28 Kasım 2014 Cuma

The Holiday Headscratcher Puzzle

The Holiday Headscratcher Puzzle

The Puzzle:


"What day do you go back to school, Horace?" asked his grandmother one day.

"Well," Horace replied, "Nine days ago, the day before yesterday was three weeks before the second day of term."

If Horace had this conversation on a Sunday, what day of the week did he start school?


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-\./^\./-


Do you have the answer? Check against our solution!

Our Solution:

The reasoning runs as follows:

Today is Sunday.

So seven days ago was Sunday too. That means that NINE days ago was Friday.

On Friday, the day before yesterday was Wednesday.

Three weeks later is Wednesday again.

That is the second day of term.

So the term began on Tuesday. Wasn't that easy?


Puzzle Author: Stephen Froggatt

The End Of Year Party Puzzle

The End Of Year Party Puzzle

The Puzzle:


Party Time!

Several people of different ages brought things for the party: Charlie, who's 12, brought the POTATO CRISPS. Wayne brought the FIZZY LEMONADE (he's 13). Helen (11) brought the PAPER PLATES, and her brother Peter (9) brought the PAPER CUPS. Sheila is the same age as Charlie: she brought the PARTY POPPERS. Young Horace brought the PEANUTS: how old is he? Have a great party!


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-\./^\./-


Do you have the answer? Check against our solution!

r Solution:

7

(Count the letters!)


Puzzle Author: Stephen Froggatt

The Dual Cabbage Way Puzzle

The Dual Cabbage Way Puzzle

The Puzzle:


Using three straight lines, divide the cabbage patch up into six sections with two cabbages in each section.



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-\./^\./-


Do you have the answer? Check against our solution!

Our Solution:

Here is one way:


Puzzle Author: Stephen Froggatt

Sticker-bility Puzzle

Sticker-bility Puzzle

The Puzzle:


The addition sum below is a puzzle I've been trying to solve. The idea is that each type of sticker stands for a different number, but that this number is the same wherever that sticker occurs.



So far I've got it to the picture shown in the second diagram. Can you finish it off for me?


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-\./^\./-


Do you have the answer? Check against our solution!

Our Solution:

Look again at the diagram:



The only numbers which haven't been used so far are the 0, 6, 7, and 9. The first one to work out is the brick wall, which has to be the 7, since none of the others fit, which means the shaded square is the 9, and the completed puzzle is:

8357 + 792 = 9149.

Sum-Things Missing Puzzle

Sum-Things Missing Puzzle

The Puzzle:


An old Mathematics book contained this addition sum which had been marked correct by the teacher:



The three squares in the diagram are where the paper was so bad I couldn't read them.

What were the three missing numbers?


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-\./^\./-


Do you have the answer? Check against our solution!


Our Solution:

The most obvious place to start was the 0 on the bottom (don't forget to carry the 1!).

Working towards the left, the next one has to be 9 to make the total 16 (+1 = 17) so we carry another 1.

That means the top left digit must be empty (or a zero):


Santa Has A Bad Code Puzzle

Santa Has A Bad Code Puzzle

The Puzzle:


Father Christmas caught a cold. Fortunately, he did manage to get a message through via one of his elves. Unfortunately, the naughty elf has hidden the message in a mysterious code. Can you crack it?


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-\./^\./-


Do you have the answer? Check against our solution!

Our Solution:

Start at the middle and spiral outwards.

Merry Christmas everybody and a Happy New Year too.

Placing Sheets Puzzle

Placing Sheets Puzzle

The Puzzle:


Eight squares of paper, all exactly the same size, have been placed on top of each other so that they overlap as shown:



In what order were the sheets placed?


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-\./^\./-


Do you have the answer? Check against our solution!



Our Solution:

The paper squares were placed in this order: CEBFHGDA

Order! Order! Puzzle

Order! Order! Puzzle

The Puzzle:


The story is told of the enterprising young farmer who crossed a sheep with a frog. Before long he had a sign outside the farm: "Woolly Jumpers For Sale". I digress. Let's have some order round here!

Write out the numbers from 1 to 20 in words:

One, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen, twenty.

Now put them in ALPHABETICAL order. Which number stays where it is?


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-\./^\./-


Do you have the answer? Check against our solution!

