What you'll learn
Adding whole numbers means putting amounts together to find a total; subtracting means taking one amount away, or finding the difference between two amounts. At SEA level this is done with numbers up to one million, using the column method, and it has to be quick and accurate because so much of the rest of the paper depends on it. This topic covers setting out columns, carrying and exchanging, subtracting across zeros, checking by estimating, and — the part that actually earns the marks — deciding which operation a word problem is asking for. Number is 19 of the 40 items on the SEA Mathematics paper, and addition and subtraction appear inside almost all of them.
Key terms and definitions
Sum — the answer to an addition. The sum of 34 and 21 is 55.
Addend — a number being added. In 34 + 21, both 34 and 21 are addends.
Difference — the answer to a subtraction, and also the gap between two numbers.
Carrying (regrouping up) — when a column totals ten or more, ten are exchanged for one of the next column along.
Exchanging (regrouping down, or borrowing) — when the top digit is too small to subtract from, one from the next column left is broken into ten.
Inverse operation — the operation that undoes another. Subtraction undoes addition.
Estimate — a rough answer found by rounding first, used to check the exact answer is sensible.
Core concepts
Lining up the columns
Every mistake in this topic that is not a careless slip comes from digits sitting in the wrong column. Numbers must be lined up by place value, with the ones under the ones, tens under tens, hundreds under hundreds. This matters most when the numbers have different lengths.
| Ten thousands | Thousands | Hundreds | Tens | Ones |
|---|---|---|---|---|
| 2 | 4 | 6 | 0 | 3 |
| 5 | 8 | 1 |
Here 24 603 + 581 is set out properly: the 5 of 581 sits under the 6, not under the 4. Writing 581 flush to the left would turn it into 58 100 and the answer would be wildly wrong. A useful habit is to write the longest number on top.
Carrying in addition
Work from the right. Add the ones column first. If the total is ten or more, write the ones digit in the answer and carry the tens digit into the next column.
For 4 876 + 2 359 the ones give 6 + 9 = 15, so write 5 and carry 1. The tens give 7 + 5 + 1 = 13, so write 3 and carry 1. The hundreds give 8 + 3 + 1 = 12, so write 2 and carry 1. The thousands give 4 + 2 + 1 = 7.
The answer is 7 235. The carried digits should be written small above the next column, not held in the head, because a forgotten carry is the single most common way to lose a mark here.
Exchanging in subtraction
Again work from the right. If the top digit is smaller than the bottom digit, take one from the column to the left and add ten to the digit you are working on.
For 6 342 − 1 875:
- Ones: 2 is smaller than 5. Exchange from the tens: the 4 becomes 3 and the 2 becomes 12. Then 12 − 5 = 7.
- Tens: 3 is smaller than 7. Exchange from the hundreds: the 3 becomes 2 and the 3 becomes 13. Then 13 − 7 = 6.
- Hundreds: 2 is smaller than 8. Exchange from the thousands: the 6 becomes 5 and the 2 becomes 12. Then 12 − 8 = 4.
- Thousands: 5 − 1 = 4.
The answer is 4 467. Check it by adding back: 4 467 + 1 875 = 6 342. That check takes about fifteen seconds and catches nearly every error.
Subtracting across zeros
This is the hardest case. In 5 000 − 1 246 there is nothing in the tens or hundreds to exchange from, so you must go further left. Take one thousand from the 5. That leaves 4 thousands and gives 10 hundreds. Break one hundred into 10 tens, leaving 9 hundreds. Break one ten into 10 ones, leaving 9 tens.
The top number now reads 4 thousands, 9 hundreds, 9 tens, 10 ones. So the ones give 10 − 6 = 4, the tens 9 − 4 = 5, the hundreds 9 − 2 = 7 and the thousands 4 − 1 = 3. The answer is 3 754. The pattern to remember is that a row of zeros all become nines except the last one, which becomes ten.
Adding several numbers at once
When three or four numbers are added, the ones column can total more than 20, so the carry can be a 2 or a 3. For 47 + 68 + 95: the ones give 7 + 8 + 5 = 20, so write 0 and carry 2. The tens give 4 + 6 + 9 + 2 = 21, giving 210. Looking for pairs that make ten speeds this up: in 7 + 8 + 3 + 2, spotting 7 + 3 and 8 + 2 gives 20 straight away.
Choosing the operation from the words
Addition is signalled by altogether, total, sum, in all and combined. Subtraction is signalled by how many more, how many fewer, difference, left, remaining and change.
But the words are not a rule. "Anisa has 12 more marbles than Kern, who has 30" is an addition even though it says more. The safe method is to picture the situation: are two amounts being joined, or is one being taken from another? That picture beats keyword-hunting every time.
Estimating to check
Round each number to a sensible place, then do the easy sum. For 4 876 + 2 359, round to 5 000 + 2 000 = 7 000. The exact answer 7 235 is close, so it is believable. Had it come out as 723 or 72 350, the estimate would have exposed it immediately.
Worked examples
Example 1: A two-step total
A school library has 3 458 books. During the year it receives 1 276 new books and loses 389 damaged ones. How many books does it have at the end of the year?
Add the new books first: 3 458 + 1 276. Ones give 14, write 4 carry 1; tens give 5 + 7 + 1 = 13, write 3 carry 1; hundreds give 4 + 2 + 1 = 7; thousands give 4. That is 4 734.
