Every evening Ross arrives at a random time to the bus stop. Two bus routes stop at this bus stop. One of the routes takes Ross home, and the other takes him to visit his friend Rachel. Ross is waiting for the first bus and depending on which bus arrives, he goes either home or to his friend’s house. After a while, Ross noticed that he is twice as likely to visit Rachel than to be at home. Based on this, Ross concludes that one of the buses runs twice as often as the other. Is he right? Can buses run at the same frequency when the condition of the task is met? (It is assumed that buses do not run randomly, but on a certain schedule).
Father Christmas has an infinite number of sweets. A minute before the New Year, Father Christmas gives some children 100 sweets, while the Snow Maiden takes one sweet from them. Within half a minute before the New Year, Father Christmas gives the children 100 more sweets, and the Snow Maiden again takes one sweet. The same is repeated for 15 seconds, for 7.5 seconds, etc. until the new Year. Prove that the Snow Maiden will be able to take away all the sweets from the children by the New Year.
Izzy wrote a correct equality on the board: \(35 + 10 - 41 = 42 + 12 - 50\), and then subtracted 4 from both parts: \(35 + 10 - 45 = 42 + 12 - 54\). She noticed that on the left hand side of the equation all of the numbers are divisible by 5, and on the right hand side by 6. Then she took 5 outside of the brackets on the left hand side and 6 on the right hand side and got \(5(7 + 2 - 9)4 = 6(7 + 2 - 9)\). Having simplified both sides by a common multiplier, Izzy found that \(5 = 6\). Where did she go wrong?
The old shoemaker Carl sewed some boots and sent his son Hans to the market to sell them for £25. Two disabled people came to the boy’s market stall (one without a left leg, the other without a right one) and was asked to sell each of them a boot. Hans agreed and sold each boot for £12.50.
When the boy came home and told the whole story to his father, Carl decided that his son should have sold the boots to the disabled buyers for less – each for £10. He gave Hans £5 and ordered him to return £2.50 to each disabled buyer.
While the boy was looking for the disabled people at the market, he saw that someone was selling sweets and as could not resist, spent £3 on sweets. After that, he found the disabled buyers and gave them the remaining money – each got £1. Returning home, Hans realised how badly he had acted. He told his father and asked for forgiveness. The shoemaker was very angry and punished his son by sending him to his room.
Sitting in his room, Hans thought about the day’s events. It turned out that since he returned £1 to each buyer, they paid £11.50 for each boot: \(12.50 - 1 = 11.50\). So, the boots cost £23: \(2 \times 11.50 = 23\). And Hans spent £3 on sweets, therefore, it total, there were £26: \(23 + 3 = 26\). But there were only £25! Where did the extra pound come from?