---
title: "Addition and Subtraction"
book: "Primary & Middle School Mathematics"
subject: math
language: en
chapter: 15
exercises: 11
source: https://one-course.com/books/math/1/en/chapter/15-addition-and-subtraction
---

# Chapter 15 — Addition and Subtraction

Adding means putting together; subtracting means taking away, or finding what is missing. This chapter trains the two column methods — with their famous carries and borrows — and the mental tricks that often make columns unnecessary.

## 15.1 Column addition

**Method 15.1 (Adding in columns).**

1. Write the numbers one under the other, units under units, tens under tens (line up on the *right* );
2. add the units; if the [sum](https://one-course.com/books/math/1/en/chapter/2-addition-first-steps#def-g1-addition-def) passes $9$ , write its units digit and *carry* the ten to the next column (a small $1$ );
3. continue with the tens, the hundreds, …, adding the carry each time.

**Example 15.2.**

Compute $457 + 368$:

$$
\begin{array}{r}
{\scriptstyle 1}\ \ {\scriptstyle 1}\ \ \phantom{0} \\[-3pt]
4\,5\,7 \\
+\ 3\,6\,8 \\
\hline
8\,2\,5
\end{array}
$$

Units: $7 + 8 = 15$: write $5$, carry $1$. Tens: $5 + 6 + 1 = 12$: write $2$, carry $1$. Hundreds: $4 + 3 + 1 = 8$. Result: $825$.

**Example 15.3 (Mental addition).**

Look for friendly numbers before reaching for columns:

- *make a ten* : $47 + 8 = 47 + 3 + 5 = 50 + 5 = 55$ ;
- *jump by parts* : $256 + 130 = 256 + 100 + 30 = 386$ ;
- *reorder* : $17 + 59 + 3 = (17 + 3) + 59 = 79$ .

![Mental addition as jumps on the number line: 256 + 130 in two hops, first the hundreds, then the tens.](https://one-course.com/images/onecourse/chapters/math-1/g3-addsub/fig-6e58cf64cb35.svg)

*Mental addition as jumps on the number line: $256 + 130$ in two hops, first the hundreds, then the tens.*

## 15.2 Column subtraction

**Method 15.4 (Subtracting in columns).**

1. Write the bigger number on top, aligned on the right;
2. subtract column by column, starting from the units;
3. if the top digit is too small, *borrow* : take one from the next place on the top row (it is worth ten in the current column);
4. *check* : the result plus the subtracted number must give back the top number.

**Example 15.5.**

Compute $603 - 147$:

$$
\begin{array}{r}
6\,0\,3 \\
-\ 1\,4\,7 \\
\hline
4\,5\,6
\end{array}
$$

Units: $3 - 7$ impossible; borrow through: $603$ becomes $5$ hundreds, $9$ tens, $13$ units. Then $13 - 7 = 6$; tens $9 - 4 = 5$; hundreds $5 - 1 = 4$. Result: $456$. Check: $456 + 147 = 603$.

![The borrow of 603 - 147 in pictures: one hundred is untied into ten tens, and one of those tens into ten units. Same amount, new clothes — 6\, \,0\, \,3 becomes 5\, \,9\, \,13, and every column can now subtract.](https://one-course.com/images/onecourse/chapters/math-1/g3-addsub/fig-be9dee719246.svg)

*The borrow of $603 - 147$ in pictures: one hundred is untied into ten tens, and one of those tens into ten units. Same amount, new clothes — $6\,\vert\,0\,\vert\,3$ becomes $5\,\vert\,9\,\vert\,13$, and every column can now subtract.*

**Example 15.6 (Subtraction as a missing part).**

“Lila has $35$ marbles, Tom has $52$: how many more has Tom?” is the subtraction $52 - 35$. Mentally, count *up* from $35$: $35 \to 40$ is $5$, then $40 \to 52$ is $12$; in total $5 + 12 = 17$. Counting up is often easier than borrowing.

![Computing 52 - 35 by counting up: first to the friendly 40, then to 52. The answer is the total of the hops: 5 + 12 = 17.](https://one-course.com/images/onecourse/chapters/math-1/g3-addsub/fig-360f80ecc6ec.svg)

*Computing $52 - 35$ by counting up: first to the friendly $40$, then to $52$. The answer is the total of the hops: $5 + 12 = 17$.*

## 15.3 Orders of magnitude

**Method 15.7 (Estimating before computing).**

Before any column computation, round the numbers to friendly ones and compute an *estimate*. After computing, compare: a result far from the estimate signals a mistake.

