A Peek Beyond the Point · Lesson 9 of 11
Addition and Subtraction of Decimals
“Connect decimal calculations with place-value exchanges, patterns, and estimates.”
• Add and subtract decimals by aligning corresponding places. • Explain regrouping through tenths, hundredths, and thousandths. • Solve word problems and check calculations by estimation. • Continue decimal sequences using their changes. • Give sensible bounds for nonnegative decimal sums and differences.
Addition and Subtraction of Decimals
Priya needs 2.7 m of cloth and Shylaja needs 3.5 m. These measurements contain whole metres and tenths of a metre, just as the earlier fractional measurements did. Decimal notation changes how we record the parts; it does not change what addition means. Combine whole metres with whole metres and tenths with tenths.
Problem
Find the total cloth required, and how much more 3.5 m is than 2.7 m.
- 1.For the sum, add whole metres: 2 + 3 = 5. Add tenths: 7 + 5 = 12.
- 2.Exchange 10 of the 12 tenths for a metre. The total is 6.2 m.
- 3.For the difference, rewrite 3.5 m as 2 metres and 15 tenths of a metre.
- 4.Subtract 2 metres and 7 tenths to obtain 8 tenths, or 0.8 m. Thus Shylaja needs 0.8 m more.
Align the decimal points when writing calculations vertically. This places units under units, tenths under tenths, and so on. If one number has fewer fractional digits, write trailing zeros to make the empty places clear. For example, 0.6 = 0.600. It is the value of the places, not the ends of the written strings, that must line up.
| Number | Whole units | Tenths | Hundredths | Thousandths |
|---|---|---|---|---|
| 0.934 | 0 | 9 | 3 | 4 |
| 0.600 | 0 | 6 | 0 | 0 |
| Sum: 1.534 | 1 | 5 | 3 | 4 |
Problem
Find 0.934 + 0.6.
- 1.Write 0.6 as 0.600 to align corresponding places.
- 2.Thousandths give 4 + 0 = 4 and hundredths give 3 + 0 = 3.
- 3.Tenths give 9 + 6 = 15. Exchange ten tenths for a unit, leaving five tenths.
- 4.The sum is 1.534. A rough check is that 0.9 + 0.6 is about 1.5.
Problem
Find 75.345 + 86.691.
- 1.Thousandths: 5 + 1 = 6.
- 2.Hundredths: 4 + 9 = 13. Record 3 hundredths and exchange 10 hundredths for 1 tenth.
- 3.Tenths: 3 + 6 + 1 = 10. Record 0 tenths and exchange these tenths for 1 unit.
- 4.Units: 5 + 6 + 1 = 12. Record 2 units and exchange 10 units for 1 ten.
- 5.Tens: 7 + 8 + 1 = 16, which is 1 hundred and 6 tens. Sum = 162.036.
Subtraction and Empty Places
Subtraction uses the same place alignment, but an exchange may travel through several empty places. Write a whole number such as 17 as 17.000 when subtracting thousandths. The zeros mean no separate fractional parts are recorded initially; whole units can still be exchanged to supply them.
Problem
Find 17 − 16.198.
- 1.Write the starting quantity as 17.000.
- 2.Exchange 1 unit for 10 tenths: 16 units and 10 tenths.
- 3.Exchange 1 of those tenths for 10 hundredths, then 1 hundredth for 10 thousandths. The starting quantity is now 16 units, 9 tenths, 9 hundredths, and 10 thousandths.
- 4.Subtract 16 units, 1 tenth, 9 hundredths, and 8 thousandths. The remainder is 0 units, 8 tenths, 0 hundredths, and 2 thousandths.
- 5.The difference is 0.802. Check by adding: 16.198 + 0.802 = 17.000.
Problem
Find 84.691 − 77.345.
- 1.At thousandths, 1 cannot supply 5. Exchange a hundredth: the fractional parts become 6 tenths, 8 hundredths, and 11 thousandths.
- 2.Subtract fractional parts: 11 − 5 = 6 thousandths, 8 − 4 = 4 hundredths, and 6 − 3 = 3 tenths.
- 3.For whole units, 4 cannot supply 7. Exchange a ten: 84 units become 7 tens and 14 units.
- 4.Subtract 7 tens and 7 units. The remaining whole-number part is 7.
- 5.Difference = 7.346. A compact written subtraction records these same exchanges above the changed digits.
Do not line up the final digits when adding numbers of different lengths. For 0.75 + 0.03, both final digits are hundredths; for 0.934 + 0.6, they are not. Align decimal points and therefore corresponding places.
