# Place Value, Rounding and Estimation

> Read and write numbers up to millions, find the value of a digit, write numbers in expanded form, round them and estimate answers.

From Selective School Maths on WhizzNotes, free study notes for NSW students: https://whizznotes.com/selective-school-maths/place-value-and-rounding/. Updated 4 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The crowd at the stadium
The news says 48 362 people came to the football final. Mia's brother says "about 48 000". Her dad says "nearly 50 000". Mia asks, "Who is right?"

"Both of you," says Dad. "Your brother rounded to the nearest thousand. I rounded to the nearest ten thousand. The 4 in 48 362 is worth 40 000, and the 8 is worth 8000. Once you know what each digit is worth, rounding is easy."

**In short:** Each digit's value depends on its place: in 48 362 the 8 is worth 8000. To round, look at the digit just to the right of the place you are rounding to: 5 or more rounds up, 4 or less rounds down. Estimate by rounding first, then working out the easy sum.

## Before you start

- Counting in ones, tens, hundreds and thousands.
- Multiplying by 10, 100 and 1000 (see Multiplying and Dividing by 10, 100 and 1000).

## Places and values

Big numbers are written in groups of three digits, with a space between groups: 1 278 356.

| Millions | Hundred thousands | Ten thousands | Thousands | Hundreds | Tens | Ones |
|---|---|---|---|---|---|---|
| 1 | 2 | 7 | 8 | 3 | 5 | 6 |

### Key idea: Place and value
The **place** of a digit is its column. The **value** is what it is worth.

In 1 278 356, the 7 is in the ten thousands place, and its value is 70 000.

One million is a 1 with six zeros: 1 000 000. It is a thousand thousands, because $1000 \times 1000 = 1\,000\,000$.

### Worked example: Reading and writing
905 040 is "nine hundred and five thousand and forty".

"Two million, thirty thousand and six" is 2 030 006. Put a 0 in every empty place.

### Watch out: Missing zeros
"Six million and twenty five thousand" is 6 025 000, not 625 000. Count the places: a million has seven digits.

## Expanded form

Expanded form breaks a number into the value of each digit.

$$352\,407 = 300\,000 + 50\,000 + 2000 + 400 + 7$$

You can also use multiplication, and powers of ten:

$$352\,407 = (3 \times 100\,000) + (5 \times 10\,000) + (2 \times 1000) + (4 \times 100) + 7$$

$$4038 = (4 \times 10^3) + (3 \times 10^1) + (8 \times 1)$$

Here $10^3$ means $10 \times 10 \times 10 = 1000$. The small number says how many tens are multiplied.

## Comparing and ordering

To compare two numbers, first count the digits. More digits means bigger. If they have the same number of digits, compare from the left until two digits differ.

### Worked example: Which is bigger?
342 468 and 324 648 both have six digits. The first digits are both 3. The second digits are 4 and 2, so 342 468 is bigger.

## Rounding

### Key idea: The rounding rule
1. Find the place you are rounding to.
2. Look at the **next digit to the right**.
3. If it is 5 or more, add 1 to the rounding place. If it is 4 or less, leave it.
4. Change every digit after the rounding place to 0.

### Worked example: Rounding 48 362
- Nearest ten: the ones digit is 2, so 48 360.
- Nearest hundred: the tens digit is 6, so 48 400.
- Nearest thousand: the hundreds digit is 3, so 48 000.
- Nearest ten thousand: the thousands digit is 8, so 50 000.

### Watch out: Rounding up past a 9
Round 6970 to the nearest hundred. The tens digit is 7, so the 9 hundreds go up by 1 to make 10 hundreds. Ten hundreds is one more thousand, so the answer is 7000. It is not "6 and 10 hundreds" or 61 000.

## Estimating

To estimate, round each number to its first digit, then work it out in your head. An estimate tells you if a real answer is sensible.

### Worked example: Quick estimates
$4865 + 3120$ is about $5000 + 3000 = 8000$.

$612 \times 48$ is about $600 \times 50 = 30\,000$. (The real answer is 29 376.)

$7820 \div 39$ is about $8000 \div 40 = 200$.

### Challenge: Selective level: digits and places
In 1 461 203, two digits are 1. Their values are 1 000 000 and 100. The difference in their values is $999\,900$.

The biggest six digit number using each of 0, 2, 5, 7, 8 and 9 once is 987 520. The smallest is 205 789, because a number cannot start with 0.

A number rounds to 3500 when rounded to the nearest hundred. The smallest whole number it could be is 3450, and the biggest is 3549.

**Mind map**

```text
Place value
  Places
    Ones, tens, hundreds
    Thousands to millions
    Value = digit x its place
  Expanded form
    Add the values
    Powers of ten
  Rounding
    Look one digit right
    5 or more: up
    4 or less: stay
  Estimating
    Round first
    Check real answers
```

## Quick summary

- A digit's value depends on its place: in 1 278 356, the 7 is worth 70 000.
- One million is 1 000 000, a thousand thousands.
- Expanded form adds the value of each digit: $3058 = 3000 + 50 + 8$.
- To round, look at the next digit to the right: 5 or more rounds up, 4 or less stays.
- Estimate by rounding each number first, then check that real answers are close.

## Quiz

Try these questions from the chapter quiz.

1. In 1 278 356, what is the value of the 7?
   - A. 7
   - B. 7000
   - C. 70 000
   - D. 700 000
   - Answer: C. 70 000. The 7 is in the ten thousands place, so it is worth 70 000.

2. Round 4865 to the nearest hundred.
   - A. 4800
   - B. 4900
   - C. 5000
   - D. 4870
   - Answer: B. 4900. The tens digit is 6, which is 5 or more, so 48 hundreds becomes 49 hundreds.

3. Which number is "two million, thirty thousand and six"?
   - A. 230 006
   - B. 2 300 006
   - C. 2 030 006
   - D. 2 003 006
   - Answer: C. 2 030 006. Two million is 2 000 000, then add 30 000 and 6. Put zeros in the empty places.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/selective-school-maths/place-value-and-rounding/#quiz