Our Solution:

The only number which stays in the same place after they have all been put in alphabetical order is the number .... FIVE !

The new order is as follows:

EIGHT, EIGHTEEN, ELEVEN, FIFTEEN, FIVE, FOUR, FOURTEEN, NINE, NINETEEN, ONE, SEVEN, SEVENTEEN, SIX, SIXTEEN, TEN, THIRTEEN, THREE, TWELVE, TWENTY, TWO.

Mirror, Mirror Puzzle

Mirror, Mirror Puzzle

The Puzzle:


If you continue shading the squares so that the two dotted lines become lines of symmetry (mirror lines) of the completed diagram, how many squares will be left unshaded?




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-\./^\./-


Do you have the answer? Check against our solution!

Our Solution:

If you continue shading the squares so that the two lines become mirror lines, we get this:



(Note: I have shaded the new squares differently just so that you can see where they go.)

The number of squares left is then 9.

It Must Be Matchic Puzzle

It Must Be Matchic Puzzle

The Puzzle:


What's the difference between a match, and the gun used at the beginning of a race?
One starts the fire, the other fires the start!

If you had six matches, it would be easy (wouldn't it?) to make two identical triangles.

But your task is to take six matches and make FOUR identical triangles.

Do you think you can do it? The matches don't have to be the same length. If you can do it, then you can call yourself a Match-ician. If you can't do it, you'll have to call yourself a lump of Match Potato!


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-\./^\./-


Do you have the answer? Check against our solution!

Our Solution:



Or you could go 3D and have them as edges of a tetrahedron:



(A tetrahedron has 6 edges and 4 triangular faces)

Hilda The Builder Puzzle

Hilda The Builder Puzzle

The Puzzle:


Hilda was playing with her building bricks when she made a tower like the one below:



How many bricks did Hilda use altogether?


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-\./^\./-


Do you have the answer? Check against our solution!

Our Solution:

While I make no claims to be an expert on perspective drawing, the diagram shows the building made of 22 blocks.



In fact it is possible to add up to 3 other blocks without affecting the three views.

Fund Raising Fun Puzzle

Fund Raising Fun Puzzle

The Puzzle:


Hungry Horace recently went on a Sponsored Walk to raise funds for new equipment at the local Hospital. His sponsor form looked like this:

Sponsor's NameAmount
Per km
Total
Fat Freda10c
Dim Jim15c
Tall Tanya$3
Silly Cilla5c
C.U.Jimmy5c
Albert Bodge$2,50
Roland Heap20c
Nice Nigel$4,50
Merlin Shriek20c
Carol Singer25c



Altogether Horace collected $22,00 from his sponsors. How far did he walk?


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-\./^\./-


Do you have the answer? Check against our solution!

Our Solution:

Altogether Horace collected $22,00 from his sponsors. $10,00 of that came from Tanya, Albert and Nigel, so that means the others paid him $12,00. Adding up the Amount Per km column we see that Horace was sponsored $1,00 per km altogether. That means he walked 12 km to raise his $12,00, giving him the required $22,00 in total.


Puzzle Author: Stephen Froggatt

Eating Pies Puzzle

Eating Pies Puzzle

The Puzzle:


Father Christmas has to visit a lot of homes on Christmas Eve when he is out delivering presents. He usually gets a glass of sherry and a mince pie at each house too. Now, Father Christmas can manage the sherry, but finds eating all those mince pies quite difficult. So Hungry Horace helps him out.

Last year Father Christmas finished his rounds with lots of mince pies to spare. He gave 25 million (25 000 000) to Hungry Horace. It takes Horace just 10 seconds to eat a mince pie. How long did it take him to eat all 25 million? An hour? A day? A week?


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-\./^\./-


Do you have the answer? Check against our solution!

Our Solution:

I made it about 7 years, 338 days, 12 hours, 26 minutes and 40 seconds!


Puzzle Author: Stephen Froggatt

Easter Eggs - Eggsactly Puzzle

Easter Eggs - Eggsactly Puzzle

The Puzzle:


This  is Hungry Horace's favourite type of Easter Egg.

He went to the shop to buy one but found that the Easter Eggs had been packed up in boxes of three like this:



"How much does this one egg cost?" Horace asked the shopkeeper.

"That's easy", he replied "It's just ....."

Then the telephone rang and he had to answer it.

