Decomposing Fractions Worksheets - Free Printable

Explanation:
We are asked to decompose each fraction into a sum of unit fractions — that is, fractions with numerator 1 (like 1/2, 1/3, 1/4, etc.).

Let’s go one by one.

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(a)
We need to write each given fraction as a sum of unit fractions.

1. 5/6
We want to write 5/6 as sum of unit fractions. One common way is to use repeated unit fractions:
5/6 = 1/6 + 1/6 + 1/6 + 1/6 + 1/6
That’s five copies of 1/6 — all unit fractions. ✔
(Other decompositions exist, like 1/2 + 1/3, since 1/2 = 3/6 and 1/3 = 2/6 → total 5/6. But the problem seems to expect using same denominator, especially since the example shows repeated 1/6s.)

Looking at the example in part (b), they used repeated unit fractions with same denominator (e.g., 1/4 + 1/4 + ...), so we’ll follow that pattern.

So:
- 5/6 = 1/6 + 1/6 + 1/6 + 1/6 + 1/6

2. 3/4
= 1/4 + 1/4 + 1/4

3. 5/8
= 1/8 + 1/8 + 1/8 + 1/8 + 1/8

4. 7/9
= 1/9 + 1/9 + 1/9 + 1/9 + 1/9 + 1/9 + 1/9

5. 4/5
= 1/5 + 1/5 + 1/5 + 1/5

6. 2/3
= 1/3 + 1/3

All of these are sums of unit fractions (numerator = 1), and they add up correctly.

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(b)
Now we’re given expressions and asked to rewrite them as a sum of unit fractions.

1.
Given:
1/2 + 1/4 + 1/4 + 1/4 + 1/4
Let’s simplify:
1/2 = 2/4, and then four 1/4’s = 4/4
So total = 2/4 + 4/4 = 6/4 = 3/2
But the question says: *decompose into a sum of unit fractions* — not simplify first. So we should just list all terms as unit fractions:

1/2, 1/4, 1/4, 1/4, 1/4 — all are already unit fractions!
So answer is just:
1/2 + 1/4 + 1/4 + 1/4 + 1/4
But maybe they want it as *only* unit fractions with same denominator? No — the instruction says “sum of unit fractions”, and 1/2 is a unit fraction too. So this is fine.

Wait — but look at the second part:
They give:
1/3 + 1/6 + 1/6 + 1/6
Again, all are unit fractions. So likely the task is just to *write the expression as a sum of unit fractions*, which it already is — unless they want to combine or rewrite using only one denominator? Hmm.

Actually, re-reading:
“Write each expression below as a sum of unit fractions.”
But the expressions are already sums of unit fractions. So perhaps the intention is to express the *total value* as a sum of unit fractions — possibly using the same denominator (like in part a).

Let’s test with first one:
1/2 + 1/4 + 1/4 + 1/4 + 1/4
Convert all to twelfths or fourths:

Use fourths:
1/2 = 2/4
Then total = 2/4 + 1/4 + 1/4 + 1/4 + 1/4 = 6/4 = 3/2
Now write 3/2 as sum of unit fractions.
3/2 = 1 + 1/2 = 1/1 + 1/2 — but 1/1 is a unit fraction (yes, numerator 1).
Alternatively, using halves: 3/2 = 1/2 + 1/2 + 1/2
That’s three copies of 1/2 — all unit fractions. That matches the style of part (a).

Similarly, second:
1/3 + 1/6 + 1/6 + 1/6
Convert to sixths:
1/3 = 2/6, plus three 1/6 = 3/6 → total = 5/6
Then 5/6 = five 1/6’s → 1/6 + 1/6 + 1/6 + 1/6 + 1/6

Third:
1/4 + 1/8 + 1/8 + 1/8 + 1/8
1/4 = 2/8, plus four 1/8 = 4/8 → total = 6/8 = 3/4
3/4 = three 1/4’s → 1/4 + 1/4 + 1/4
Or as eighths: 6/8 = six 1/8’s → 1/8 ×6

Which form is expected? Looking back at part (a), they used the *same denominator* as original fraction (e.g., 5/6 → five 1/6s). So likely we should express the total as sum of unit fractions with denominator equal to the LCM or original denominator.

