Simplifying Proper Fractions to Lowest Terms (Harder Questions) (A) - Free Printable
Educational worksheet: Simplifying Proper Fractions to Lowest Terms (Harder Questions) (A). Download and print for classroom or home learning activities.
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Step-by-step solution for: Simplifying Proper Fractions to Lowest Terms (Harder Questions) (A)
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Show Answer Key & Explanations
Step-by-step solution for: Simplifying Proper Fractions to Lowest Terms (Harder Questions) (A)
You're working on “Simplifying Proper Fractions (A)” — a worksheet that asks you to reduce each fraction to its lowest terms (also called simplest form). This means dividing both the numerator and denominator by their greatest common divisor (GCD) until no further simplification is possible.
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To simplify any fraction:
1. Find the GCD of the numerator and denominator.
2. Divide both by that GCD.
3. Write the new fraction.
You can also simplify step-by-step by dividing by small primes (2, 3, 5, etc.) until no more common factors exist.
---
Let’s solve all 40 problems with explanations where needed.
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## 🔢 Solutions:
1. 14/21 → GCD = 7 → 14÷7=2, 21÷7=3 → 2/3 ✔ *(already shown)*
2. 7/287 → 287 ÷ 7 = 41 → 1/41
3. 76/384 → GCD = 4 → 76÷4=19, 384÷4=96 → 19/96
4. 4/44 → GCD = 4 → 4÷4=1, 44÷4=11 → 1/11
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5. 203/350 → GCD = 7 → 203÷7=29, 350÷7=50 → 29/50
6. 10/55 → GCD = 5 → 10÷5=2, 55÷5=11 → 2/11
7. 30/770 → GCD = 10 → 30÷10=3, 770÷10=77 → 3/77
8. 99/225 → GCD = 9 → 99÷9=11, 225÷9=25 → 11/25
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9. 5/30 → GCD = 5 → 5÷5=1, 30÷5=6 → 1/6
10. 40/44 → GCD = 4 → 40÷4=10, 44÷4=11 → 10/11
11. 312/468 → GCD = 156 → 312÷156=2, 468÷156=3 → 2/3
*Alternatively: divide by 12 → 26/39, then by 13 → 2/3*
12. 44/104 → GCD = 4 → 44÷4=11, 104÷4=26 → 11/26
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13. 100/450 → GCD = 50 → 100÷50=2, 450÷50=9 → 2/9
14. 20/25 → GCD = 5 → 20÷5=4, 25÷5=5 → 4/5
15. 12/18 → GCD = 6 → 12÷6=2, 18÷6=3 → 2/3
16. 40/44 → GCD = 4 → 40÷4=10, 44÷4=11 → 10/11
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17. 98/148 → GCD = 2 → 98÷2=49, 148÷2=74 → 49/74
18. 155/225 → GCD = 5 → 155÷5=31, 225÷5=45 → 31/45
19. 830/850 → GCD = 10 → 830÷10=83, 850÷10=85 → 83/85
20. 12/114 → GCD = 6 → 12÷6=2, 114÷6=19 → 2/19
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21. 10/12 → GCD = 2 → 10÷2=5, 12÷2=6 → 5/6
22. 8/72 → GCD = 8 → 8÷8=1, 72÷8=9 → 1/9
23. 20/290 → GCD = 10 → 20÷10=2, 290÷10=29 → 2/29
24. 75/230 → GCD = 5 → 75÷5=15, 230÷5=46 → 15/46
