Number · Number Operations · Lesson 4
Adding and Subtracting Whole Numbers and Decimals
Every receipt at the market, every MoMo balance and every football attendance figure involves adding and subtracting. In this lesson you will use expanded form (partitioning) to see exactly what happens to each place value, then use the column method confidently with carrying, exchanging and decimal points.
By the end you will be able to
- ✔Partition whole numbers and decimals into expanded form and use it to add and subtract.
- ✔Add and subtract numbers of up to four digits (and beyond) in columns, carrying and exchanging correctly.
- ✔Line up decimal points and use placeholder zeros when adding and subtracting decimals.
- ✔Check answers using estimation and the inverse operation.
Key words
- Partitioning
- Splitting a number into the values of its digits, e.g. 785 = 700 + 80 + 5. Also called expanded form.
- Carrying (regrouping)
- When a column adds up to 10 or more, writing the ones digit and moving the ten to the next column on the left.
- Exchanging (borrowing)
- Changing 1 from a higher place into 10 of the next lower place so that you can subtract.
- Placeholder zero
- A zero written to fill an empty place, e.g. writing 327.6 as 327.60 to match 54.13.
- Inverse operation
- The operation that undoes another. Addition and subtraction are inverses, so you can check a subtraction by adding.
Curriculum reference
Content standard: B7.1.2.2 Demonstrate an understanding of addition, subtraction, multiplication and division of (i) whole numbers, and (ii) decimal numbers, to solve problems.
Indicator: Add and subtract up to four-digit numbers.
Starter
The shop's end-of-day account
Auntie Abena runs a provisions shop at Madina. On Monday she sold goods worth GH₵785 and on Tuesday goods worth GH₵9,342 (a big customer bought cartons of milk for a wedding). Her MoMo statement says GH₵10,127 came in. Is that right? And if she then pays a supplier GH₵7.85 out of GH₵93.60 in her drawer, how much is left? Let's find out how careful adding and subtracting answers these questions.
1. Adding by partitioning (expanded form)
Adding is putting together like with like: hundreds with hundreds, tens with tens, ones with ones. Partitioning makes this visible. Split each number into the values of its digits, add the matching parts, then put the parts back together.
| Number | Thousands | Hundreds | Tens | Ones |
|---|---|---|---|---|
| 785 | 700 | 80 | 5 | |
| 9,342 | 9,000 | 300 | 40 | 2 |
| Sum of parts | 9,000 | 1,000 | 120 | 7 |
Now recombine: 9,000 + 1,000 + 120 + 7 = 10,127. Notice that the hundreds added up to 1,000 and the tens to 120 — more than one place can hold. In the column method, these extra amounts are what we carry into the next place. So Auntie Abena's MoMo statement is correct.
2. The column method: carrying
The column method is a short way of writing partitioning. Write the numbers under each other with the ones lined up. Start from the ones column (the right) and work left. If a column adds up to 10 or more, write the ones digit of that total and carry the ten as 1 into the next column. Step through the animation.
▶ Watch each column being added from right to left, and see where a 1 is carried into the next column.
The 1 that is carried from the tens to the hundreds is really one hundred (ten tens), and the 1 carried from the hundreds to the thousands is really one thousand (ten hundreds). That is why we carry to the left: ten of any place makes one of the next place.
3. Adding decimals
Decimals follow exactly the same idea: tenths with tenths, hundredths with hundredths. The easiest way to keep the places lined up is to line up the decimal points, one exactly under the other. If one number has fewer decimal places, fill the gap with a placeholder zero: 327.6 becomes 327.60. This does not change its value.
By partitioning: 327.6 = 300 + 20 + 7 + 0.6 and 54.13 = 50 + 4 + 0.1 + 0.03. Add like parts: 300 + 70 + 11 + 0.7 + 0.03 = 381.73. The column method shows the same thing more quickly.
▶ The decimal points stay in one straight line. A placeholder zero fills the empty hundredths place of 327.6.
4. Subtracting with exchanging
In subtraction we again work from the right, place by place. When the top digit is smaller than the digit below it, we exchange: take 1 from the next place on the left and turn it into 10 of the current place. For example, 1 ten becomes 10 ones, and 1 tenth becomes 10 hundredths.
