---
title: "Subtraction"
book: "Primary & Middle School Mathematics"
subject: math
language: en
chapter: 9
exercises: 10
source: https://one-course.com/books/math/1/en/chapter/9-subtraction
---

# Chapter 9 — Subtraction

When the units on top are too few to take from, we break open a ten — the *borrow*, the mirror of the carry. This chapter explains it with sticks, then trains the written subtraction, with counting up as a friendly shortcut.

## 9.1 Why we borrow

**Example 9.1 (Unbundling a ten).**

$42 - 17$: we want to take $7$ units from $2$ units — too few! So untie one of the $4$ tens into ten loose units: now the top has $3$ tens and $12$ units. Take away: $12 - 7 = 5$ units, and $3 - 1 = 2$ tens. So $42 - 17 = 25$.

![The borrow is unbundling: one ten becomes ten units, so the subtraction can go through.](https://one-course.com/images/onecourse/chapters/math-1/g2-subtraction/fig-0a2f51550dc8.svg)

*The borrow is unbundling: one ten becomes ten units, so the subtraction can go through.*

## 9.2 The written method

**Method 9.2 (Column subtraction).**

1. Write the bigger number on top, aligned on the right;
2. subtract the units; if the top digit is too small, borrow a ten (it becomes ten units in the current column);
3. subtract the tens, then the hundreds;
4. *check* : the answer plus the number taken away must give back the top number.

**Example 9.3.**

Compute $503 - 168$:

$$
\begin{array}{r}
5\,0\,3 \\
-\ 1\,6\,8 \\
\hline
3\,3\,5
\end{array}
$$

Units: $3 - 8$ too small; borrow through the tens (there are none, so borrow from the hundreds first): the top becomes $4$ hundreds, $9$ tens, $13$ units. Then $13 - 8 = 5$; $9 - 6 = 3$; $4 - 1 = 3$. Check: $335 + 168 = 503$. ✓

## 9.3 Counting up

**Method 9.4 (Subtracting by counting up).**

For numbers that are close, climb from the small one to the big one and add the hops:

$$
83 - 78: \quad 78 \to 80 \text{ is } 2, \ 80 \to 83 \text{ is } 3,
\text{ so } 83 - 78 = 5 .
$$

Counting up is quicker than borrowing when the two numbers are near each other — and it is exactly how a shopkeeper gives change.

![83 - 78 by counting up: two hops, 2 + 3 = 5.](https://one-course.com/images/onecourse/chapters/math-1/g2-subtraction/fig-03448847481e.svg)

*$83 - 78$ by counting up: two hops, $2 + 3 = 5$.*

**Example 9.5 (Giving change).**

A toy costs $34$; you pay with a $50$. Change: count up, $34 \to 40$ is $6$, $40 \to 50$ is $10$, so the change is $16$. The subtraction $50 - 34 = 16$, done the shopkeeper’s way.

## 9.4 Exercises

**Exercise 9.1 ★.**

Compute (no borrow needed): $58 - 23$; $176 - 45$; $389 - 164$.

**Solution of Exercise 9.1.**

$58 - 23 = 35$; $176 - 45 = 131$; $389 - 164 = 225$.

**Exercise 9.2 ★.**

Compute with a borrow: $52 - 17$; $73 - 28$; $60 - 24$.

**Solution of Exercise 9.2.**

$52 - 17 = 35$; $73 - 28 = 45$; $60 - 24 = 36$.

**Exercise 9.3 ★.**

Compute in columns and check each one: $324 - 167$; $500 - 236$; $703 - 158$.

**Solution of Exercise 9.3.**

$324 - 167 = 157$ (check $157 + 167 = 324$); $500 - 236 = 264$ (check $264 + 236 = 500$); $703 - 158 = 545$ (check $545 + 158 = 703$).

**Exercise 9.4 ★.**

Compute by counting up: $62 - 58$; $91 - 85$; $100 - 96$.

**Solution of Exercise 9.4.**

$62 - 58 = 4$; $91 - 85 = 6$; $100 - 96 = 4$.

**Exercise 9.5 ★.**

Compute the change (count up): pay $50$ for a $42$ book; pay $20$ for a $13$ toy; pay $100$ for a $75$ bag.

**Solution of Exercise 9.5.**

$50 - 42 = 8$; $20 - 13 = 7$; $100 - 75 = 25$.

**Exercise 9.6 ★.**

“A tree is $156$ cm tall. It grew $28$ cm this year. How tall was it last year?” Write the subtraction and answer.

**Solution of Exercise 9.6.**

Last year: $156 - 28 = 128$ cm.

**Exercise 9.7 ★.**

Complete with the twin addition to check: $85 - 39 = 46$; is it right? $214 - 68 = 156$; is it right?

**Solution of Exercise 9.7.**

$85 - 39 = 46$: check $46 + 39 = 85$ — right. $214 - 68 = 156$: check $156 + 68 = 224 \neq 214$ — wrong (the correct answer is $146$).

**Exercise 9.8 ★.**

For each story, choose $+$ or $-$: “$47$ birds, $19$ fly away”; “$47$ apples, $19$ more picked”; “Sam has $47$, Lea has $19$: how many more has Sam?”.

**Solution of Exercise 9.8.**

“fly away”: $47 - 19 = 28$. “more picked”: $47 + 19 = 66$. “how many more”: $47 - 19 = 28$.

**Exercise 9.9 ★.**

Fill the hole: $63 - \;?\; = 45$. (Count up from $45$ to $63$.)

**Solution of Exercise 9.9.**

From $45$ up to $63$ is $18$, so $63 - 18 = 45$: the hole is $18$.

**Exercise 9.10 ★★.**

A game: think of a number, add $30$, then take away $18$. The result is $58$. What was the number? (Work backward!)

**Solution of Exercise 9.10.**

Work backward from $58$: before taking away $18$, the number was $58 + 18 = 76$; before adding $30$, it was $76 - 30 = 46$. The starting number was $46$ (check: $46 + 30 - 18 = 58$).