Math worksheet focused on reducing fractions and finding equivalent fractions, designed for educational practice.

Let’s work through each problem step by step.

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Part 1: Reduce the fractions

We need to simplify each fraction by dividing both numerator and denominator by their greatest common factor (GCF).

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Problem 1: 44/212

Find GCF of 44 and 212.

Factors of 44: 1, 2, 4, 11, 22, 44
Factors of 212: 1, 2, 4, 53, 106, 212
→ GCF = 4

Divide top and bottom by 4:

44 ÷ 4 = 11
212 ÷ 4 = 53
→ 11/53

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Problem 2: 36/38

GCF of 36 and 38 is 2.

36 ÷ 2 = 18
38 ÷ 2 = 19
→ 18/19

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Problem 3: 56/72

GCF of 56 and 72.

56 = 8×7, 72 = 8×9 → GCF = 8

56 ÷ 8 = 7
72 ÷ 8 = 9
→ 7/9

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Problem 4: 40/116

GCF of 40 and 116.

40 = 4×10, 116 = 4×29 → GCF = 4

40 ÷ 4 = 10
116 ÷ 4 = 29
→ 10/29

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Problem 5: 35/55

GCF of 35 and 55 is 5.

35 ÷ 5 = 7
55 ÷ 5 = 11
→ 7/11

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Problem 6: 58/68

GCF of 58 and 68 is 2.

58 ÷ 2 = 29
68 ÷ 2 = 34
→ 29/34

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Part 2: Complete the equivalent fractions

We find what number was multiplied to get from the first fraction to the others.

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Problem 7: 4/5 = _/50 = _/75

To get from 5 to 50 → multiply by 10 → so numerator: 4 × 10 = 40
To get from 5 to 75 → multiply by 15 → so numerator: 4 × 15 = 60
→ 40/50 = 60/75

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Problem 8: _/37 = _/74 = _/111

Look at denominators: 37, 74, 111

74 ÷ 37 = 2 → so if first denominator is 37, second is ×2, third is ×3

So numerators must also be ×1, ×2, ×3 → let’s assume the base numerator is x.

But we don’t have a starting numerator — wait, look again.

Actually, the problem is written as:

_ / 37 = _ / 74 = _ / 111

This means all three fractions are equal. So if we pick any numerator for 37, say 1, then:

1/37 = 2/74 = 3/111

Check: 2÷2=1, 74÷2=37 → yes; 3÷3=1, 111÷3=37 → yes.

So answer: 1/37 = 2/74 = 3/111

But actually, since it's “complete”, and no starting numerator given, we can use 1 as base.

Alternatively, maybe they want us to fill in blanks assuming the first blank is unknown? But looking at format, it’s likely we’re to find numerators that make them equivalent with those denominators.

Since 74 = 37×2, 111=37×3, then numerators must be n, 2n, 3n.

The simplest is n=1 → 1, 2, 3.

So: 1, 2, 3

But written as fractions: 1/37 = 2/74 = 3/111

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Problem 9: 1/5 = _/25 = _/45

5 → 25: ×5 → numerator: 1×5 = 5
5 → 45: ×9 → numerator: 1×9 = 9
→ 5/25 = 9/45

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Problem 10: 6/9 = _/15 = _/21

First, reduce 6/9 → divide by 3 → 2/3

Now, 2/3 = ?/15 → 3×5=15 → 2×5=10 → 10/15
2/3 = ?/21 → 3×7=21 → 2×7=14 → 14/21

Or directly from 6/9:

6/9 = x/15 → cross multiply: 6×15 = 9x → 90 = 9x → x=10
6/9 = y/21 → 6×21 = 9y → 126 = 9y → y=14

→ 10/15 = 14/21

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Problem 11: 1/4 = _/36 = _/12

4 → 36: ×9 → numerator: 1×9 = 9
4 → 12: ×3 → numerator: 1×3 = 3
→ 9/36 = 3/12

Wait — but 3/12 reduces to 1/4, yes. And 9/36 also reduces to 1/4.

But note: the problem says “complete the equivalent fractions” — so we fill in the blanks.

So: 9/36 = 3/12

But order: 1/4 = 9/36 = 3/12? That’s correct numerically, but usually we write in increasing or decreasing order? Doesn’t matter — mathematically correct.

Actually, 3/12 is simpler than 9/36, but both are equivalent.

We’ll go with: 9, 3

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Problem 12: _/62 = _/31 = _/93

Denominators: 62, 31, 93

Note: 62 = 31×2, 93 = 31×3

So if we let the middle fraction be x/31, then:

First: x/31 = ?/62 → multiply numerator and denominator by 2 → 2x/62
Third: x/31 = ?/93 → multiply by 3 → 3x/93

So to make them equal, we can choose x=1 → then:

2/62 = 1/31 = 3/93

Check: 2/62 = 1/31 ✓, 3/93 = 1/31 ✓

So numerators: 2, 1, 3

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Now, compiling all answers:

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Final Answers:

1. 11/53
2. 18/19
3. 7/9
4. 10/29
5. 7/11
6. 29/34
7. 40, 60
8. 1, 2, 3
9. 5, 9
10. 10, 14
11. 9, 3
12. 2, 1, 3

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Final Answer:
1. $\frac{11}{53}$
2. $\frac{18}{19}$
3. $\frac{7}{9}$
4. $\frac{10}{29}$
5. $\frac{7}{11}$
6. $\frac{29}{34}$
7. $40$, $60$
8. $1$, $2$, $3$
9. $5$, $9$
10. $10$, $14$
11. $9$, $3$
12. $2$, $1$, $3$