HOW MUCH DOES HORACE'S FAVOURITE EGG COST?


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-\./^\./-


Do you have the answer? Check against our solution!

Our Solution:

$2,50


Do Not Pay Any Less, Mrs Mess Puzzle

Do Not Pay Any Less, Mrs Mess Puzzle

The Puzzle:


Mrs Mess was buying a set of garden furniture. The bill was seventy dollars.

She gave the attendant what she thought were two $50 notes, (actually two $100 notes).

The attendant was sleepy and didn't notice either, so he gave Mrs Mess what he thought were three $10 notes (actually three $50 notes).

Who ended up better off than they should?


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-\./^\./-


Do you have the answer? Check against our solution!

Our Solution:

Mrs Mess comes away better off than she should, by $20 in fact. Despite the warning in the title, Mrs Mess did pay less than she should have done!

Cost of furniture: $70

Mrs Mess paid: $100 x 2 = $200

Change received: $50 x 3 = $150

Net payment: $200 - $150 = $50

Profit for Mrs Mess: $70 - $50 = $20

Count the Shapes Puzzle

Count the Shapes Puzzle

The Puzzle:


How many triangles and quadrilaterals are there in this diagram?



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-\./^\./-


Do you have the answer? Check against our solution!

Our Solution:

64 Triangles
36 Quadrilaterals
Total: 100 shapes.

Christmas Pudding Puzzle

Christmas Pudding Puzzle

The Puzzle:


Father Christmas spends 364 days of the year as a taster of Christmas puddings (which is why he is so round and jolly). Recently he came across a Magic Pudding.

If you eat some of this pudding, the next thing you say comes out in a magic code.

Father Christmas tried it, liked it, and said:

"IP! IP! IP! XIBU B MPWFMZ QVEEJOH! NFSSZ DISJTUNBT BOE B IBQQZ OFX ZFBS UP ZPV BMM!"

What did he say?


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-\./^\./-


Do you have the answer? Check against our solution!

Our Solution:

Go one letter down in the alphabet: "HO! HO! HO! ..."

Now see if you can figure out the rest!

Catastrophe Puzzle

Catastrophe Puzzle

The Puzzle:


Draw a circle with radius 5 cm.

Draw a second circle with radius 3 cm just touching the most North-East part of the first circle.

Draw two small triangles, 4 cm apart, resting on top of the second circle.

Inside the second circle, arranged so that they are symmetrical about a vertical line through the centre, draw these four things:
a small triangle at the centre
two small circles 3 cm apart, above the triangle
the bottom half of a circle, below the triangle.

WHAT'S MISSING?


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-\./^\./-


Do you have the answer? Check against our solution!

Our Solution:

The instructions as given would have produced something like this:



What's missing? Feet, tail and whiskers at least!

Calendar Confusion Puzzle

Calendar Confusion Puzzle

The Puzzle:


If I said that in three days' time it would be a Thursday, I am sure that most of you would have no difficulty telling me that today was a Monday.

Try this one then. Yesterday was two days before Monday. What day is it today? Yes, you're right again. It's Sunday. Do you get the idea?

Now let's tackle a similar question from The National Mathematics Contest (1991) Paper:

Three days ago, yesterday was the day before Sunday. What day will it be tomorrow
?


Our Solution:

When I tested this, the order of popularity of answers handed in were: Sunday (9), Thursday (4), Monday (3), Friday (2), Wednesday (1), Saturday (1). THE RIGHT ANSWER WAS THURSDAY. Why?

Three days ago, yesterday was the day before Sunday, so three days ago was itself Sunday.
That means today is Wednesday, so tomorrow is Thursday

Blockslide Puzzle

Blockslide Puzzle

The Puzzle:


The diagram below shows a cross-shaped box containing three numbered blocks.



The puzzle is to slide the blocks around the box until the numbers read 1,2,3 as you go down.

How do you do it?

Our Solution:

.
-\./^\./-


Do you have the answer? Check against our solution!

Birthday Smarties Puzzle

Birthday Smarties Puzzle

The Puzzle:


The Birthday Cake has all gone but there are twelve piles of Smarties left. Each pile is held together by icing so can't be split up. Most of the guests have gone, but Hungry Horace and his two friends want to share out the Smarties equally.



Can you share out the piles so that everybody gets 25 Smarties each?


