Year 10 and 11 Challenges

1. I was practising my division and unfortunately I smudged one of the questions as shown. Can you work out what the original question and answer was?

 

2.

Billy needs to fill his bath with water. The hot water tap would completely fill the bath in 8 minutes if the plug is left in the plughole. The cold water tap would take 4 minutes to do the same. With the bath full and the taps off, the time taken for the water to completely drain away through the plughole is 3 minutes. Assuming Billy has left both taps on and left the plug out, how long would it take for the bath to fill completely?

 

3. Bert travels to work on the London Underground and always uses the same escalator (which descends at a constant speed) in the morning. He has discovered that if he walks down 26 steps he needs 30 seconds to get to the bottom; but if he makes 34 steps he only needs 18 seconds to reach the bottom. If the time is measured from the instant that the top step begins to descend to the time that Bert steps off the last step at the bottom, what is the height of the escalator in steps?

 

4. The diagram represents a sheet of 12 pictures. I wish to give 4 of them to a friend by cutting them out, but so that the 4 pictures are still joined by common edges (but not corners). For example ABCD, EFGH, BCGL, FGHL would be fine but not EFLM. How many different ways could I do this?

A B C D E F G H J K L M

 

5. Nine travellers, each possessing a car, meet at a town on the eastern edge of a desert. They wish to explore the desert, always travelling due west. Each car can travel forty miles on the contents of the car tank, which holds a gallon of fuel, and each can carry nine extra gallon cans of fuel and no more. Only unopened cans can be transferred from car from car. What is the greatest distance from the town to which any of them can enter the desert without depositing any cans in the desert for the return journey and they must all return back to the town?

 

6. The following shapes are called pentominoes. There are twelve different pentominoes each formed by using five squares. Select one of the pentominoes. Using nine of the remaining pentominoes (i.e. do not use the one you have chosen) form a large scale version of the selected pentomino, enlarged by a factor of three.

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