Look at 93.6 − 7.85. First write 93.6 as 93.60 so that both numbers have hundredths. In the hundredths, 0 − 5 is impossible, so exchange 1 tenth for 10 hundredths: 10 − 5 = 5. Now the tenths are 5 − 8, so exchange 1 one for 10 tenths: 15 − 8 = 7. The ones are now 2 − 7, so exchange 1 ten: 12 − 7 = 5. Finally the tens: 8 − 0 = 8. The answer is 85.75.
▶ Notice every exchange: a crossed-out digit loses 1, and the column on its right gains 10.
Zeros in the top number need extra care. In 5,003 − 1,768 there are no tens or hundreds to exchange from, so you go to the thousands: 5 thousands become 4 thousands and 10 hundreds; one of those hundreds becomes 10 tens; one ten becomes 10 ones. The top number is now 4 thousands, 9 hundreds, 9 tens and 13 ones, and you can subtract: 3,235.
Worked examples
Try each one yourself first, then open the solution and compare every step.
Example 1
Adding by partitioning
Show the solution
- Step 1Partition: 785 = 700 + 80 + 5 and 9,342 = 9,000 + 300 + 40 + 2.
- Step 2Add like parts: thousands 9,000; hundreds 700 + 300 = 1,000; tens 80 + 40 = 120; ones 5 + 2 = 7.
- Step 3Recombine: 9,000 + 1,000 + 120 + 7 = 10,127.
Answer: 10,127
Example 2
Subtracting four-digit numbers
▶ See how the exchange passes across the zero in the tens place.
Show the solution
- Step 1Write 4,507 on top and 2,689 below, ones under ones.
- Step 2Ones: 7 − 9 is impossible. The tens digit is 0, so exchange from the hundreds: 5 hundreds → 4 hundreds and 10 tens; then 10 tens → 9 tens and 10 ones. Ones: 17 − 9 = 8.
- Step 3Tens: 9 − 8 = 1.
- Step 4Hundreds: 4 − 6 is impossible, so exchange 1 thousand: 14 − 6 = 8. Thousands: 3 − 2 = 1.
- Step 5Check: 1,818 + 2,689 = 4,507 ✓
Answer: 1,818
Example 3
Adding decimals
Show the solution
- Step 1Line up the decimal points and write 327.6 as 327.60.
- Step 2Hundredths: 0 + 3 = 3. Tenths: 6 + 1 = 7.
- Step 3Ones: 7 + 4 = 11, write 1 carry 1. Tens: 2 + 5 + 1 = 8. Hundreds: 3.
- Step 4Answer: 381.73. Estimate check: 330 + 50 = 380 ✓
Answer: 381.73
Example 4
Subtracting decimals
Show the solution
- Step 1"Subtract 7.85 from 93.6" means 93.6 − 7.85. Write 93.6 as 93.60.
- Step 2Hundredths: exchange 1 tenth, 10 − 5 = 5.
- Step 3Tenths: 5 − 8 is impossible, exchange 1 one: 15 − 8 = 7.
- Step 4Ones: 2 − 7 is impossible, exchange 1 ten: 12 − 7 = 5. Tens: 8 − 0 = 8.
- Step 5Answer: 85.75. Check: 85.75 + 7.85 = 93.60 ✓
Answer: 85.75
Example 5
A two-step market problem
Show the solution
- Step 1After paying: 2,500.00 − 1,286.45. Exchange across the zeros: 2,500.00 − 1,286.45 = 1,213.55.
- Step 2Check: 1,213.55 + 1,286.45 = 2,500.00 ✓
- Step 3After receiving: 1,213.55 + 350.80 = 1,564.35.
- Step 4Estimate: 2,500 − 1,300 + 350 = 1,550, close to 1,564.35 ✓
Answer: GH₵1,564.35
Summary
- Partitioning (expanded form) shows that we always add or subtract like place values.
- Line up whole numbers by their ones digits; line up decimals by their decimal points.
- Fill empty decimal places with placeholder zeros (327.6 = 327.60).
- Carry when a column adds to 10 or more; exchange 1 from the left when the top digit is too small.
- Estimate before and check after: addition and subtraction are inverse operations.
Practice: 15 questions
5 easy, 5 medium, 5 hard. Each one is marked straight away, with a full worked solution.
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