Primary & Middle School Mathematics · Grades 1–9

8Addition with Carrying

When the units of a sum pass nine, ten of them bundle into a new ten — and that little bundle, moved to the next column, is the famous carry. This chapter explains the carry with sticks, then trains the written method.

8.1 Why we carry

Example 8.1 (Bundling ten units)

$27 + 15$ : put the units together, $7 + 5 = 12$ units. But ten units make a ten! Bundle ten of them: that leaves $2$ loose units and one new ten to carry. Now the tens: $2 + 1 + 1 = 4$ tens. So $27 + 15 = 42$ .

The carry is a bundle: ten loose units become one ten, moved to the tens column.

8.2 The written method

Method 8.2 (Column addition)

Write the numbers one under the other, units under units, tens under tens;
add the units; if the answer passes $9$ , write the units digit and carry a small $1$ above the tens;
add the tens (with the carry), then the hundreds;
check with an estimate: round each number and add roughly.

Example 8.3

Compute $268 + 156$ :

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

Units: $8 + 6 = 14$ : write $4$ , carry $1$ . Tens: $6 + 5 + 1 = 12$ : write $2$ , carry $1$ . Hundreds: $2 + 1 + 1 = 4$ . Estimate check: about $270 + 160 = 430$ , close to $424$ . ✓

8.3 Adding in your head

Example 8.4 (Make a ten)

For small sums, jump to a round number first:

48 + 6 = 48 + 2 + 4 = 50 + 4 = 54 .

Add tens and units separately: $35 + 23 = (30 + 20) + (5 + 3) = 50 + 8 = 58$ . And a near-hundred trick: $99 + 46 = 100 + 46 - 1 = 145$ .

Example 8.5 (A little problem)

“The library has $185$ books; $57$ new ones arrive. How many books now?” Adding: $185 + 57 = 242$ (units $5 + 7 = 12$ , carry …). The library has $242$ books.

8.4 Exercises

Exercise 8.1 ★

Compute (no carry needed): $34 + 25$ ; $126 + 43$ ; $507 + 82$ .

Solution

Solution of Exercise 8.1.

$34 + 25 = 59$ ; $126 + 43 = 169$ ; $507 + 82 = 589$ .

Exercise 8.2 ★

Compute with a carry: $47 + 8$ ; $56 + 27$ ; $65 + 19$ .

Solution

Solution of Exercise 8.2.

$47 + 8 = 55$ ; $56 + 27 = 83$ ; $65 + 19 = 84$ .

Exercise 8.3 ★

Compute in columns: $268 + 145$ ; $374 + 158$ ; $429 + 293$ .

Solution

Solution of Exercise 8.3.

$268 + 145 = 413$ ; $374 + 158 = 532$ ; $429 + 293 = 722$ .

Exercise 8.4 ★

Compute in your head by making a ten: $38 + 5$ ; $67 + 6$ ; $54 + 8$ .

Solution

Solution of Exercise 8.4.

$38 + 5 = 38 + 2 + 3 = 43$ ; $67 + 6 = 67 + 3 + 3 = 73$ ; $54 + 8 = 54 + 6 + 2 = 62$ .

Exercise 8.5 ★

Compute by adding tens and units separately: $42 + 36$ ; $53 + 24$ ; $61 + 28$ .

Solution

Solution of Exercise 8.5.

$42 + 36 = 70 + 8 = 78$ ; $53 + 24 = 70 + 7 = 77$ ; $61 + 28 = 80 + 9 = 89$ .

Exercise 8.6 ★

Estimate first (round to the nearest ten or hundred), then compute exactly: $58 + 34$ ; $296 + 187$ .

Solution

Solution of Exercise 8.6.

$58 + 34$ : estimate $60 + 30 = 90$ ; exactly $92$ . $296 + 187$ : estimate $300 + 200 = 500$ ; exactly $483$ .

Exercise 8.7 ★

“Ana has $128$ marbles, Tom has $95$ . How many together?” Write the addition and the answer sentence.

Solution

Solution of Exercise 8.7.

$128 + 95 = 223$ . Together they have $223$ marbles.

Exercise 8.8 ★

Complete the units hole: $3\,?\, + 27 = 64$ (find the missing units digit of the first number).

Solution

Solution of Exercise 8.8.

The first number plus $27$ is $64$ , so it is $64 - 27 = 37$ : the missing units digit is $7$ (the number is $37$ ).

Exercise 8.9 ★

Two classes go to the show: $27$ and $28$ children, plus $6$ grown-ups. How many people in all? (Add the two classes first.)

Solution

Solution of Exercise 8.9.

Children: $27 + 28 = 55$ ; with the grown-ups, $55 + 6 = 61$ people.

Exercise 8.10 ★★

Find three numbers on the list $27$ , $33$ , $40$ , $65$ that add up to $100$ . (Look for two whose units are ten friends first.)

Solution

Solution of Exercise 8.10.

$27$ and $33$ have units $7$ and $3$ (ten friends): $27 + 33 = 60$ , then $60 + 40 = 100$ . The three numbers are $27$ , $33$ and $40$ .