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Math · Decimals
Add & Subtract Decimals: Align Place Value, Estimate, Check
Move from decimal place value to accurate addition and subtraction through alignment, regrouping, estimation, money contexts, missing values and reasonableness checks.
The idea
Decimal addition and subtraction use the same place-value structure as whole-number operations. The decimal points line up because ones must combine with ones, tenths with tenths, and hundredths with hundredths.
Equivalent zeros are useful placeholders. Writing 5.6 as 5.60 does not change its value, but it makes the tenths and hundredths columns visible before you calculate.
Estimate before or after the exact calculation. If 4.86 + 2.97 is close to 5 + 3, an answer near 8 is reasonable. An estimate does not replace the exact answer; it helps catch a misplaced decimal or regrouping error.
Worked examples
Example 1 · Align the decimal points
Find 3.47 + 2.6.
- Rewrite 2.6 as 2.60.
- Add hundredths: 7 + 0 = 7.
- Add tenths: 4 + 6 = 10 tenths, so write 0 tenths and regroup 1 one.
- Add ones: 3 + 2 + 1 = 6.
3.47 + 2.60 = 6.07.
Example 2 · Subtract with regrouping
Find 8.2 − 3.75.
- Rewrite 8.2 as 8.20.
- Regroup so there are enough hundredths to subtract 5 hundredths.
- Continue through the tenths and ones columns.
- Check with addition: 4.45 + 3.75 = 8.20.
8.20 − 3.75 = 4.45.
Example 3 · Money is decimal place value
A book costs $7.68. You pay with $12.00. What is the change?
- Write 12.00 − 7.68.
- Keep dollars under dollars and cents under cents.
- Subtract with regrouping.
- Check that $7.68 + $4.32 = $12.00.
The change is $4.32.
Example 4 · Estimate, then calculate
Find 4.86 + 2.97 and decide whether the answer is reasonable.
- Estimate 4.86 ≈ 5 and 2.97 ≈ 3, so expect about 8.
- Calculate exactly with decimal points aligned.
- 4.86 + 2.97 = 7.83.
- 7.83 is close to 8, so the exact answer passes the estimate check.
7.83; estimate ≈ 8.
Example 5 · Solve a missing addend
3.45 + x = 8.20. Find x.
- Undo addition by subtracting 3.45 from 8.20.
- Align place values: 8.20 − 3.45.
- The difference is 4.75.
- Check: 3.45 + 4.75 = 8.20.
x = 4.75.
Example 6 · Use two methods to check
Find 10.00 − 6.48.
- Exact subtraction gives 3.52.
- Estimate 10 − 6.5 ≈ 3.5.
- Add back: 6.48 + 3.52 = 10.00.
- Both checks support the result.
10.00 − 6.48 = 3.52.
Common mistakes
These are tempting because they use familiar operations. Check what the question is actually measuring.
Why this fails: Digits must be aligned by place value. In 5.6 + 0.38, write 5.60 above 0.38.
Why this fails: A zero can be added only to the right as an equivalent placeholder: 5.6 = 5.60, not 5.06.
Why this fails: The decimal point does not block regrouping. Ten tenths make one whole, just as ten ones make one ten.
Why this fails: A quick estimate can expose an answer such as 0.783 when the addends are about 5 and 3.
Why this fails: In 3.45 + x = 8.20, subtraction isolates the missing addend; the equation tells you which inverse operation to use.
Foundation
Foundation practice
Align place values and use equivalent zeros before regrouping.
1. 3.4 + 2.5
2. 1.27 + 0.42
3. 5.6 + 0.38
4. 0.75 + 0.09
5. 8.2 − 3.1
6. 4.00 − 1.35
7. 6.5 − 0.78
8. 1.03 − 0.47
9. Rewrite 7.4 with two decimal places.
10. Rewrite 9 as a decimal with hundredths.
Core
Core practice
Mix addition and subtraction, including money and estimation checks.
11. 2.46 + 3.7
12. 9.04 − 2.6
13. 12.75 + 4.89
14. 20.00 − 6.48
15. 3.95 + 2.08
16. 7.2 − 4.65
17. 0.99 + 0.36
18. 10.5 − 0.87
19. A snack costs $4.36 and a drink costs $2.19. What is the total?
20. You have $10.02 and spend $3.78. How much remains?
Think
Think practice
Use estimates, reverse operations and error analysis instead of doing a single routine.
21. Estimate 4.8 + 1.6 to the nearest whole before finding the exact sum.
22. Estimate 9.5 − 2.75 to the nearest whole before finding the exact difference.
23. A student writes 5.6 + 0.38 = 5.44. What likely went wrong?
24. A student says 8.00 − 3.64 = 5.64 because 8 − 3 = 5 and then copied .64. Explain.
25. 2.75 + x = 6.20. Find x.