Let’s verify with the examples in (b):

First expression:
1/2 + 1/4 + 1/4 + 1/4 + 1/4
Total = 3/2
If we write as unit fractions with denominator 2: 3/2 = 1/2 + 1/2 + 1/2 ✔ simple.

Second:
1/3 + 1/6 + 1/6 + 1/6 = 5/6 → write as five 1/6s.

Third:
1/4 + 1/8×4 = 1/4 + 4/8 = 2/8 + 4/8 = 6/8 = 3/4 → write as three 1/4s, or six 1/8s. Since original had eighths, maybe prefer 1/8s: six 1/8s.

But the problem likely expects the simplest repeated-unit-fraction form, matching part (a). In part (a), they wrote 5/6 as five 1/6s — so denominator matches the original fraction’s denominator.

So for (b), we compute the total value, then express it as sum of unit fractions with denominator = denominator of simplified result.

Let’s do cleanly:

(b) i.
1/2 + 1/4 + 1/4 + 1/4 + 1/4
= 1/2 + 4×(1/4) = 1/2 + 1 = 3/2
Now write 3/2 as sum of unit fractions:
3/2 = 1/2 + 1/2 + 1/2 ✔ (three halves)

(b) ii.
1/3 + 1/6 + 1/6 + 1/6
= 1/3 + 3×(1/6) = 1/3 + 1/2 = 2/6 + 3/6 = 5/6
→ 5/6 = 1/6 + 1/6 + 1/6 + 1/6 + 1/6

(b) iii.
1/4 + 1/8 + 1/8 + 1/8 + 1/8
= 1/4 + 4×(1/8) = 1/4 + 1/2 = 2/8 + 4/8 = 6/8 = 3/4
→ 3/4 = 1/4 + 1/4 + 1/4
(or 6/8 = six 1/8s — both valid, but 1/4s match denominator of simplified fraction 3/4)

Since in part (a) they used denominator of the original fraction (e.g., 5/6 → /6), and 3/4 simplifies to denominator 4, using 1/4 is consistent.

So final answers:

(a)
5/6 = 1/6 + 1/6 + 1/6 + 1/6 + 1/6
3/4 = 1/4 + 1/4 + 1/4
5/8 = 1/8 + 1/8 + 1/8 + 1/8 + 1/8
7/9 = 1/9 + 1/9 + 1/9 + 1/9 + 1/9 + 1/9 + 1/9
4/5 = 1/5 + 1/5 + 1/5 + 1/5
2/3 = 1/3 + 1/3

(b)
i. 1/2 + 1/2 + 1/2
ii. 1/6 + 1/6 + 1/6 + 1/6 + 1/6
iii. 1/4 + 1/4 + 1/4

Let me double-check arithmetic:

- 1/2 ×3 = 3/2 ✔
- 1/6 ×5 = 5/6 ✔
- 1/4 ×3 = 3/4 ✔

All correct.

Final Answer:
(a)
5/6 = 1/6 + 1/6 + 1/6 + 1/6 + 1/6
3/4 = 1/4 + 1/4 + 1/4
5/8 = 1/8 + 1/8 + 1/8 + 1/8 + 1/8
7/9 = 1/9 + 1/9 + 1/9 + 1/9 + 1/9 + 1/9 + 1/9
4/5 = 1/5 + 1/5 + 1/5 + 1/5
2/3 = 1/3 + 1/3

(b)
i. 1/2 + 1/2 + 1/2
ii. 1/6 + 1/6 + 1/6 + 1/6 + 1/6
iii. 1/4 + 1/4 + 1/4