / (11/4)) / 5

Let me confirm with a reliable source in my mind: in Glencoe Math or similar, when they write:

a
b/c
d

it usually means (a / (b/c)) / d or a / ((b/c)/d)? I think it's ambiguous, but commonly, the fraction is read from top to bottom with the longest bar.

To resolve this, let's look at the spacing. In the user's message, for problem 5, it's:

"5.
8
--
11/4
--
5"

The "--" under 8 is probably the first fraction bar, then "11/4" is below it, then another "--" and "5" below that. This suggests that the entire thing is 8 over (11/4 over 5), so 8 / ( (11/4) / 5 )

Yes, that makes sense. So denominator is (11/4) divided by 5.

So (11/4) / 5 = 11/20

Then 8 / (11/20) = 8 * 20/11 = 160/11

And 160/11 can be left as improper fraction or mixed number. Since other answers are proper fractions, but 160/11 is fine.

But let's see problem 6: it's 1/3 over 8/9, which is (1/3)/(8/9) = 3/8, a proper fraction.

For problem 5, 160/11 is greater than 1, which is ok.

Perhaps the worksheet expects simplified form.

Another thought: maybe "11/4" is a single entity, and the fraction is 8 divided by that, and then the result is over 5, but that would be written differently.

I recall that in some formats, this is called a continued fraction, but for grade school, it's likely (8) / ( (11/4) / 5 )

Let's calculate it as 8 ÷ (11/4 ÷ 5) = 8 ÷ (11/20) = 160/11

I think that's correct.

So for now, I'll go with 160/11.

But let's move to problem 6 and come back.

Problem 6:
  1/3
  —
  8/9

Meaning: (1/3) ÷ (8/9)

Dividing by a fraction: multiply by reciprocal.

So: (1/3) × (9/8) = (1×9)/(3×8) = 9/24

Simplify: divide numerator and denominator by 3 → 3/8

✔ Final Answer for #6: 3/8

Now back to problem 5. Given that problem 6 is straightforward, and problem 5 has three levels, perhaps it's intended to be:

The expression is \frac{8}{\frac{11}{4}} \times \frac{1}{5} or something.

Let's think differently. In the context of "complex fractions", a complex fraction has a fraction in numerator or denominator or both. Here, for problem 5, the denominator itself is a complex fraction: (11/4) over 5.

So yes, the whole thing is 8 divided by [ (11/4) / 5 ]

So [ (11/4) / 5 ] = 11/20

Then 8 / (11/20) = 160/11

And 160/11 is already simplified.

So I'll go with that.

But to match the style, perhaps they want it as a mixed number, but the instruction is to simplify, and 160/11 is simplified.

So for problem 5: 160/11

But let's double-check with calculation:

(11/4) / 5 = 11/4 * 1/5 = 11/20

8 / (11/20) = 8 * 20/11 = 160/11

Yes.

Some might interpret it as 8 / (11/4) / 5 = (8 * 4/11) / 5 = (32/11) / 5 = 32/55

Which is also possible.

Given that 32/55 is a nicer number and matches the size of other answers, and in many online sources, when written with two bars, it's often (top/middle)/bottom, I think 32/55 is more likely intended.

Let me search my memory: in Pearson or McGraw-Hill worksheets, for example, if they write:

6
2/3
4

it usually means (6 / (2/3)) / 4 = (6*3/2)/4 = 9/4 / 4 = 9/16

Or 6 / ((2/3)/4) = 6 / (2/12) = 6 / (1/6) = 36

Both are possible, but typically, the convention is that the fraction is evaluated from top to bottom with the main bar being the first one.

To resolve this, let's look at the position. In the user's text, for problem 5, it's listed as:

"5.
8
--
11/4
--
5"

The "--" after 8 is likely the primary fraction bar, so 8 is numerator, and the denominator is "11/4 over 5", which is (11/4)/5.

So I think 160/11 is correct.

But to be safe, let's see the answer choices or common practices.

Since this is a skills practice, and other problems are basic, perhaps problem 5 is meant to be \frac{8}{\frac{11}{4}} divided by 5, but that's not what's written.

Another idea: perhaps "11/4" is a typo, and it's supposed to be 11 over 4, but in the context, I think we have to go with what's given.

I recall that in some systems, the expression a/b/c is interpreted as (a/b)/c, not a/(b/c).

For example, in programming, 8/ (11/4) / 5 would be (8 / (11/4)) / 5 = (32/11) / 5 = 32/55

Whereas 8 / ((11/4)/5) = 8 / (11/20) = 160/11

In mathematics, without parentheses, a/b/c is usually (a/b)/c, because division is left-associative.

Is that true? Let me confirm.

In standard order of operations, division and multiplication are left-associative, so a/b/c = (a/b)/c

For example, 8/2/2 = (8/2)/2 = 4/2 = 2, not 8/(2/2) = 8/1 = 8.

Yes! That's key.

So for problem 5, if it's written as 8 / (11/4) / 5, then it should be (8 / (11/4)) / 5

Because division is left-associative.

So first, 8 divided by (11/4) = 8 * 4/11 = 32/11

Then, (32/11) divided by 5 = 32/11 * 1/5 = 32/55

And 32/55 is already simplified.

This makes sense, and it's a nice fraction like the others.

Whereas 160/11 is large and less likely for this level.

Also, in the list, problem 3 is 2/45, problem 4 is 5/72, so 32/55 fits.

So I think the intended interpretation is (8 / (11/4)) / 5 = 32/55

Therefore, for problem 5: 32/55

Okay, I'll go with that.

So summary:

1. 1/10
2. 1/6
3. 2/45
4. 5/72
5. 32/55
6. 3/8

Now, let's write the final answers.

Final Answer:
1. \frac{1}{10}
2. \frac{1}{6}
3. \frac{2}{45}
4. \frac{5}{72}
5. \frac{32}{55}
6. \frac{3}{8}