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-\./^\./-


Do you have the answer? Check against our solution!

Our Solution:

The diagram below shows one way to group the Smarties so that Horace and his friends get 25 each.



Another way would be: 12+9+3+1, 12+13 and 4+1+3+3+8+6.

And Mint Sauce Puzzle

And Mint Sauce Puzzle

The Puzzle:


Six wolves catch six lambs in six minutes.

How many wolves will be needed to catch sixty lambs in sixty minutes?

No, the answer is not sixty. Try again!


Do you have the answer? Check against our solution!

Our Solution:

I should also have said that the answer was not 10 either! That is a popular answer, but I'm afraid it is not correct. The right answer was SIX WOLVES.

Let's see why:

6 wolves catch 6 lambs in 6 minutes. Multiply by 10:
The same 6 wolves catch 60 lambs in 60 minutes.

(Give them 10 times as long and they'll catch 10 times as many lambs.)

Another way of seeing the answer is to note that each wolf catches 1 lamb in six minutes. In 60 minutes, therefore, each wolf will catch 10 lambs (ten times as many). With 6 wolves, six times as many lambs will be caught.

Ancestrally Speaking Puzzle

Ancestrally Speaking Puzzle

The Puzzle:


Hungry Horace was looking through the family photograph album, which has a photo of each of his parent, each of his grandparents, all the way up to each of his great-great-great-grandparents.

How many photos is that?


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-\./^\./-


Do you have the answer? Check against our solution!

Our Solution:

2 parents
4 grandparents
8 great-grandparents
16 great-great-grandparents
32 great-great-great-grandparents

2 + 4 + 8 + 16 + 32 = 62 photos in all.

And Mint Sauce Puzzle

And Mint Sauce Puzzle

The Puzzle:

Six wolves catch six lambs in six minutes.

How many wolves will be needed to catch sixty lambs in sixty minutes?

No, the answer is not sixty. Try again!



Do you have the answer? Check against our solution!

Our Solution:

I should also have said that the answer was not 10 either! That is a popular answer, but I'm afraid it is not correct. The right answer was SIX WOLVES.

Let's see why:

6 wolves catch 6 lambs in 6 minutes. Multiply by 10:
The same 6 wolves catch 60 lambs in 60 minutes.

(Give them 10 times as long and they'll catch 10 times as many lambs.)

Another way of seeing the answer is to note that each wolf catches 1 lamb in six minutes. In 60 minutes, therefore, each wolf will catch 10 lambs (ten times as many). With 6 wolves, six times as many lambs will be caught.

Ancestrally Speaking Puzzle

Ancestrally Speaking Puzzle

The Puzzle:


Hungry Horace was looking through the family photograph album, which has a photo of each of his parent, each of his grandparents, all the way up to each of his great-great-great-grandparents.

How many photos is that?


.



Do you have the answer? Check against our solution!

Our Solution:

2 parents
4 grandparents
8 great-grandparents
16 great-great-grandparents
32 great-great-great-grandparents

2 + 4 + 8 + 16 + 32 = 62 photos in all.

Alphabet Spaghetti Puzzle

Alphabet Spaghetti Puzzle

The Puzzle:


Spaghetti is famous for the way it all gets tangled up on the plate. Those of you who think they know their alphabet are bound to get all tangled up with this puzzle too, unless you read it and think about it very carefully!

What letter of the alphabet is the one which comes eight letters before the letter which comes five letters after the fourth appearance of the first letter to occur four times in this sentence?


Our Solution:

"r"

You get this by working backwards through the puzzle: the first letter to occur four times is t: WhaT leTTer of T....

Five letters after the fourth "t" is the letter "p", and eight letters before that is "r".

Alphabet Numbers Puzzle

Alphabet Numbers Puzzle

The Puzzle:


Given only one of each letter in the alphabet, what are
the smallest and largest numbers that you could write down?


Our Solution:

Using only one of each letter in the alphabet, you can spell:

ZERO or NOUGHT
MINUS FORTY (allowing negative numbers)
FIVE THOUSAND

These are the smallest and largest possible numbers.

A Brave Puzzle

A Brave Puzzle

The Puzzle:


Only for the brave, this one!

This square has eleven letters missing, which you have to replace:



Every row, column AND the main diagonals contain all the letters in the word "BRAVE".

