What is place value?
Place value refers to the value of each digit within a number.
In the number \({3,147,286}\) the digit \({2}\) has a value of \({200}\) (two hundred).
The digit \({3}\) has a value of \({3,000,000}\) (three million).
Question
What is the value of the digit \({8}\) in \({3,147,286}\)?
Answer
The digit \({8}\) has a value of eight tens, or \({80}\).
Writing and describing whole numbers
When numbers are written down, the digits are often separated in groups of 3 by commas or spaces.
This helps us to describe and interpret the number by saying it out loud.
Any number with a single comma or space will be described as ‘thousand’.
For any number with 2 commas or spaces, the first space will be described as million and the second space as thousand.
Question
Describe the number \({4,235,225}\) in words.
Write twenty-three thousand and fifty six in figures.
Answers
- Four million, two hundred and thirty five thousand, two hundred and twenty five.
- \({23,056}\)
Ordering whole numbers
When you have a series of large numbers that are not in number order, it is sometimes difficult to make sense of them.
Here is a table showing the daily profits of an online music store, written in order of days:
| Day | Mon | Tue | Wed | Thurs | Fri | Sat | Sun |
|---|---|---|---|---|---|---|---|
| Profit | \(\pounds{5,934}\) | \(\pounds{7,656}\) | \(\pounds{7,573}\) | \(\pounds{8,678}\) | \(\pounds{10,834}\) | \(\pounds{14,976}\) | \(\pounds{5,004}\) |
If these numbers were put into a place value table, it would be easier to arrange them in order.
Look at each column in turn.
The figures for Friday and Saturday will be the largest as these have figures in the tens of thousands column.
Looking at the thousands column shows that there is a \({4}\) in the thousand column for Saturday and a \({0}\) in the thousands column for Friday.
Therefore Saturday has the largest number.
Here is the table written in order of amounts, starting with the greatest:
| Day | Sat | Fri | Thurs | Tue | Wed | Mon | Sun |
|---|---|---|---|---|---|---|---|
| Profit | \(\pounds{14,976}\) | \(\pounds{10,834}\) | \(\pounds{8,678}\) | \(\pounds{7,656}\) | \(\pounds{7,573}\) | \(\pounds{5,934}\) | \(\pounds{5,004}\) |
Multiplying by 10, 100, 1,000
Multiplying by 10
When we multiply by \({10}\), every digit moves one place to the left.
Units become tens, tens become hundreds and hundreds become thousands.
Multiply by 10
Image caption, What is 23 × 10?
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Multiplying by 100
When we multiply by \({100}\), every digit moves two places to the left.
Units become hundreds, tens become thousands and hundreds become ten-thousands.
Multiply by 100
Image caption, What is 12 × 100?
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Multiplying by 1,000
When we multiply by \({1,000}\), every digit moves three places to the left.
Units become thousands, tens become ten-thousands, and hundreds become hundred-thousands.
Multiply by 1000
Image caption, What is 7 × 1,000?
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Dividing by 10, 100, and 1,000
Dividing by 10
When we divide by \({10}\), every digit moves one place to the right.
Thousands become hundreds, hundreds become tens and tens become units.
Divide by 10
Image caption, What is 270 ÷ 10?
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Dividing by 100
When we divide by \({100}\), every digit moves two places to the right.
Thousands become tens, hundreds become units, tens become tenths and units become hundredths.
Divide by 100
Image caption, What is 1,300 ÷ 100?
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Dividing by 1,000
When we divide by \({1,000}\), every digit moves three places to the right.
Thousands become units.
Hundreds, tens and units become fractions of a unit.
Divide by 1000
Image caption, What is 4,000 ÷ 1,000?
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Winning the race: Place value
Follow one runner as she uses place value to plan her training and to make sure she can beat her racing rival.
Trainer On your marks!
Jackie Concentrate
Trainer Get Set
Jackie You got this.
Oh Noooooo!
She is sooo annoying.
Every training session since the summer she’s beaten me!
OK, so calm down. Let’s look at this.
Wow! 15:35 seconds, that’s not a bad time for me.
I just need to work out the maths here.
So my time at the start of the season was 16.44 seconds.