Decimal Sequences
Decimal sequences can increase or decrease by a constant quantity, or follow a repeated pattern of changes. Begin by finding differences between consecutive terms. Check that the proposed rule explains every term supplied. Then apply that rule, keeping place values aligned during mental or written computation.
Problem
Continue 10.56, 10.67, 10.78 by three terms.
- 1.10.67 − 10.56 = 0.11 and 10.78 − 10.67 = 0.11.
- 2.Using the constant-change rule, add 0.11 repeatedly.
- 3.The next values are 10.89, 11.00, and 11.11.
- 4.The step from 10.89 to 11.00 crosses a whole-unit boundary but keeps the same change.
One final source question lists 5.5, 6.4, 6.39, 7.29, 7.28, 6.18, 6.17. The appended answer instead uses 8.18 and 8.17. Those versions disagree. In the corrected practice version below, the last two supplied terms are explicitly 8.18 and 8.17, giving alternating changes of +0.90 and −0.01. A finite list without a stated rule can have more than one continuation.
Estimating Sums and Differences
Estimate a result before calculating it exactly. This gives a range against which the answer can be checked. For a nonnegative decimal with whole-number part 25, the value is at least 25 and less than 26. For another with whole-number part 8, it is at least 8 and less than 9. Their sum is therefore at least 33 and less than 35.
Problem
Estimate the range of 25.936 + 8.202, then calculate it.
- 1.The whole-number parts sum to 25 + 8 = 33. The two fractional parts together contribute less than 2.
- 2.Because the fractions are nonzero, the sum lies strictly between 33 and 35.
- 3.Add 0.936 + 0.202 = 1.138 and combine with 33 to obtain 34.138.
- 4.The exact result fits the estimated range. The same bounds apply if 25.936 is replaced by 25.93603259.
For a difference, the two fractional parts can alter the whole-part difference by less than one in either direction. If x has whole part a and y has whole part b, then x − y lies strictly between a − b − 1 and a − b + 1. For example, 25.936 − 8.202 lies between 16 and 18. Its exact value is 17.734. These are broad checks, not replacements for the calculation.
Make an increasing sequence with change 0.05 and a decreasing sequence with change 0.5. Include a step across a whole-unit boundary in each. Estimate a range for one sum and one difference from your sequences, then calculate the exact values.
Check Your Understanding
Use the ideas from this lesson to choose an answer. Explain your choice to yourself before opening the explanations below.
Quiz
What is 0.75 + 0.03?
What is 9.9 − 9.09?
What is 6.236 + 0.487?
Continue 5, 4.95, 4.90 with the constant-change rule.
Which bound is always valid for the sum of two nonnegative decimals with whole parts 25 and 8?
The quantities contain 75 and 3 hundredths, giving 78 hundredths.
Practice Problems
- Find 29.19 + 9.91 and 6.236 − 0.487.
- Find 18 − 8.8, 17 − 0.05, and 34.505 − 18.1.
- Mahi buys 0.25 kg, 0.3 kg, 0.5 kg, 0.2 kg, and 0.05 kg of vegetables. Find the total mass.
- Pinto supplies 3.79 L, 4.2 L, and 4.25 L in three days, and 25 L over six days. What does he supply over the other three days?
- A person’s mass changes from 35.75 kg to 34.50 kg. Describe the change.
- Extend each constant-change sequence by three terms: (a) 4.4, 4.45, 4.5; (b) 25.75, 26.25, 26.75; (c) 13.5, 16, 18.5; (d) 8.5, 9.4, 10.3.
- Extend each decreasing sequence by three terms: (a) 12.45, 11.95, 11.45; (b) 36.5, 33, 29.5.
- Corrected pattern version: continue 5.5, 6.4, 6.39, 7.29, 7.28, 8.18, 8.17 using alternating changes +0.90, −0.01.
- Give bounds for 12.45 + 3.8 and 12.45 − 3.8, then calculate both.
For the sum, 29 + 9 = 38 and 0.19 + 0.91 = 1.10, giving 39.10. For the difference, count thousandths: 6236 − 487 = 5749, so the result is 5.749.
Key Takeaways
• Decimal arithmetic combines or removes corresponding place values. • Align decimal points, and use trailing zeros when helpful. • Ten parts of one place can be exchanged for one part of the next larger place. • Subtraction from a whole number may require successive exchanges through zeros. • Sequence rules should account for every supplied term. • Estimate a range before accepting an exact result. • A sum of two nonnegative decimals is at least the sum of their whole parts and less than two more.