26. x − 1.68 = 5.03. Find x.
27. 14.2 − x = 6.95. Find x.
28. Which is greater: 3.7 + 2.46 or 9.04 − 2.6?
29. Without exact calculation, is 15.6 + 3.27 closer to 18, 19 or 20?
30. Why is 30.00 − 12.47 = 17.53 reasonable?
Challenge
Challenge practice
Connect decimal operations to multi-step contexts and early algebra thinking.
31. A runner covers 1.48 km, then 2.57 km. How much farther is needed to reach 5.00 km?
32. A $20.00 budget pays $6.48 and $4.89. How much remains?
33. x + 3.96 = 10.04. Find x.
34. 11.75 − x = 7.46. Find x.
35. x − 3.64 = 4.36. Find x.
36. Find 13.04 − 5.8, then check by addition.
37. Find 7.77 + 1.23 without a written algorithm.
38. A calculator shows 86.4 for 18.5 − 9.86. Give one quick reason this cannot be correct.
39. A container has 5.00 L. You use 2.99 L, then add 0.58 L. How much is in it now?
40. Why can you append a zero to 6.5 before adding hundredths?
Answer key
- 1. 5.9
Tenths align with tenths: 34 tenths + 25 tenths = 59 tenths. - 2. 1.69
Add hundredths, tenths and ones in aligned columns. - 3. 5.98
Write 5.6 as 5.60, then add 0.38. - 4. 0.84
75 hundredths + 9 hundredths = 84 hundredths. - 5. 5.1
Write both to tenths and subtract each place. - 6. 2.65
Regroup through the tenths and hundredths places. - 7. 5.72
Rewrite 6.5 as 6.50 before subtracting. - 8. 0.56
Regroup one whole into tenths and hundredths as needed. - 9. 7.40
A trailing zero does not change the value. - 10. 9.00
9 wholes equals 9.00. - 11. 6.16
Rewrite 3.7 as 3.70, align decimal points, and add. - 12. 6.44
Rewrite 2.6 as 2.60, then subtract. - 13. 17.64
Add hundredths and tenths with regrouping into the ones place. - 14. 13.52
Use zeros as placeholders and regroup from the whole-number places. - 15. 6.03
95 hundredths + 8 hundredths crosses a whole, so regroup. - 16. 2.55
Write 7.2 as 7.20, then subtract. - 17. 1.35
The hundredths and tenths regroup to make an additional whole. - 18. 9.63
Write 10.5 as 10.50 and regroup carefully. - 19. $6.55
4.36 + 2.19 = 6.55. - 20. $6.24
10.02 − 3.78 = 6.24. - 21. Estimate: 7; exact: 6.4.
5 + 2 = 7 is a useful benchmark; exact addition gives 6.4. - 22. Estimate: about 7; exact: 6.75.
10 − 3 ≈ 7, and 6.75 is close to that estimate. - 23. The student probably failed to align place values.
Write 5.60 + 0.38; the correct sum is 5.98. - 24. Decimal subtraction must be done by place value with regrouping; the correct difference is 4.36.
The fractional part cannot simply be copied when subtracting. - 25. x = 3.45.
6.20 − 2.75 = 3.45; add back to check. - 26. x = 6.71.
Undo subtraction by adding: 5.03 + 1.68 = 6.71. - 27. x = 7.25.
The missing amount removed is 14.20 − 6.95 = 7.25. - 28. 9.04 − 2.6 is greater.
The values are 6.16 and 6.44; compare after calculating or estimating. - 29. 19.
15.6 is about 16 and 3.27 is about 3, giving about 19; exact is 18.87. - 30. Because 30 − 12.5 is about 17.5.
The estimate closely matches the exact difference and confirms the decimal is in a sensible place. - 31. 0.95 km.
1.48 + 2.57 = 4.05; then 5.00 − 4.05 = 0.95. - 32. $8.63.
6.48 + 4.89 = 11.37; 20.00 − 11.37 = 8.63. - 33. x = 6.08.
10.04 − 3.96 = 6.08. - 34. x = 4.29.
11.75 − 7.46 = 4.29. - 35. x = 8.00.
Add 3.64 to both sides: 4.36 + 3.64 = 8.00. - 36. 7.24; check: 7.24 + 5.80 = 13.04.
The inverse-operation check rebuilds the starting value. - 37. 9.00.
1.23 is exactly what 7.77 needs to reach 9.00. - 38. The difference must be less than 18.5, and an estimate is about 9.
18.5 − about 10 is about 8.5, so 86.4 has an impossible size. - 39. 2.59 L.
5.00 − 2.99 = 2.01; 2.01 + 0.58 = 2.59. - 40. Because 6.5 and 6.50 are equivalent decimals.
The zero creates a hundredths placeholder without changing the value.