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25. 18/33 → GCD = 3 → 18÷3=6, 33÷3=11 → 6/11
26. 6/82 → GCD = 2 → 6÷2=3, 82÷2=41 → 3/41
27. 80/190 → GCD = 10 → 80÷10=8, 190÷10=19 → 8/19
28. 24/140 → GCD = 4 → 24÷4=6, 140÷4=35 → 6/35
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29. 270/1000 → GCD = 10 → 270÷10=27, 1000÷10=100 → 27/100
30. 102/276 → GCD = 6 → 102÷6=17, 276÷6=46 → 17/46
31. 9/108 → GCD = 9 → 9÷9=1, 108÷9=12 → 1/12
32. 7/42 → GCD = 7 → 7÷7=1, 42÷7=6 → 1/6
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33. 252/369 → GCD = 9 → 252÷9=28, 369÷9=41 → 28/41
34. 260/970 → GCD = 10 → 260÷10=26, 970÷10=97 → 26/97
35. 100/360 → GCD = 20 → 100÷20=5, 360÷20=18 → 5/18
36. 9/108 → Same as #31 → 1/12
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37. 8/48 → GCD = 8 → 8÷8=1, 48÷8=6 → 1/6
38. 56/80 → GCD = 8 → 56÷8=7, 80÷8=10 → 7/10
39. 60/134 → GCD = 2 → 60÷2=30, 134÷2=67 → 30/67
40. 2/12 → GCD = 2 → 2÷2=1, 12÷2=6 → 1/6
---
## 📝 Final Answer Key:
Here’s the completed worksheet in order:
```
1. 2/3 11. 2/3 21. 5/6 31. 1/12 41. (none)
2. 1/41 12. 11/26 22. 1/9 32. 1/6
3. 19/96 13. 2/9 23. 2/29 33. 28/41
4. 1/11 14. 4/5 24. 15/46 34. 26/97
5. 29/50 15. 2/3 25. 6/11 35. 5/18
6. 2/11 16. 10/11 26. 3/41 36. 1/12
7. 3/77 17. 49/74 27. 8/19 37. 1/6
8. 11/25 18. 31/45 28. 6/35 38. 7/10
9. 1/6 19. 83/85 29. 27/100 39. 30/67
10. 10/11 20. 2/19 30. 17/46 40. 1/6
```
*(Note: Problem numbers go from 1–40; I listed them in rows for clarity.)*
---
## 💡 Tips for Simplifying Fractions:
- Always check if both numbers are even → divide by 2.
- If digits add up to multiple of 3 → divisible by 3.
- Ends in 0 or 5 → divisible by 5.
- Use prime factorization for harder ones.
- Double-check your answer: Can you divide numerator and denominator again? If not, it’s simplified!
---
✔ You’ve now fully solved and understood how to simplify all fractions on this worksheet! Great job practicing foundational math skills.
---
✔ General Strategy:
To simplify any fraction:
1. Find the GCD of the numerator and denominator.
2. Divide both by that GCD.
3. Write the new fraction.
You can also simplify step-by-step by dividing by small primes (2, 3, 5, etc.) until no more common factors exist.
---
Let’s solve all 40 problems with explanations where needed.
---
## 🔢 Solutions:
Row 1:
1. 14/21 → GCD = 7 → 14÷7=2, 21÷7=3 → 2/3 ✔ *(already shown)*
2. 7/287 → 287 ÷ 7 = 41 → 1/41
3. 76/384 → GCD = 4 → 76÷4=19, 384÷4=96 → 19/96
4. 4/44 → GCD = 4 → 4÷4=1, 44÷4=11 → 1/11
---
Row 2:
5. 203/350 → GCD = 7 → 203÷7=29, 350÷7=50 → 29/50
6. 10/55 → GCD = 5 → 10÷5=2, 55÷5=11 → 2/11
7. 30/770 → GCD = 10 → 30÷10=3, 770÷10=77 → 3/77
8. 99/225 → GCD = 9 → 99÷9=11, 225÷9=25 → 11/25
---
Row 3:
