Mathematics from zero
Subtraction
You have 5 apples. You eat 2. Three are left. You did not destroy the apples you ate — you just took them out of the group and counted what remained. That taking-away is subtraction.
After this lesson you can say what subtraction means, name the result, see the two everyday questions it answers, subtract small numbers by counting back, and subtract larger numbers by borrowing from the next place.
Subtraction takes one amount away from another. When you remove some of a group
and ask “how many are left?”, that is subtraction. We write it with the minus
sign, −. So 5 − 2 = 3 is read “five take away two makes three”. You start with
the first number, remove the second number, and read what remains.
The result of a subtraction is called the difference. In 5 − 2 = 3, the 3 is
the difference. The word fits its second meaning too: the difference is the gap
between the two numbers — how far apart they sit. Either way you describe it, the
answer to a subtraction is the difference.
Subtraction is addition run backwards. Addition jumps right along the number line;
subtraction jumps left. To work out 7 − 3, stand on 7 and count back three steps:
6, 5, 4. You land on 4. And because it is addition reversed, you can always check a
subtraction by adding: 4 + 3 = 7, so 7 − 3 = 4 is correct.
Subtraction answers two everyday questions. One is take-away: “I had 7, I lost
3, what is left?” The other is difference: “Sara has 7 sweets, Tom has 3 — how many
more does Sara have?” Both are 7 − 3 = 4. The same operation answers “how many
remain” and “how far apart” — it helps to notice which question you are really
asking.
For larger numbers, subtract one place at a time — and borrow when a place is too small. Line the numbers up by place and subtract ones from ones, tens from tens. If the top digit in a place is smaller than the bottom one, you cannot take away enough. Then you borrow: take one unit from the next place up and regroup it into ten of the current place. Borrowing is carrying in reverse — one ten traded back into ten ones.
Subtract 42 − 17.
Line them up by place: 42 is 4 tens and 2 ones; 17 is 1 ten and 7 ones.
Start with the ones: you need 2 − 7, but 2 is smaller than 7 — not enough. Borrow:
take 1 ten from the 4 tens, leaving 3 tens, and regroup it into 10 ones. The ones
place now holds 2 + 10 = 12. Now 12 − 7 = 5. Write 5 in the ones place.
Now the tens: after the borrow there are 3 tens on top, 1 ten below. 3 − 1 = 2.
Write 2 in the tens place.
The difference is 25. Check by adding: 25 + 17 = 42. Correct.
Why this works
Why does borrowing work? Because one ten and ten ones are the same amount — exactly
the place-value rule. Borrowing does not change the top number; it only rewrites it so
a place that ran short can be paid from the place above. 42 is still 42 whether you
read it as “4 tens 2 ones” or “3 tens 12 ones”.
Common mistake
The most common mistake is subtracting the small digit from the large one regardless
of which is on top. In 42 − 17 the ones column is 2 − 7, not 7 − 2. The order
matters in subtraction: you must take the bottom digit from the top digit, and if the
top is too small, you borrow. Never silently flip the column.
Subtract by counting back: 8 − 3. Type the difference.
Sara has 9 stickers, Tom has 9. How many more does Sara have? Type the difference.
Subtract: 15 − 7. Type the difference.
Subtract: 53 − 28. Type the difference.
Subtract: 60 − 24. Type the difference.
In 42 − 17, the ones column asks 2 − 7. The 2 is too small. What do you do?
Subtraction takes one amount away from another, and its result is the difference — which is also the gap between the two numbers. It is addition reversed: count back on the number line, and check any subtraction by adding. It answers both “how many are left” and “how far apart”. For larger numbers, subtract one place at a time; when a place on top is too small, borrow one unit from the next place and regroup it into ten.