That reminds me, I must see the Postman about all those missing letters.








Our Solution:

12 Days Of Christmas Puzzle

12 Days Of Christmas Puzzle

The Puzzle:


According to the traditional song, on the first day of Christmas (25th December), my true love sent to me:

. A partridge in a pear tree

On the second day of Christmas (26th December), my true love sent to me THREE presents:

. Two turtle doves
. A partridge in a pear tree

On the third day of Christmas (27th December and so on) my true love sent to me SIX presents:

. Three French hens
. Two turtle doves
. A partridge in a pear tree

This carries on until the the twelfth day of Christmas, when my true love sends me:

Twelve drummers drumming
Eleven pipers piping
Ten lords a-leaping
Nine ladies dancing
Eight maids a-milking
Seven swans a-swimming
Six geese a-laying
Five gold rings
Four calling birds
Three French hens
Two turtle doves
A partridge in a pear tree

After the twelve days of Christmas are over, how many presents has my true love sent me altogether
?


Our Solution:

1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 + 45 + 55 + 66 + 78 = 364 presents

Which is really interesting when you think there are 365 days in a typical year!

Numbers Skip Count

Example: You Skip Count by 2 like this:

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ...

Skip Counting by 10s

Skip Counting by 10s is like normal counting, except there is an extra "0":
10, 20, 30, 40, 50, 60, 70, 80, 90, 100, ...

Skip Counting by 5s

Skip Counting by 5s has a nice pattern:
5, 10, 15, 20, 25, 30, 35, 40, 45, 50, ...
That pattern should make it easy for you!
a

Skip Counting by 3s and 4s

Skip Counting by 3s is:
3, 6, 9, 12, 15, 18, 21, 24, 27, 30, ...
Skip Counting by 4s is:
4, 8, 12, 16, 20, 24, 28, 32, 36, 40, ...

What is World Maths Day

World Maths Day (World Math Day in American English) is an online international mathematics competition, powered by educational resource provider 3P Learning (the same organisation behind the school resources Mathletics, Spellodrome and IntoScience). Smaller elements of the wider Mathletics program effectively power the World Maths Day event.

The first World Maths Day was held on March 14 2007 (Pi Day), and has been held on the 1st Wednesday in March in subsequent years. Despite these origins, the phrases "World Maths Day" and "World Math Day" are trademarks and not to be confused with other competitions such as the International Mathematical Olympiad or days such as Pi Day. In 2010, World Maths Day created a Guinness World Record for the Largest Online Maths Competition.


In 2011, the team behind the competition added a second event World Spelling Day - and officially rebranded as the World Education Games. In 2012, a third event was added - World Science Day. The World Maths Day and World Education Games games are now sponsored by Samsung, and supported by UNICEF as global charity partner.

Number of Estimating and rounding

Estimating and rounding

Estimating is an important part of mathematics and a very handy tool for everyday life. Get in the habit of estimating amounts of money, lengths of time, distances, and many other physical quantities.
Rounding off is a kind of estimating.
To round off decimals:
  1. Find the place value you want (the "rounding digit") and look at the digit just to the right of it.
  2. If that digit is less than 5, do not change the rounding digit but drop all digits to the right of it.
  3. If that digit is greater than or equal to five, add one to the rounding digit and drop all digits to the right of it.
To round off whole numbers:
  1. Find the place value you want (the "rounding digit") and look to the digit just to the right of it.
  2. If that digit is less than 5, do not change the "rounding digit" but change all digits to the right of the "rounding digit" to zero.
  3. If that digit is greater than or equal to 5, add one to the rounding digit and change all digits to the right of the rounding digit to zero.
Estimating, or being able to guess and come close to a correct answer, is an important part of mathematics and a very handy tool for everyday life. You should get in the habit of estimating amounts of money, lengths of time, distances, and many other physical quantities. Rounding is a kind of estimating.
To round a number you must first find the rounding digit, or the digit occupying the place value you're rounding to. Then look at the digit to the right of the rounding digit. If it is less than 5, then leave the rounding digit unchanged. If it is more than five, add one to the rounding digit. If it is five, the rule is to always round up (add one to the rounding digit). This rule was created to "break the tie" when you are rounding a number that is exactly between two other numbers. These kinds of rules are called "conventions", and are important so we all get the same answer when doing the same problems.
If you're dealing with a decimal number, drop all of the digits following the rounding digit.
If you're dealing with a whole number, all the digits to the right of the rounding digit become zero.
This sounds a lot more complicated than it really is!
It's easiest to learn rounding by studying examples.

