Primary & Middle School Mathematics · Grades 1–9

22Multiplying Larger Numbers

Grade 3 multiplied by one digit (Chapter 16); this chapter multiplies by two-digit numbers — and explains the mysterious “shifted line” of the column method, which is nothing but a splitting into tens and units.

22.1 Multiplying by tens

Proposition 22.1 (Times 10, 20, 300 …)

Multiplying by $10$ appends one zero, by $100$ two zeros. To multiply by $20$ , multiply by $2$ , then by $10$ :

36 \times 20 = 36 \times 2 \times 10 = 72 \times 10 = 720 .

Likewise $14 \times 300 = 14 \times 3 \times 100 = 4\,200$ .

Proof. Admitted at this level. ∎

22.2 The column method with two digits

Example 22.2 (Why the shifted line?)

To compute $23 \times 12$ , split $12$ into $10 + 2$ :

23 \times 12 = 23 \times 2 + 23 \times 10 = 46 + 230 = 276 .

The column method writes exactly these two products, one under the other — the second shifted left because it is a number of tens:

\begin{array}{r} 2\,3 \\ \times\ 1\,2 \\ \hline 4\,6 \\ 2\,3\,\cdot \\ \hline 2\,7\,6 \end{array}

(the dot marks the units place of the shifted line, often left blank).

The rectangle picture of 23 × 12: four easy products, 200 + 40 + 30 + 6 = 276. The column method groups them into two lines: 46 (the “× 2” row) and 230 (the “× 10” row). — The rectangle picture of $23 \times 12$ : four easy products, $200 + 40 + 30 + 6 = 276$ . The column method groups them into two lines: $46$ (the “ $\times 2$ ” row) and $230$ (the “ $\times 10$ ” row).

Method 22.3 (Column multiplication by a two-digit number)

To compute $457 \times 36$ :

first line: $457 \times 6 = 2\,742$ ;
second line: $457 \times 3$ tens $= 1\,371$ tens — write $1\,371$ shifted one place left;
add the two lines: $2\,742 + 13\,710 = 16\,452$ ;
estimate to check: $457 \times 36 \approx 500 \times 36 = 18\,000$ ? Better: $450 \times 36$ is about $16\,000$ — plausible.

\begin{array}{r} 4\,5\,7 \\ \times\ \ \,3\,6 \\ \hline 2\,7\,4\,2 \\ 1\,3\,7\,1\,\cdot \\ \hline 1\,6\,4\,5\,2 \end{array}

Example 22.4 (Mental tricks)

Split the friendly way: $18 \times 11 = 18 \times 10 + 18 = 198$ .
Use a near-hundred: $25 \times 99 = 25 \times 100 - 25 = 2\,475$ .
Double and halve: $16 \times 35 = 8 \times 70 = 560$ (halving one factor and doubling the other keeps the product).

22.3 Multiplication in problems

Example 22.5

A school orders $28$ boxes of $145$ sheets of paper.

Operation: $145 \times 28$ .
Estimate: $150 \times 30 = 4\,500$ , a bit too much on both counts.
Columns: $145 \times 8 = 1\,160$ ; $145 \times 2$ tens $= 2\,900$ ; total $4\,060$ .
Sentence: the school receives $4\,060$ sheets.

22.4 Exercises

Exercise 22.1 ★

Compute mentally: $47 \times 10$ ; $23 \times 100$ ; $36 \times 20$ ; $15 \times 300$ ; $42 \times 50$ .

Solution

Solution of Exercise 22.1.

$470$ ; $2\,300$ ; $720$ ; $4\,500$ ; $2\,100$ .

Exercise 22.2 ★

Compute $34 \times 21$ by splitting as in Example 22.2 ( $34 \times 21 = 34 \times 20 + 34$ ), then check in columns.

Solution

Solution of Exercise 22.2.

$34 \times 21 = 34 \times 20 + 34 = 680 + 34 = 714$ ; the columns give the same lines: $34$ and $680$ .

Exercise 22.3 ★

Compute in columns: $63 \times 24$ ; $85 \times 47$ ; $126 \times 53$ .

Solution

Solution of Exercise 22.3.

$63 \times 24 = 1\,512$ ; $85 \times 47 = 3\,995$ ; $126 \times 53 = 6\,678$ .

Exercise 22.4 ★

Compute in columns, with an estimate first: $208 \times 34$ ; $517 \times 62$ .

Solution

Solution of Exercise 22.4.

$208 \times 34$ : estimate $200 \times 34 = 6\,800$ ; exactly $7\,072$ .

$517 \times 62$ : estimate $500 \times 60 = 30\,000$ ; exactly $32\,054$ .

Exercise 22.5 ★

Compute with a trick from Example 22.4: $45 \times 11$ ; $32 \times 99$ ; $24 \times 45$ (double and halve — twice if you like).

Solution

Solution of Exercise 22.5.

$45 \times 11 = 450 + 45 = 495$ .

$32 \times 99 = 3\,200 - 32 = 3\,168$ .

$24 \times 45 = 12 \times 90 = 1\,080$ (or once more: $6 \times 180 = 1\,080$ ).

Exercise 22.6 ★

A cinema has $26$ rows of $18$ seats. How many seats in all?

Solution

Solution of Exercise 22.6.

$26 \times 18 = 468$ seats.

Exercise 22.7 ★

A truck carries $32$ pallets of $48$ crates. How many crates? Give an estimate first, then the exact answer.

Solution

Solution of Exercise 22.7.

Estimate: $30 \times 50 = 1\,500$ . Exactly: $32 \times 48 = 1\,536$ crates.

Exercise 22.8 ★

Find the mistake: Ana computes $57 \times 23$ as $57 \times 3 = 171$ and $57 \times 2 = 114$ , then adds $171 + 114 = 285$ . The correct answer is $1\,311$ . What did she forget?

Solution

Solution of Exercise 22.8.

She forgot that the $2$ of $23$ counts tens: the second line must be $57 \times 20 = 1\,140$ , not $114$ . Correct total: $171 + 1\,140 = 1\,311$ .

Exercise 22.9 ★★

A gardener plants $14$ rows of $35$ tulips and $12$ rows of $28$ daffodils. How many flowers in all? (Two products, then a sum.)

Solution

Solution of Exercise 22.9.

Tulips: $14 \times 35 = 490$ . Daffodils: $12 \times 28 = 336$ . Total: $490 + 336 = 826$ flowers.

Exercise 22.10 ★★

Every day, a bakery uses $67$ kg of flour. About how much flour is that in a $30$ -day month — estimate with rounded numbers, then compute exactly.

Solution

Solution of Exercise 22.10.

Estimate: $70 \times 30 = 2\,100$ kg. Exactly: $67 \times 30 = 2\,010$ kg.

Exercise 22.11 ★★

Explain, with a rectangle picture like the one of this chapter, why $15 \times 12$ can be computed as $15 \times 10 + 15 \times 2$ . Then invent another splitting of $15 \times 12$ that also works.

Solution

Solution of Exercise 22.11.

A $15 \times 12$ rectangle of unit squares can be cut by a vertical line into a $15 \times 10$ part and a $15 \times 2$ part: counting each part gives $150 + 30 = 180$ . Another cut works just as well, for instance $15 = 10 + 5$ : $10 \times 12 + 5 \times 12 = 120 + 60 = 180$ . Any way of cutting the rectangle leaves the total number of squares unchanged.