So after eight weeks of training I’ve got it down to 15.35 seconds.
So how much have I knocked off?
I can do this in my head using place value.
15.35 seconds is made up of 1 ten, 5 ones, 3 tenths and 5 hundredths.
So I need to work out the difference between 15.35 and 16.44.
If I call 15.35, 15 and 35 hundredths, to get to 16, I need an additional 65 hundredths.
Then from 16 to 16.44 hundredths is another 44 hundredths.
Add 65 and 44 and you get 109 hundredths.
Which is one second, zero tenths and nine hundredths.
So 1.09 seconds. Cool!
But I know who I need to beat. It’d be good if I knew her time.
Trainer Jackie! Come on Jackie!
Jackie No, you go ahead.
I’m just going to sit this one out. My knee’s aching.
So I get a chance to time her and then I can plan my training strategy!
(WHISTLE)
So if it took me eight weeks to knock just over a second
from my time and I’ve got four more weeks to go.
This is going to be a real incentive for my training.
There!
14.98.
OK let’s do the maths.
My time was 15.35, hers is 14.98.
So what’s the difference?
From 14.98 to 15 is two hundredths of a second and then 15 to 15.35 is 35 hundredths.
So 2 + 35 is 37.
So I still need to shave off 37 hundredths of a second from my time to go as fast as her.
Well that seems very doable, yeah! I can easily beat her.
Let’s get some water.
Hang on!
Coach always says that she runs much faster when she’s in front of a big crowd.
I hate big crowds but she could knock off half a second.
I have to think about the crowd.
Well here we normally get about 850 people.
Coach says at the championships there’ll be 100 times as many.
Wow that seems a lot!
I need to check this. I can use place value again.
The places are 10,000’s, 1000’s, 100’s, 10’s, 1’s there’s the decimal point 10th’s and 100th.
So 850 multiplied by 100 you move the digits two places to the left.
85,000! It’s not the Olympics?!
That can’t be right. Start again.
Let’s just move the digits one place to the left.
That looks more like it!
Coach must have meant ten times as many but that’s still a lot of people she’s definitely going to add on half a second.
I’d better get training!
Yeah, yeah I’m coming!
The knee’s better now. No problem.
Test section
Question 1
What is the value of the digit \({4}\) in the number \({54,062}\)?
Answer
The value of the digit \({4}\) in the number \({54,062}\) is \({4,000}\).
Question 2
How do you say the number \({34,023}\)?
Answer
to count how many digits are in the number.
The number includes \({34}\) thousands, no hundreds and \({23}\).
So the correct answer is thirty-four thousand and twenty-three.
Question 3
Look at the cost of these three cars:
Car \({A}\): \(\pounds{18,500}\)
Car \({B}\): \(\pounds{1,890}\)
Car \({C}\): \(\pounds{15,600}\)
Put the cars in order of price from the most expensive to the cheapest.
Answer
'Car \({A}\), Car \({C}\), Car \({B}\)' is the correct order.
Question 4
What is the value of \(37\times100\)?
Answer
When a number is multiplied by \({100}\), every digit moves two places to the left.
So, the correct answer is: \({3,700}\).
Question 5
What is the value of \(170\times1,000\)?
Answer
When a number is multiplied by \({1,000}\), every digit moves three places to the left.
So the correct answer is \({170,000}\)
Question 6
What is the value of \(8,500\div10\)?
Answer
When a number is divided by \({10}\), every digit moves one place to the right.
So the correct answer is: \({850}\)
Question 7
What is the value of \({4}~{million}\div{100}\)?
Answer
\({4}~{million}\) is \({4,000,000}\).
When it is divided by \({100}\), every digit moves two places to the right.
So the correct answer is \({40,000}\)
Question 8
Packs of flour cost \({47}{p}\) each.
A baker buys ten packs.
What is the total cost of the flour?
Answer
\({47}{p}\) multiplied by \({10}\) gives \({470}~pence\), which is \(\pounds{4.70}\).
Question 9
The thickness of \({10}\) pieces of wood placed on top of each other is \({250}~{cm}\).
What is the thickness of each piece?
Answer
\({250}~{cm}\) divided by \({10}\) is \({25}~{cm}\).