To round the number 16,745.2583 to the nearest thousandth

First find the rounding digit. This is the "8". You are trying to get rid of the all the digits to the right of the 8, but you want the result to be as accurate as possible.
Now look one digit to the right, at the digit in the ten-thousandths place which is "3". See that 3 is less than 5, so leave the number "8" as is, and drop the digits to the right of 8. This gives 16,745.258.

To round 14,769.3352 to the nearest hundredFind the rounding digit, "7". Look at the digit one place to right, "6". Six is more than 5, so this number needs to be rounded up. Add one the rounding digit and change all the rest of the digits to the right of it to zero. You can remove the decimal part of the number too. The result is 14,800.

To round 365 to the nearest ten
Find the rounding digit, "6". Look at the digit to the right of the six, "5". Since 365 is exactly halfway between 360 and 370, the two nearest multiples of ten, we need the rule to decide which way to round. The rule says you round up, so the answer is 370.

Decimal numbers

The zero and the counting numbers (1,2,3,...) make up the set of whole numbers. But not every number is a whole number. Our decimal system lets us write numbers of all types and sizes, using a clever symbol called the decimal point.
As you move right from the decimal point, each place value is divided by 10.

Our decimal system of numbers lets us write numbers as large or as small as we want, using a secret weapon called the decimal point. In our number system, digits can be placed to the left and right of a decimal point, to indicate numbers greater than one or less than one. The decimal point helps us to keep track of where the "ones" place is. It's placed just to the right of the ones place. As we move right from the decimal point, each number place is divided by 10.
We can read the decimal number 127.578 as "one hundred twenty seven and five hundred seventy-eight thousandths". But in daily life, we'd usually read it as "one hundred twenty seven point five seven eight."
Here is another way we could write this number:
Notice that the part to the right of the decimal point, five hundred seventy-eight thousandths, can be written as a fraction: 578 over 1000. However, you will hardly ever see a decimal number written like this.
Why do you think this is? You can see that our decimal code is a very handy and quick way to write a number of any size!
Examples

Here's how to write these numbers in decimal form:
Three hundred twenty-one and seven tenths 321.7
(6 x 10) + (3 x 1) + (1 x 1/10) + (5 x 1/100)63.15
Five hundred forty-eight thousandths0.548
Five hundred and forty-eight thousandths 500.048
Hint #1: Remember to read the decimal point as "and" -- notice in the last two problems what a difference that makes!
Hint #2: When writing a decimal number that is less than 1, a zero is normally used in the ones place:
0.526 not .526

Numbers place value

place value

In our decimal number system, the value of a digit depends on its place, or position, in the number. Each place has a value of 10 times the place to its right.
A number in standard form is separated into groups of three digits using commas. Each of these groups is called a period.


The idea of place value is at the heart of our number system. First, however, a symbol for nothing--our zero--had to be invented. Zero "holds the place" for a particular value, when no other digit goes in that position. For example, the number "100" in words means one hundred, no tens, and no ones. Without a symbol for nothing, our decimal number system wouldn't work.

Beginning with the ones place at the right, each place value is multiplied by increasing powers of 10. For example, the value of the first place on the right is "one", the value of the place to the left of it is "ten," which is 10 times 1. The place to the left of the tens place is hundreds, which is 10 times 10, and so forth.
For easier readability, commas are used to separate each group of three digits, which is called a period. When a number is written in this form, it is said to be in "standard form."
Examples
Numbers can be represented in many ways, but standard form is usually the easiest and shortest way. Here are some numbers expressed in different forms, with their standard form shown alongside. Which form do you think is the best?
Example 1
one billion, sixty million, five hundred twenty thousand
1,060,520,000
Example 2
four hundred sixteen thousand, seven hundred thirty-one
416,731
Example 3
6,000,000 + 70,000 + 20 + 1
6,070,021
Try It!
In each of these numbers, what value does the digit 5 have?
In 17,526,010 the 5 represents " five hundred thousands"
In 2,110,735,000 the 5 represents "five thousands"