**Example 15.8.**

For $487 + 312$: the estimate is $500 + 300 = 800$. The column result, $799$, is close: plausible. If you had found $1\,099$ (a missed alignment), the estimate would have warned you.

## 15.4 Exercises

**Exercise 15.1 ★.**

Compute in columns: $346 + 253$; $478 + 265$; $1\,536 + 2\,876$.

**Solution of Exercise 15.1.**

$346 + 253 = 599$; $478 + 265 = 743$; $1\,536 + 2\,876 = 4\,412$.

**Exercise 15.2 ★.**

Compute in columns, then check by an addition: $587 - 234$; $805 - 348$; $1\,000 - 463$.

**Solution of Exercise 15.2.**

$587 - 234 = 353$ (check: $353 + 234 = 587$).

$805 - 348 = 457$ (check: $457 + 348 = 805$).

$1\,000 - 463 = 537$ (check: $537 + 463 = 1\,000$).

**Exercise 15.3 ★.**

Compute mentally by making tens: $38 + 7$; $56 + 9$; $45 + 28$ (add $30$, take away $2$).

**Solution of Exercise 15.3.**

$38 + 7 = 38 + 2 + 5 = 45$; $56 + 9 = 56 + 4 + 5 = 65$; $45 + 28 = 45 + 30 - 2 = 73$.

**Exercise 15.4 ★.**

Compute mentally by jumps: $340 + 220$; $675 + 210$; $512 - 300$; $840 - 220$.

**Solution of Exercise 15.4.**

$340 + 220 = 560$; $675 + 210 = 885$; $512 - 300 = 212$; $840 - 220 = 620$.

**Exercise 15.5 ★.**

Compute $63 - 47$ by counting up (as in [Example 15.6](#ex-g3-addsub-missing)), then check in columns.

**Solution of Exercise 15.5.**

From $47$ to $50$: $3$; from $50$ to $63$: $13$; total $3 + 13 = 16$. Columns confirm: $63 - 47 = 16$.

**Exercise 15.6 ★.**

Estimate first (round to hundreds), then compute exactly: $492 + 218$; $913 - 388$.

**Solution of Exercise 15.6.**

$492 + 218$: estimate $500 + 200 = 700$; exactly $710$.

$913 - 388$: estimate $900 - 400 = 500$; exactly $525$.

**Exercise 15.7 ★.**

A school has $348$ girls and $296$ boys. How many students in all? Write the operation, compute, and answer with a sentence.

**Solution of Exercise 15.7.**

$348 + 296 = 644$. The school has $644$ students.

**Exercise 15.8 ★.**

A book has $224$ pages; Nora has read $178$. How many pages are left to read?

**Solution of Exercise 15.8.**

$224 - 178 = 46$. Nora still has $46$ pages to read.

**Exercise 15.9 ★.**

Complete the holes (work column by column):

$$
\begin{array}{r}
3\,?\,7 \\
+\ ?\,2\,? \\
\hline
8\,6\,2
\end{array}
$$

Is there only one way to fill the three holes? Explain.

**Solution of Exercise 15.9.**

Units: $7 + \text{?} = 2$ is impossible, so $7 + 5 = 12$: the units hole is $5$, carry $1$. Tens: $\text{?} + 2 + 1 = 6$: the tens hole is $3$ (no carry). Hundreds: $3 + \text{?} = 8$: the hundreds hole is $5$. So $337 + 525 = 862$, and each hole had exactly one possible digit — there is only one way.

**Exercise 15.10 ★★.**

A bakery sold $145$ croissants in the morning and $89$ in the afternoon; $17$ croissants were left unsold at closing time. How many croissants were baked that day? (Two steps.)

**Solution of Exercise 15.10.**

Sold: $145 + 89 = 234$ croissants. Baked: sold plus unsold, $234 + 17 = 251$ croissants.

**Exercise 15.11 ★★.**

Theo says: “to compute $500 - 199$, I compute $500 - 200$ and then add $1$.” Is he right? Compute both ways, and explain his trick.

**Solution of Exercise 15.11.**

He is right: $500 - 200 = 300$, plus $1$ gives $301$; in columns, $500 - 199 = 301$ too. Taking away $200$ removes *one too many*, so one must be given back — subtracting a friendly neighbor and correcting is often faster than borrowing.