Primary & Middle School Mathematics · Grades 1–9

9Subtraction

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.

9.2 The written method

Method 9.2 (Column subtraction)

Write the bigger number on top, aligned on the right;
subtract the units; if the top digit is too small, borrow a ten (it becomes ten units in the current column);
subtract the tens, then the hundreds;
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. — $83 - 78$ 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

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

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

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

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

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

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

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

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

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

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: $46 + 30 - 18 = 58$ ).