A Peek Beyond the Point · Lesson 4 of 11
Adding and Subtracting Hundredths
“Understand the exchanges behind calculations with units, tenths, and hundredths.”
• Add corresponding units, tenths, and hundredths. • Regroup hundredths into tenths and tenths into whole units. • Explain single and successive exchanges in subtraction. • Check a calculation by counting entirely in hundredths.
Adding Corresponding Parts
A measurement may contain whole units, tenths, and hundredths. To combine two such measurements, count each size of part separately. Just as adding hundreds to ones without accounting for their sizes would be confusing, we must not mix tenths with hundredths as if they were equal parts.
Start with the smallest parts. If the hundredths total reaches ten or more, group ten hundredths into one tenth. Add that tenth to the tenths already present. If the tenths total then reaches ten, group ten tenths into one whole unit. The exchanges let us write a grouped answer with fewer than ten parts of each fractional size.
Problem
Add 15 units, 3 tenths, and 4 hundredths to 2 units, 6 tenths, and 8 hundredths.
- 1.Whole units total 15 + 2 = 17. Tenths total 3 + 6 = 9. Hundredths total 4 + 8 = 12.
- 2.Twelve hundredths are 1 tenth and 2 hundredths. Add the new tenth to the 9 tenths to get 10 tenths.
- 3.Exchange those 10 tenths for 1 whole unit. The 17 units become 18.
- 4.The sum is 18 units and 2 hundredths, or 18 + 2/100 units.
We can check this by converting the entire measurements into hundredths. The first contains 1500 + 30 + 4 = 1534 hundredths and the second contains 200 + 60 + 8 = 268 hundredths. Their sum is 1802 hundredths. Grouping gives 18 units and 2 hundredths, exactly as before.
Problem
Add 15 + 6/10 + 4/100 and 14 + 3/10 + 6/100 units.
- 1.The whole-unit total is 29, the tenths total is 9, and the hundredths total is 10.
- 2.Exchange 10 hundredths for 1 tenth. The tenths now total 10.
- 3.Exchange 10 tenths for 1 whole unit. The result is 30 units.
- 4.Check in hundredths: 1564 + 1436 = 3000 hundredths, or 30 units.
Subtracting with an Exchange
Subtraction removes parts from the first quantity. When that quantity has too few of a required small part, exchange one larger part for ten smaller parts. Be careful to reduce the larger-part count at the same time. An exchange creates no new quantity; it reorganises a quantity already present.
Problem
Find (25 + 9/10) − (6 + 4/10 + 7/100) units.
- 1.The first quantity has 25 units, 9 tenths, and 0 hundredths.
- 2.Exchange 1 tenth for 10 hundredths. It now has 25 units, 8 tenths, and 10 hundredths.
- 3.Subtract hundredths: 10 − 7 = 3. Subtract tenths: 8 − 4 = 4. Subtract units: 25 − 6 = 19.
- 4.The result is 19 + 4/10 + 3/100 units. Check: 2590 − 647 = 1943 hundredths.
Successive Exchanges
Sometimes one exchange is not enough. If a hundredths subtraction uses a tenth and the remaining tenths are still too few, exchange a whole unit as well. Keeping a record of the updated counts avoids subtracting from a larger-part count that has already changed.
Problem
Find (15 + 3/10 + 4/100) − (2 + 6/10 + 8/100) units.
- 1.Four hundredths cannot supply eight. Exchange one tenth: 15 units, 3 tenths, 4 hundredths becomes 15 units, 2 tenths, 14 hundredths.
- 2.Two tenths cannot supply six. Exchange one unit: the first quantity becomes 14 units, 12 tenths, 14 hundredths.
- 3.Subtract like parts: 14 − 8 = 6 hundredths; 12 − 6 = 6 tenths; 14 − 2 = 12 units.
- 4.The difference is 12 + 6/10 + 6/100 units. Check: 1534 − 268 = 1266 hundredths.
| Stage | Units | Tenths | Hundredths |
|---|---|---|---|
| Starting quantity | 15 | 3 | 4 |
| After exchanging one tenth | 15 | 2 | 14 |
| After exchanging one unit | 14 | 12 | 14 |
| Quantity removed | 2 | 6 | 8 |
| Remainder | 12 | 6 | 6 |
Whole-number arithmetic uses the same structure. In 653 − 268, an exchange changes 5 tens and 3 ones into 4 tens and 13 ones. A further exchange changes 6 hundreds and 4 tens into 5 hundreds and 14 tens. The idea is identical: ten of a smaller place make one of the place immediately to its left.
An exchanged tenth must be removed from the tenths count. Giving the hundredths ten extra parts while leaving all original tenths in place increases the quantity and produces a wrong answer.
Solve one addition and one subtraction in two ways: first keep units, tenths, and hundredths separate; then convert both numbers entirely into hundredths. Compare your results. Describe every exchange in words before writing the final answer.
Check Your Understanding
Use the ideas from this lesson to choose an answer. Explain your choice to yourself before opening the explanations below.
Quiz
Twelve hundredths can be regrouped as what?
What is (15 + 34/100) + (2 + 68/100)?
After exchanging one tenth, which equals 25 units and 9 tenths?
What is (15 + 34/100) − (2 + 68/100)?
Why can we check an addition by counting entirely in hundredths?
Ten of the twelve hundredths make one tenth; two hundredths remain.
Practice Problems
- Find 3/10 + (3 + 4/100) units.
- Add 9 + 5/10 + 7/100 and 2 + 1/10 + 3/100 units.
- Subtract 4 + 4/100 units from 7 + 7/100 units.
- Find (8 + 6/100) − (5 + 3/100) units.
- Find (12 + 6/10 + 2/100) − (9/10 + 9/100) units.
- Explain the connection between regrouping in 483 + 268 and in the first worked addition of this lesson.
Convert 3/10 into 30 hundredths. Add 30/100 + 4/100 = 34/100 to the 3 units. Sum = 3 + 34/100 units.
Key Takeaways
• Add and subtract parts of the same size. • Ten hundredths equal one tenth. • Ten tenths equal one whole unit. • An exchange changes the counts in two neighbouring places. • Successive exchanges may be needed in a subtraction. • Converting everything to hundredths provides an independent arithmetic check.