Humanising Language Teaching
Year 6; Issue 3; September 04

Three Logical Mathematical problems

Marie Agrell
France

Hair-splitting

Until age 25 I had a magnificent head of hair, but by my 26th birthday, I had lost one 26th of my hair. By age 27, I had lost one 27th (of my youthful hair), by 28 one 28th, and so forth until my 50th birthday when I lost one 50th of the hair I had before I began to loose it. Today is my 51st birthday. Without using a calculator, can you prove that I have less than one hair out of two but that I still have more than one out of four?

At the swimming pool

In the 1st lane, François begins to swim the crawl from the deep end of the pool but without diving in. Vincent starts swimming the breaststroke from the shallow end at the same time. They swim at a constant speed. They pass each other the first time 5 meters from the shallow end of the pool. When they arrive at the opposite ends of the pool, they each turn and swim in the other direction at the same speed. They pass each other the second time 2.5 meters from the deep end. What is the length of the pool and the ratio between the speed of François's crawl and Vincent's breaststroke?

Watch out! It's melting

On the beach, when it's very hot, my sister Roberta and I love to eat ice cream. The ice cream man sells big ice cream cones for only 1 dollar. But the cone isn't very strong, and after 7 minutes it breaks, and everything falls into the sand (ugh!). For my sister Roberta, this is not a problem, because she always manages to eat her ice cream in 5 minutes. But I can't, and so every time I lose 10% of my ice cream!

Today, Roberta and I have only 60 cents each. That's ok, she says, let's buy one ice cream together and we can take turns licking it. That way, we will be able to eat half each, and we'll each have 10 cents left in our pockets. This is certainly an appealing proposition, but it doesn't seem altogether fair. What do you think?

SOLUTIONS

Hair-splitting

Draw the corresponding function and compare with the integral with the areas of inside and outside quadrilaterals.

At the swimming pool

The second observation is happening after having swum 3 times more: the swimming-pool is 12.5 meters. François is one and a half times faster than Vincent.

Watch out! It's melting

Roberta is eating 60%. I am eating 40%. Roberta should pay 60 cents. I should pay 40 cents.