Now subtract the damaged books: 4 734 − 389.
- Ones: 4 − 9 needs an exchange. The 3 tens become 2 and the 4 becomes 14. 14 − 9 = 5.
- Tens: 2 − 8 needs an exchange. The 7 hundreds become 6 and the 2 becomes 12. 12 − 8 = 4.
- Hundreds: 6 − 3 = 3, and the thousands give 4.
The answer is 4 345 books.
Estimate to check: 3 500 + 1 300 − 400 = 4 400. Close enough to trust.
Example 2: A comparison question
In one week a doubles vendor in Arima sold 2 105 doubles. The following week he sold 1 847. How many more did he sell in the first week?
How many more between two amounts means subtract the smaller from the larger: 2 105 − 1 847.
- Ones: 5 − 7 needs an exchange, but the tens digit is 0. Go to the hundreds: the 1 hundred becomes 0 and the 0 tens become 10. Now take one of those tens: the tens become 9 and the ones become 15. 15 − 7 = 8.
- Tens: 9 − 4 = 5.
- Hundreds: 0 − 8 needs an exchange. The 2 thousands become 1 and the 0 becomes 10. 10 − 8 = 2.
- Thousands: 1 − 1 = 0.
The answer is 258 doubles.
Check by adding back: 1 847 + 258 = 2 105. Correct.
Example 3: Finding a missing addend
Devon and Riya together collected 1 500 bottle caps. Devon collected 862. How many did Riya collect?
The total is known and one part is known, so the missing part is found by subtracting: 1 500 − 862.
The zeros need care. Take one hundred from the 5, leaving 4 hundreds and giving 10 tens. Take one ten from those, leaving 9 tens and giving 10 ones.
- Ones: 10 − 2 = 8
- Tens: 9 − 6 = 3
- Hundreds: 4 − 8 needs an exchange. The 1 thousand becomes 0 and the 4 becomes 14. 14 − 8 = 6.
- Thousands: 0
The answer is 638 bottle caps.
Check: 862 + 638 = 1 500. Correct.
Common mistakes and how to avoid them
Misaligned columns. Writing 24 603 + 581 with the 581 pushed left. Fix: always start writing from the right-hand edge and work leftwards, or rule faint vertical lines.
Forgetting to add the carried digit. Getting 4 876 + 2 359 = 7 225. Fix: write every carry down as a small digit and tick it off as you use it.
Exchanging but not reducing the column you took from. Turning the 4 into 14 but leaving the neighbouring digit unchanged. Fix: cross out the digit you took from and write the new value above it, every single time.
Subtracting the smaller digit from the bigger one regardless of position. Doing 5 − 2 = 3 in a column where the question is 2 − 5. Fix: if the top digit is smaller, you must exchange — there is no choice.
Losing the second step. Answering 4 734 in Example 1 and stopping. Fix: reread the final sentence of the question before writing the answer.
How parents can help at home
You do not need to remember any method yourself. What helps is asking questions and insisting on the layout.
Ask for the columns. When your child is stuck on an addition or subtraction written across the page, the prompt is almost always "draw the columns". Rewriting 3 458 + 1 276 vertically, with the digits stacked, solves most of the difficulty on its own.
Use shopping receipts. A supermarket bill is a ready-made addition exercise. Ask "if I had given the cashier $200 TT, what change should I get?"
Ask for the estimate first. Before your child works out an exact answer, ask "roughly, what should it be?" A child who says "about 7 000" and then writes 723 will spot the problem without being told.
Ask them to check by adding back. After every subtraction, "add your answer to the number you took away — do you get back to where you started?"
Talk through the words. For a word problem, ask "are we putting things together or taking something away?" Do not let a keyword decide it.
If a mistake appears, resist giving the correct answer. Ask which column went wrong. Finding their own error teaches far more than being told.
Exam technique for addition and subtraction of whole numbers
Set out every calculation in columns, even ones that look easy. The few seconds spent lining up digits are cheaper than a lost mark.
Estimate before you calculate on any question with numbers over a thousand. It catches place-value errors, which are the errors most likely to make a wrong answer look plausible.
Check every subtraction by adding the answer back to the number you subtracted. If you do not return to the starting number, the error is in your working.
Read multi-step questions twice and underline the final question. Many Number items are built so that the answer to step one appears among the options.
In multiple-choice items, the distractors are built from the standard errors — a forgotten carry, a missed exchange, the subtraction done the wrong way round. Matching an option exactly is not proof you are right. With 40 items and 75 marks, the plain arithmetic questions are the ones to finish fastest, leaving time for the problem-solving items.
Quick revision summary
- Line digits up by place value: ones under ones, tens under tens.
- Always work from the right-hand column leftwards.
- Carrying: when a column totals ten or more, write the ones digit and carry the ten.
- Exchanging: if the top digit is too small, take one from the left and add ten.
- Across zeros, the zeros become nines and the last one becomes ten.
- Check a subtraction by adding the answer back to the subtrahend.
- Check an addition by estimating with rounded numbers first.
- Altogether, total, in all usually mean add; difference, how many more, left usually mean subtract — but picture the situation rather than trusting the word.
- Answer the question that was asked, not the first step of it.