---
title: "Multiplying Larger Numbers"
book: "Primary & Middle School Mathematics"
subject: math
language: en
chapter: 22
exercises: 11
source: https://one-course.com/books/math/1/en/chapter/22-multiplying-larger-numbers
---

# Chapter 22 — Multiplying Larger Numbers

Grade 3 multiplied by one digit ([Chapter 16](https://one-course.com/books/math/1/en/chapter/16-multiplication#ch-g3-mult)); 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](https://one-course.com/books/math/1/en/chapter/1-counting-to-20#def-g1-counting-zero), by $100$ two [zeros](https://one-course.com/books/math/1/en/chapter/1-counting-to-20#def-g1-counting-zero). 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](https://one-course.com/books/math/1/en/chapter/10-multiplication-first-steps#def-g2-mult-def), 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).](https://one-course.com/images/onecourse/chapters/math-1/g4-mult/fig-cb4114601663.svg)

*The rectangle picture of $23 \times 12$: four easy [products](https://one-course.com/books/math/1/en/chapter/10-multiplication-first-steps#def-g2-mult-def), $200 + 40 + 30 + 6 = 276$. 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 $457 \times 36$:

1. first line: $457 \times 6 = 2\,742$ ;
2. second line: $457 \times 3$ *tens* $= 1\,371$ tens — write $1\,371$ shifted one place left;
3. add the two lines: $2\,742 + 13\,710 = 16\,452$ ;
4. 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](https://one-course.com/books/math/1/en/chapter/10-multiplication-first-steps#def-g2-mult-def) ).

## 22.3 Multiplication in problems

**Example 22.5.**

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

1. Operation: $145 \times 28$ .
2. Estimate: $150 \times 30 = 4\,500$ , a bit too much on both counts.
3. Columns: $145 \times 8 = 1\,160$ ; $145 \times 2$ tens $= 2\,900$ ; total $4\,060$ .
4. 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 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](#ex-g4-mult-why) ($34 \times 21 = 34 \times 20 + 34$), then check in columns.

**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 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 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](#ex-g4-mult-tricks): $45 \times 11$; $32 \times 99$; $24 \times 45$ (double and halve — twice if you like).

**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 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 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 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](https://one-course.com/books/math/1/en/chapter/10-multiplication-first-steps#def-g2-mult-def), then a [sum](https://one-course.com/books/math/1/en/chapter/2-addition-first-steps#def-g1-addition-def).)

**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 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 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.