Counting washers by singles ~ envelopes ~ bundles ~ and cartons

Washers are small metal discs

Once upon a time a shopkeeper sold a great many small items to lots of customers. Here I have chosen to tell the story ~ and show pictures ~ of just one of the many items sold ~ steel washers.

Four steel washers
A steel washer ~ three more on the table

Many steel washers
A pile of washers about to be counted

A bundle of ten washers
A handful of ten washers ~ a convenient number

A bundle of 34 washers
A collection of counted washers ~
three piles of ten ~ four singles

If any shopkeeper spends too much time counting items for every customer ~ one ~ two ~ three ~ four ~ on ~ and on ~ then a queue will form.

Customers get grumpy if they have to wait in a queue. The shopkeeper decided to pack the washers into envelopes. Each envelope contains ten washers. Pictorially it looks like this ~

Counting ten washers
Counting ten washers

Put washers into an envelope
Put the washers into an envelope

Seal the envelope with washers inside
Seal the envelope

Because of lots of things ~ the speedy service being one of them ~ the shopkeeper's trade flourished. People were buying a great many items at every visit.

If someone wanted forty-three washers the shopkeeper would hand over four envelopes and also three single washers ~ that was much quicker than counting up to forty-three.

Washers sold in envelopes and singly
43 washers ~ four envelopes of ten ~ and three singles

Everyone thought using envelopes was a good idea. Trade increased. The shopkeeper bundled ten envelopes into a package and tied it with string. Ten envelopes ~ each with ten washers ~ that means one hundred washers.

Ten envelopes full of washers
A bundle of ten envelopes ~
How many washers?

Twenty envelopes full of washers
Two bundles ~ How many envelopes? ~
How many washers?

Here are three quick questions to check how things are going for you ~

Five envelopes contain fifty washers
How many envelopes? ~
How many washers?

2600 washers in 26 envelopes
How many envelopes? ~ and washers?

14 envelopes is 140 washers
How many washers?

After some time the shopkeeper had to send someone to go to the stationery shop to buy more envelopes ~ stocks kept running low.

Boxed envelopes
A box of envelopes ~ 1000 in a box ~
The '100 GSM' is a measure of quality ~ not quantity

Someone on the staff pointed out that the number of envelopes being used could be saved. The suggestion was made to put the washers straight into boxes ~ one hundred to a box. In that way there would also be a handy storage facility for the end-user.

Counting 100 washers into a box
Count 100 washers into a box

Closing a box lid over 100 washers
Prepare to close the lid

Replace the lid
Shake the box to let the lid fit down

Three boxes ~ 300 washers ~ ready for collection
300 washers ready for collection ~ and three useful boxes ~ extra ~ free

Here is a short summary of the maths that was going on.

Ten washers fit into one envelope ~
or 1 envelope = 10 washers

Ten envelopes tied together make a bundle ~
or 1 bundle = 10 envelopes ~
or 1 bundle = 100 washers

Change to boxes instead of tied~up~bundles
One box is equivalent to ten envelopes and contains 100 washers.

To return to our story ~ the shopkeeper had obviously had a good mathematical education. He chose to work in tens ~ ten washers ~ ten envelopes ~ maybe next year it will be ten boxes fit into one carton.

As a further update to the story ~ the shopkeeper is now a manufacturer of washers ~ with a huge factory ~ and living happily ever after.

Back to thinking about maths ~ hmmm ~ with ten boxes to one carton ~ how many washers in one carton?

Does a van take ten cartons?

Will ten vanloads fit into one container?

How many containers fit onto a train?

How many trainloads fit into a ship?

Finally ~ how many washers in a shipload?

The answer is ~ quite a lot!

Working in tens is what mathematicians do. Sometimes they are not very imaginative. Instead of parcelling things into envelopes ~ bundles ~ cartons ~ whatever ~ they use a different system.

They use labels ~ ones ~ tens ~ hundreds ~ thousands ~ tens-of-thousands ~ and so on. Some people call the 'ones' units. It means the same as ones or singles.

In shorthand ~ or in maths-code if you wish ~ we could have ~

cartons ~ boxes ~ envelopes ~ singles
C ~ B ~ E ~ S


thousands ~ hundreds ~ tens ~ units
T ~ H ~ T ~ U


thousands ~ hundreds ~ tens ~ ones
T ~ H ~ T ~ O

The important thing to note is that any one item is ten times greater than the one beside it on the right.

Not everyone works in tens when packaging ~ sometimes it is convenient to work in some other quantity [114067].

I have a note on Packing Eggs which develops this topic [114049].

There is a commentary for staff ~ or parents ~ or carers ~ available for subscribers to BusyBusy [114043].

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Last updated 2021/01/30
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