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Add & Subtract Fractions: Common Denominators, Mixed Numbers, Estimate, Check
Move from equivalent fractions into addition and subtraction with common denominators, mixed numbers, estimation, reverse problems and reasonableness checks.
The idea
Fractions can be added or subtracted only when the pieces name the same-sized parts. That is why 1/2 + 1/4 becomes 2/4 + 1/4 before the numerators are combined.
A common denominator is not a trick; it is a way to rename fractions without changing their values. Choose a useful common multiple, rewrite each fraction, combine the numerators, then simplify if possible.
Mixed numbers can be handled by working with whole parts and fractional parts or by converting to improper fractions. Estimate first with benchmarks such as 0, 1/2, 1, 1 1/2 and 2 so a misplaced denominator or regrouping error is easier to catch.
Worked examples
Example 1 · Same denominator
Find 3/8 + 2/8.
- Both fractions already use eighths.
- Add the numerators: 3 + 2 = 5.
- Keep the denominator 8 because the size of each part has not changed.
3/8 + 2/8 = 5/8.
Example 2 · Related denominators
Find 1/2 + 1/4.
- Rename 1/2 as 2/4.
- Now both fractions use fourths.
- Add: 2/4 + 1/4 = 3/4.
1/2 + 1/4 = 3/4.
Example 3 · Unlike denominators
Find 2/3 + 1/4.
- A useful common denominator for 3 and 4 is 12.
- Rename 2/3 as 8/12 and 1/4 as 3/12.
- Add: 8/12 + 3/12 = 11/12.
2/3 + 1/4 = 11/12.
Example 4 · Mixed numbers
Find 1 1/2 + 2 1/4.
- Add the whole-number parts: 1 + 2 = 3.
- Rename 1/2 as 2/4.
- Add the fractional parts: 2/4 + 1/4 = 3/4.
1 1/2 + 2 1/4 = 3 3/4.
Example 5 · Subtract and regroup
Find 3 1/4 − 1 2/3.
- Use twelfths: 1/4 = 3/12 and 2/3 = 8/12.
- Because 3/12 is smaller than 8/12, regroup one whole from 3 as 12/12.
- Now 2 15/12 − 1 8/12 = 1 7/12.
3 1/4 − 1 2/3 = 1 7/12.
Example 6 · Missing value and check
x + 5/8 = 1 1/4. Find x.
- Rename 1 1/4 as 1 2/8 or 10/8.
- Subtract 5/8 from 10/8.
- x = 5/8.
- Check: 5/8 + 5/8 = 10/8 = 1 1/4.
x = 5/8.
Common mistakes
These are tempting because they use familiar operations. Check what the question is actually measuring.
Why this fails: The denominator names the size of the pieces. Rename both fractions using equal-sized pieces before adding numerators.
Why this fails: If 1/2 becomes fourths, multiply both numerator and denominator by 2: 1/2 = 2/4.
Why this fails: 6/8 is correct but can be simplified to 3/4 by dividing numerator and denominator by 2.
Why this fails: Mixed-number subtraction may require regrouping one whole into fractional parts before subtracting.
Why this fails: Benchmarks make impossible results easier to spot. For example, 7/8 + 5/6 should be close to 2, not close to 1/2.
Foundation
Foundation practice
Build fluency with same or related denominators before moving to less obvious common denominators.
1. 3/8 + 2/8
2. 5/9 − 2/9
3. 1/2 + 1/4
4. 3/4 − 1/8
5. 2/3 + 1/6
6. 5/6 − 1/3
7. 7/10 + 1/5
8. 11/12 − 1/4
9. 2/5 + 3/10
10. 7/8 − 3/16
Core
Core practice
Use common denominators with unlike fractions and mixed numbers.
11. 2/3 + 1/4
12. 5/6 − 3/8
13. 3/5 + 7/12
14. 7/9 − 2/3
15. 1/2 + 2/3
16. 5/8 + 3/4
17. 7/10 − 1/6
18. 11/12 − 5/18
19. 1 1/2 + 2 1/4
20. 3 1/4 − 1 2/3
Think
Think practice
Estimate, analyse errors, compare expressions and solve reverse problems.
21. Estimate 7/8 + 5/6, then find the exact sum.
22. Estimate 3 1/10 − 1 7/8, then find the exact difference.
23. A student says 2/5 + 1/5 = 3/10 because both numerators and denominators should be added. Explain.
24. A student writes 3/4 − 1/6 = 2/2. What should happen instead?
25. x + 5/8 = 1 1/4. Find x.
26. x − 3/4 = 1 1/6. Find x.
27. 2 1/3 − x = 5/6. Find x.
28. Which is greater: 2/3 + 3/8 or 5/6 + 1/4?
29. Which is closer to 1: 7/8 + 1/10 or 2/3 + 1/4?
30. Why is 4 1/2 − 2 3/4 = 1 3/4 a reasonable result?
Challenge
Challenge practice
Use fractions in equations, multi-step contexts and early algebra reasoning.
31. 3/4 + 5/6
32. 2 1/2 − 7/8
33. x + 5/6 = 2 1/4. Find x.
