# Yellow Reaper dragon

Tanya had just gone to sleep. Suddenly she woke up scared, and shouted grandpa. Grandpa came and asked her what had happened.

Tanya:  I saw a scary dream. In my dream I saw a Yellow Reaper dragon.  Whatever she saw, she broke it up into 10 pieces.  Then the dragon saw a mountain.  It was very big. She broke it into 10 small mountains which were still too big for its mouth. So she broke each small mountain into 10 big stones so that she had 100 stones.  She kept breaking each stone into 10 pieces. Each time she did that, the stones decreased in size to one tenth but became 10 times more in number.  One of the stones was about to hit me.  I woke up.  I am still scared.

Grandpa: Tanya, you have to stop watching dragon shows at night.

Tanya:  I am scared.

# Grandpa’s story of tens

Grandpa:  I will tell you a real story with the tens.  I remember it well because one of the math teachers in Britain told us this story in school.  Until 1971, the UK currency consisted of pounds, and the coins shilling and pence (penny).  There were 12 pennies in a shilling and 20 shillings in a pound. In 1971, the UK currency was changed.  Great Britain Pound had the same value as the pound but it contained 100 new pennies.  There was a lot of confusion for a while.  Now think of this money and go to sleep.  We will talk more about it tomorrow.

# New class of fractions

The next morning in school.
Teacher:  Today we are going to talk about a new a class of fractions.  These are fractions where the denominator will be 10, 100, 1000 and so on.  No other numbers in the denominator- only zeros and ones. These are called decimal fractions. In this system 1/10 is written as 0.1, 1/100 as 0.01 and 1/1000 as 0.001.

Tanya: Can all the other fractions be converted to decimals?

Teacher:  Most can be.  You can use long division but a simple way is to take the denominator and find a multiple that ends in a number which is one followed by zeros.  For example for 1/2, we know 5 x 2 to be 10.  Then 1/2  = 5/10 or 0.5.

Tinku:  I get it. The same way 1/4 = 25/100 or 0.25, 1/5 = 2/10 or 0.2 but I have trouble with 1/3.

# Recurring numbers in decimals

Teacher:  You can’t find an exact answer but can get an approximate one.  You could say that 10/3 = 3 +1/3 and 100/3 = 33 +1/3.  For every zero you add to the numerator, the placement value will change after the decimal will change by the same number.  So 1/3 will become 0.33.  Remember this is an approximate answer.  It will be more exact to say that 1/3 =0.3333……= 0.͞3.  The bar on the top shows that this digit will repeat for ever.

Tanya: Now I want to try 1/7, 100/7 is about 14. So 1/7 is about 1.4. You could also say that 10000/7 equals 1428, and then 1/7 will be about 1͞.4͞2͞8.

Tanya starts talking to another girl in the class.  The teacher noticed it.

Teacher:  What is so interesting that you are talking in my class?  If it is so interesting, stand up and tell everyone.

Tanya: Sir, my grandpa told me his story because I woke up with a scary dream.

Tanya told the whole class about her dream about the Yellow Reaper dragon and the UK currency change.

Teacher: Tanya, you have given us a very good description of the decimal system except that it is not a dragon. Did your grandpa show you how to convert the shillings and old pennies into the new currency?

Tanya:  No, he said he would do this today.

# Converting old British currency into new in 1971

Teacher:  May be we can do this here in class.  There were 20 shillings in the pound.

Tinku: Does that mean the shilling was 1/20th of a pound?

Teacher:  Yes, you know that 5 x 20 = 100.  So we could also say that it was 5/100th of a pound or 0.05 pounds or 5 new pence.

David:  That was easy.  What about the old pence, which was 1/12 of a shilling?

Teacher:  We just figured out that the shilling was worth 5 new pence.  So 12 old pence gave you 5 new pence or 24 old pence were worth 10 new pence.  Therefore you could say that 2.4 old pence equaled one new pence.

Tanya:  I figure that the old pence must be worth 5/12th of the new pence. I cannot get the exact conversion but with long division I get the old pence to be worth 0.41͞6 new pence. Is this correct?

Teacher:  Actually, you should say, it was worth 0.41͞6 old pence.

# Other examples of decimal system

Peter: Another example for the use of the decimal system would be for distance. In the past the distance was measured in miles and now we use kilometers. You can convert one from the other. One mile = 1.6 kilometers  1 kilometer is 1000 meters. Also, we can divide meter into smaller units: 0.01 meter = 1 centimeter and 0.001 meter = 1 millimeter.  We have these marks on the scales which you use in our geometry class.

Tanya:  What about the weights? The old system was pounds and now we use grams, kilograms etc.  It is again in the metric system. We also measure milk in liters and deciliters.

Teacher:  Great.  Tell me class.  What is easier to understand?  One taxi driver tells you that you travelled 30 km and at the rate of 20 new pence per km you need to pay 6 pounds.  The other tells you, “Sir the rate is 6 shillings and 4 pence (old) per mile and you travelled 18 miles and 1130 yards. So the bill is 6 pounds.”

# Challenge

Yards and meters: Approximately 8 kilometer equal 5 miles. A kilometer has 1000 meters and a mile has 1760 yards. Tell Peter how many kilometers in one mile?  Also Kate wants to know how many yards in 1 meter so that she can buy the right amount of fabric.

Solution: 5 miles = 8 km  or  1 mile = 8/5 = 16/10 = 1.6 km

1 mile = 1.6 km = 1760 yards or 1 km = 1760/1.6 yards or 1 meter = 1760/1600 yards = 1.1 yards

Note: 1 yard = 3 feet = 36 inches.  1 meter = 1.1 yards = 39.6 inches.

 0+1/2= 1/2 3/6+2/6= 5/6 10/12 +3/12 =13/12 65/60 + 12/60 = 77/60 77/60 + 10/60 = 87/60

### Supplementary

Tanya just came back from a running camp and is talking to her friend Tinku.

Tinku:  Welcome back  Tanya.  Tell me something interesting about the camp.

Tanya:  You will love the running program I was in.  It was a 7 day thing.  On day 1, I ran half a kilometer.  It was easy.  On day 2, they made me run one third kilometer more than day 1, on  day 3, it was one fourth kilometer more than day 2.

Tinku:  Let me guess, on day 4, it was one fifth kilometer more than day 3, on day 5, one sixth kilometer more than day 4, on day 6, one seventh kilometer more.

Tanya:  It was a lot of running.  On day 7, it was one eighth kilometer more than day 6.  I don’t believe it.  I could still run.

Tinku:  You know, I love all the details.  I want to figure out how much you ran on day 7. it is easier to do in decimals.

 Day Frac km Add km Total km 1 1/2 0.5 0 +0.5 =0.500 2 1/3 0.333 0.500+0.333=0.833 3 1/4 0.25 0.833+0.250=1.083 4 1/5 0.2 1.083+0.200=1.283 5 1/6 0.167 1.283+0.167=1.450 6 1/7 0.143 1.450+0.143=1.593 7 1/8 0.125 1.593+0.125=1.718

Tanya: As you can see, I ran 1.718 kilometers on day 7. It was more than three times on day 1 (3 x 0.5 = 1.5).