# Tanya wanted a bedtime story

Tanya was 12 year old now but she still wanted someone to tell her a bedtime story. You may say that she was spoiled brat. For a better description you would have to say she was a spoiled bright brat. Tanya’s grandpa told her a bed time story regularly but he was away on business. Tanya asked her mom but she is busy. Tanya trotted over to her grandma and started protesting.

Tanya: Grandma, you never tell me a story. Tell me one today, please.

Grandma: Okay, okay, I will tell you one. Let’s go to your room.

Tanya: I want a real story, not a short one like there were a king and a queen and they both died.

# Grandma said her story would never end

Grandma: My dear Tanya. I will tell you a story that never ends (Tanya was wide eyed hearing this). Once upon a time a rich farm owner had a big silo where he stored the maize grain from his harvest. It was a huge place filled with lots and lots of grains of maize. The silo was completely closed except for a small hole in the back that nobody noticed. One day one sparrow came and saw the hole and the grains behind it. It went in, got a grain and flew away.

Tanya: Grandma, I thought it was a long story. Then what happened?

Grandma: The sparrow told one of its friends. Immediately, both the sparrows came. They went in and each got one grain, that means they got 2 grains between them.

Tanya: And then?

Grandma: Each of the two birds told one of their friends and then 4 birds came and took a grain each.

Tanya was not wide eyed any more but said: And then?

Grandma: Each of the four birds told one of their friends and then 8 birds came and took a grain each. And then, each of the eight birds told one of their friends and then 16 birds came and took a grain each.

Grandma saw that Tanya was sleeping but to make sure she added, “And then, each of the 16 birds told one of their friends and then 32 birds came and took a grain each”.

Tanya was sound asleep.

Tanya got up in the morning as usual, got ready and went to school. Upon returning from the school, she talked to her mom.

Tanya: Mom, Grandma cheats.

Mom: Tanya (scolding tone). Never speak ill of the elders. She is your Grandma and she loves you.

Tanya: Mom, she said that she would tell me a story that would never end but the story has to end.

Tanya tells her mother the whole story.

# Increasing exponents

Mom: First let me tell you what you should get out of that story, and then we can talk about it ending or not. In your math class, you will learn about increasing exponential functions. These functions have a base and power. In this case the base is 2 and the power increases with each trip from 2 to 2 x 2, 2 x 2 x 2, 2 x 2 x 2 x 2 and so on. She could have said that the bird brought 2 (or any number) more friends each time. If this number is 3, the number of birds coming each time would have been 3 times more than the previous time.

Tanya: Two questions mom.

Mom: Yes, tell me the first one dear.

Tanya: If more and more grains are taken each time, would the silo not run out of grains sometime or the other. The poor birds will have to go back with empty beaks and Grandma’s story will end.

Mom: You are right. Grandma assumes that there is an infinite number of grains in the silo and that is an error. The number of grains is finite and hence the story will have to end.

Tanya: I guess Grandma doesn’t know that I can count to very high numbers.

Mom: You said that you had two questions. What is the second one?

# Exponents in real life

Tanya: Does this ever happen in real life?

Mom: Sort of. When you were born, your Grandma deposited $10000 in a bank for your college education. The bank said that you would get 10% per year interest compounded every year.

Tanya: What is that?

Mom: There are two types of interest. One is called simple interest. In this case each year the bank gives you an interest of $1000 which is 10% of $10000. In the 11 years you would earn Rs 11000 in interest and you would have a total of $21000 in the bank.

Tanya: That’s good mom. What is the compound interest then? Is it better?

Mom: For compound interest, they say the first year they give you the interest of $1000 but you deposit it back in the bank so that you have $11000. In the next year, they give you the 10% interest on the 11,000 which is 1100. Each year, you keep depositing the interest back into the account and earn the additional interest. This would keep going.

Tanya: How much money is there now?

Mom: You can’t have it until you reach the age of 18 and you need it for going to college. I think you should figure it out yourself. You can take Tinku’s help if you want.

Tanya went over to Tinku and said let’s calculate something. Tinku started crunching the numbers and was taking forever. Tanya stopped him saying that there has to be a better way because her mom had said this to be an example of an exponential function. Together they decided that it would be simpler first to calculate thinking that she has only $1 in the initial deposit. Then the first year increase would take it to $ 1.1^{1}. the second year to $ 1.1^{2 }and so on all the way to $ 1.1^{18 }which would be $ 5.559917. Now because she started with $ 10000, she would have the amount of $10000 x 5.559917 or 55599.17. Tanya figured that Grandma ‘s gift would be a big help but she would need much more than that to finish her college. She was not worried. She knew that everyone else in the family would also help.

*Challenge*

In 2013, Manal’s father says that she will get a pocket allowance of $2 per day because at Manal’s age, this was the allowance he used to get and he is 30 years older to Manal. Manal did not like it especially when Tanya told her that during the last 30 years, the average annual inflation rate in Canada has been 4%. That means the cost of everything increased each year by 4% Manal needs to find out what would be fair to ask her dad for the allowance. What do you calculate it to be?

*Solution: *This is an example of an exponential function. With a 4% increase, the value for each consecutive year will become Final value = Initial value x 1.04^{number of years}. Here the initial value for the daily allowance is $2 and the number of years is 30, so the final value will be $6.49. That would be a fair value with the argument that her allowance should have the same purchasing power as her dad did.