34. x − 5/12 = 2 1/3. Find x.
35. 1 1/2 − x = 7/10. Find x.
36. Find 2 1/6 − 3/4, then check by addition.
37. A recipe uses 2/5 cup, 3/4 cup and 1/10 cup of three ingredients. How many cups altogether?
38. A hiker walks 7/8 km, then 5/12 km. What total distance is that?
39. A 3 L container loses 7/10 L, then another 3/5 L. How much remains?
40. Why must 2/3 and 3/5 be renamed before adding their numerators?
Answer key
- 1. 5/8
The denominators already match, so add 3 + 2 and keep eighths. - 2. 1/3
5/9 − 2/9 = 3/9, which simplifies to 1/3. - 3. 3/4
Rename 1/2 as 2/4, then 2/4 + 1/4 = 3/4. - 4. 5/8
Rename 3/4 as 6/8, then 6/8 − 1/8 = 5/8. - 5. 5/6
2/3 = 4/6, so 4/6 + 1/6 = 5/6. - 6. 1/2
1/3 = 2/6, so 5/6 − 2/6 = 3/6 = 1/2. - 7. 9/10
1/5 = 2/10, so 7/10 + 2/10 = 9/10. - 8. 2/3
1/4 = 3/12, so 11/12 − 3/12 = 8/12 = 2/3. - 9. 7/10
2/5 = 4/10, so 4/10 + 3/10 = 7/10. - 10. 11/16
7/8 = 14/16, so 14/16 − 3/16 = 11/16. - 11. 11/12
Use twelfths: 8/12 + 3/12 = 11/12. - 12. 11/24
Use twenty-fourths: 20/24 − 9/24 = 11/24. - 13. 1 11/60
Use sixtieths: 36/60 + 35/60 = 71/60 = 1 11/60. - 14. 1/9
2/3 = 6/9, so 7/9 − 6/9 = 1/9. - 15. 1 1/6
3/6 + 4/6 = 7/6 = 1 1/6. - 16. 1 3/8
3/4 = 6/8, so 11/8 = 1 3/8. - 17. 8/15
Use thirtieths: 21/30 − 5/30 = 16/30 = 8/15. - 18. 23/36
Use thirty-sixths: 33/36 − 10/36 = 23/36. - 19. 3 3/4
1/2 = 2/4; 1 + 2 = 3 and 2/4 + 1/4 = 3/4. - 20. 1 7/12
Regroup in twelfths: 2 15/12 − 1 8/12 = 1 7/12. - 21. Estimate: about 2; exact: 1 17/24.
Both fractions are close to 1. Exactly, 21/24 + 20/24 = 41/24 = 1 17/24. - 22. Estimate: about 1; exact: 1 9/40.
3.1 − 1.875 is a little above 1; using fortieths gives 124/40 − 75/40 = 49/40. - 23. The correct sum is 3/5.
The pieces are already fifths, so only the counts of fifths are added. - 24. Use a common denominator: 3/4 − 1/6 = 9/12 − 2/12 = 7/12.
Subtracting numerators and denominators separately changes the size of the pieces. - 25. x = 5/8.
1 1/4 = 10/8; 10/8 − 5/8 = 5/8. - 26. x = 1 11/12.
Add 3/4 to 1 1/6: 7/6 + 3/4 = 14/12 + 9/12 = 23/12. - 27. x = 1 1/2.
The amount removed is 2 1/3 − 5/6 = 14/6 − 5/6 = 9/6 = 1 1/2. - 28. 5/6 + 1/4 is greater.
The sums are 25/24 = 1 1/24 and 13/12 = 1 1/12. - 29. 7/8 + 1/10.
7/8 + 1/10 = 39/40, while 2/3 + 1/4 = 11/12; 39/40 is closer to 1. - 30. Because 4.5 − 2.75 is about 1.75, which is 1 3/4.
A decimal or benchmark estimate confirms the fraction result has the right size. - 31. 1 7/12
9/12 + 10/12 = 19/12 = 1 7/12. - 32. 1 5/8
2 1/2 = 20/8; 20/8 − 7/8 = 13/8 = 1 5/8. - 33. x = 1 5/12.
2 1/4 = 27/12 and 5/6 = 10/12; the difference is 17/12. - 34. x = 2 3/4.
2 1/3 = 28/12; adding 5/12 gives 33/12 = 2 3/4. - 35. x = 4/5.
1 1/2 = 15/10; 15/10 − 7/10 = 8/10 = 4/5. - 36. 1 5/12; check: 1 5/12 + 3/4 = 2 1/6.
26/12 − 9/12 = 17/12; adding 9/12 returns 26/12. - 37. 1 1/4 cups.
Use twentieths: 8/20 + 15/20 + 2/20 = 25/20 = 1 1/4. - 38. 1 7/24 km.
21/24 + 10/24 = 31/24 = 1 7/24. - 39. 1 7/10 L.
3/5 = 6/10, so 3 − 13/10 = 30/10 − 13/10 = 17/10. - 40. Because thirds and fifths are different-sized parts; a common denominator renames them as equal-sized parts.
Once the units match, the numerators can count how many of those equal-sized parts there are.