# Multiplying and Dividing by 10, 100 and 1000

> Use zeros to multiply and divide big numbers in your head, and estimate answers quickly.

From Selective School Maths on WhizzNotes, free study notes for NSW students: https://whizznotes.com/selective-school-maths/powers-of-ten/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The flood money
After a big flood, the government gives 4000 dollars to each of 2100 families. A news reporter asks, "How much is that altogether?"

The reporter's calculator is flat, so she does it in her head. $21 \times 4 = 84$. Then she counts the zeros: 2100 has two and 4000 has three, so five zeros in total. Answer: $8\,400\,000$ dollars.

Big numbers with zeros at the end are friendly. You only need a small times table fact and careful counting.

**In short:** Multiplying by 10, 100 or 1000 adds one, two or three zeros to a whole number, and dividing takes them away. For numbers that end in zeros, multiply the small parts first, then add all the zeros back.

## Before you start

- Times tables up to $12 \times 12$.
- **Place value:** in $4300$ the 4 means 4 thousands and the 3 means 3 hundreds.
- $10 = 10^1$, $100 = 10^2$, $1000 = 10^3$. These are called **powers of ten**.

## Multiplying by 10, 100, 1000

### Key idea: Each $\times 10$ moves digits one place left
$43 \times 10 = 430$ (add 1 zero)

$43 \times 100 = 4300$ (add 2 zeros)

$43 \times 1000 = 43\,000$ (add 3 zeros)

Really the digits move one place to the left for each 10, and a zero fills the empty ones place.

## Multiplying by tens, hundreds and thousands

Split the number into a small part and a power of ten.

### Worked example: Peel off the zeros
$16 \times 70 = 16 \times 7 \times 10 = 112 \times 10 = 1120$

$48 \times 300 = 48 \times 3 \times 100 = 144 \times 100 = 14\,400$

### Key idea: Zeros on both numbers
Multiply the non-zero parts, then add **all** the zeros from both numbers.

$600 \times 900$: $6 \times 9 = 54$, and there are 4 zeros. Answer: $540\,000$.

### Watch out: Zeros inside the fact
$450 \times 60$: $45 \times 6 = 270$, and there are 2 zeros. Answer: $27\,000$.

Do not lose the zero that is already in 270. Write the fact first, then add the zeros on the end.

### Worked example: More practice
$700 \times 800$: $7 \times 8 = 56$, 4 zeros, so $560\,000$.

$280 \times 400$: $28 \times 4 = 112$, 3 zeros, so $112\,000$.

$1100 \times 700$: $11 \times 7 = 77$, 4 zeros, so $770\,000$.

## Dividing with zeros

When you divide, you can **cross off the same number of zeros** from the end of both numbers. The answer does not change.

### Worked example: Cancel zeros
$1210 \div 110$: cross off one zero from each to get $121 \div 11 = 11$.

$48\,000 \div 1600$: cross off two zeros to get $480 \div 16 = 30$.

$643\,200 \div 80$: cross off one zero to get $64\,320 \div 8 = 8040$.

### Exam tip: Why does it work?
$1210 \div 110$ asks "how many 110s are in 1210?" That is the same as "how many 11s are in 121?". Making both numbers 10 times smaller keeps the answer the same.

## Estimating

To **estimate**, round each number to one that is easy, then multiply. The sign $\approx$ means "about equal to".

### Worked example: Quick estimates
$702 \times 19 \approx 700 \times 20 = 14\,000$

$698 \times 51 \approx 700 \times 50 = 35\,000$

$198 \times 31 \approx 200 \times 30 = 6000$

Estimates help you check answers. If your exact answer to $702 \times 19$ were $1338$, the estimate of $14\,000$ tells you something went wrong.

### Challenge: Selective level: chains of operations
$600 \times 20 \times 9 \div 900$

Make it friendlier: $600 \div 900$ is messy, but $9 \div 900 = \frac{1}{100}$. So the answer is $600 \times 20 \div 100 = 12\,000 \div 100 = 120$.

"Divide the sum of 12 000 and 800 by 160." The sum is $12\,800$. Cross off a zero: $1280 \div 16 = 80$.

$17\,290 \times 110 = 17\,290 \times 11 \times 10$. Since $17\,290 \times 11 = 172\,900 + 17\,290 = 190\,190$, the answer is $1\,901\,900$.

**Mind map**

```text
Powers of ten
  Multiply by 10, 100, 1000
    Add 1, 2 or 3 zeros
    Digits move left
  Zeros on both
    Small fact first
    Then count all the zeros
  Divide
    Cross off the same zeros
    $1210 \div 110 = 121 \div 11$
  Estimate
    Round to easy numbers
    $702 \times 19 \approx 14\,000$
```

## Quick summary

- $\times 10$ adds one zero, $\times 100$ adds two, $\times 1000$ adds three.
- For numbers ending in zeros, multiply the small parts and then add all the zeros.
- Keep any zero that is part of the small fact, such as $5 \times 4 = 20$.
- To divide, cross off the same number of zeros from both numbers.
- Estimate by rounding to easy numbers. Use $\approx$ for "about".

## Quiz

Try these questions from the chapter quiz.

1. $43 \times 1000 = $
   - A. 4300
   - B. 43 000
   - C. 430 000
   - D. 4 300 000
   - Answer: B. 43 000. Times 1000 adds three zeros.

2. $48 \times 30 = $
   - A. 144
   - B. 1440
   - C. 14 400
   - D. 1200
   - Answer: B. 1440. $48 \times 3 = 144$, then add one zero.

3. $700 \times 800 = $
   - A. 5600
   - B. 56 000
   - C. 560 000
   - D. 5 600 000
   - Answer: C. 560 000. $7 \times 8 = 56$ and there are 4 zeros.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/selective-school-maths/powers-of-ten/#quiz