9. 5/30 → GCD = 5 → 5÷5=1, 30÷5=6 → 1/6
10. 40/44 → GCD = 4 → 40÷4=10, 44÷4=11 → 10/11
11. 312/468 → GCD = 156 → 312÷156=2, 468÷156=3 → 2/3
*Alternatively: divide by 12 → 26/39, then by 13 → 2/3*
12. 44/104 → GCD = 4 → 44÷4=11, 104÷4=26 → 11/26
---
Row 4:
13. 100/450 → GCD = 50 → 100÷50=2, 450÷50=9 → 2/9
14. 20/25 → GCD = 5 → 20÷5=4, 25÷5=5 → 4/5
15. 12/18 → GCD = 6 → 12÷6=2, 18÷6=3 → 2/3
16. 40/44 → GCD = 4 → 40÷4=10, 44÷4=11 → 10/11
---
Row 5:
17. 98/148 → GCD = 2 → 98÷2=49, 148÷2=74 → 49/74
18. 155/225 → GCD = 5 → 155÷5=31, 225÷5=45 → 31/45
19. 830/850 → GCD = 10 → 830÷10=83, 850÷10=85 → 83/85
20. 12/114 → GCD = 6 → 12÷6=2, 114÷6=19 → 2/19
---
Row 6:
21. 10/12 → GCD = 2 → 10÷2=5, 12÷2=6 → 5/6
22. 8/72 → GCD = 8 → 8÷8=1, 72÷8=9 → 1/9
23. 20/290 → GCD = 10 → 20÷10=2, 290÷10=29 → 2/29
24. 75/230 → GCD = 5 → 75÷5=15, 230÷5=46 → 15/46
---
Row 7:
25. 18/33 → GCD = 3 → 18÷3=6, 33÷3=11 → 6/11
26. 6/82 → GCD = 2 → 6÷2=3, 82÷2=41 → 3/41
27. 80/190 → GCD = 10 → 80÷10=8, 190÷10=19 → 8/19
28. 24/140 → GCD = 4 → 24÷4=6, 140÷4=35 → 6/35
---
Row 8:
29. 270/1000 → GCD = 10 → 270÷10=27, 1000÷10=100 → 27/100
30. 102/276 → GCD = 6 → 102÷6=17, 276÷6=46 → 17/46
31. 9/108 → GCD = 9 → 9÷9=1, 108÷9=12 → 1/12
32. 7/42 → GCD = 7 → 7÷7=1, 42÷7=6 → 1/6
---
Row 9:
33. 252/369 → GCD = 9 → 252÷9=28, 369÷9=41 → 28/41
34. 260/970 → GCD = 10 → 260÷10=26, 970÷10=97 → 26/97
35. 100/360 → GCD = 20 → 100÷20=5, 360÷20=18 → 5/18
36. 9/108 → Same as #31 → 1/12
---
Row 10:
37. 8/48 → GCD = 8 → 8÷8=1, 48÷8=6 → 1/6
38. 56/80 → GCD = 8 → 56÷8=7, 80÷8=10 → 7/10
39. 60/134 → GCD = 2 → 60÷2=30, 134÷2=67 → 30/67
40. 2/12 → GCD = 2 → 2÷2=1, 12÷2=6 → 1/6
---
## 📝 Final Answer Key:
Here’s the completed worksheet in order:
```
1. 2/3 11. 2/3 21. 5/6 31. 1/12 41. (none)
2. 1/41 12. 11/26 22. 1/9 32. 1/6
3. 19/96 13. 2/9 23. 2/29 33. 28/41
4. 1/11 14. 4/5 24. 15/46 34. 26/97
5. 29/50 15. 2/3 25. 6/11 35. 5/18
6. 2/11 16. 10/11 26. 3/41 36. 1/12
7. 3/77 17. 49/74 27. 8/19 37. 1/6
8. 11/25 18. 31/45 28. 6/35 38. 7/10
9. 1/6 19. 83/85 29. 27/100 39. 30/67
10. 10/11 20. 2/19 30. 17/46 40. 1/6
```
*(Note: Problem numbers go from 1–40; I listed them in rows for clarity.)*
---
## 💡 Tips for Simplifying Fractions:
- Always check if both numbers are even → divide by 2.
- If digits add up to multiple of 3 → divisible by 3.
- Ends in 0 or 5 → divisible by 5.
- Use prime factorization for harder ones.
- Double-check your answer: Can you divide numerator and denominator again? If not, it’s simplified!
---
✔ You’ve now fully solved and understood how to simplify all fractions on this worksheet! Great job practicing foundational math skills.
Parent Tip: Review the logic above to help your child master the concept of simplest form fraction worksheet.
