Earlier right now I set you this downside from final 12 months’s British Mathematical Olympiad (BMO), the UK’s high maths competitors for pre-university college students, which is taken by virtually 2,000 youngsters a 12 months.
The query was tried by 90 per cent of the contestants, and about 1 in 3 obtained full marks. How did you get on?
Ping pong percentages
Arun and Disha performed a number of video games of desk tennis. At 5 factors throughout the day, Arun calculated the share of the video games performed up to now that he had gained. The outcomes of those calculations had been precisely 30 per cent, precisely 40 per cent, precisely 50 per cent, precisely 60 per cent and precisely 70 per cent, however not essentially in that order.
What’s the smallest doable variety of video games they performed?
Right here’s a touch. Arun wins one of many first two video games.
Right here’s the way you may need gone about it. First, let G be the smallest doable variety of video games.
If at any stage Arun has gained 30 per cent of the video games he performed, then we are able to write the equation:
W/G = 3/10
the place W are the video games he has gained at that stage. We will rearrange this equation as 10W = 3G, from which we are able to deduce that G should be a a number of of 10.
Nonetheless we additionally know that there’s a level at which he has gained 70 per cent of video games. In different phrases there’s a w such that 10w = 7G.
We will eradicate the prospect that G = 10, since if it was then this may imply that after ten video games Arun has gained each 3 and seven video games, which is not possible.
Let’s say G = 20. Then both (i) after 10 video games he has gained 3 of them, and after 20 video games he has gained 14 of them, or (ii) after 10 video games he has gained 7 of them and after 20 video games he has gained 6 of them. [This is because we are only going to achieve the 30 per cent, and 70 per cent, of wins when the games played is divisible by 10]
Each (i) and (ii) are not possible. (i) as a result of there are usually not sufficient video games to go from 3 to 14 wins in 10 video games, and (ii) as a result of you’ll be able to’t go from 7 wins to six.
Now let’s strive with G = 30. I instructed you that Arun wins now of the primary two video games. So, after two video games his operating complete of wins/video games is 1/2, which is 50 per cent.
Let’s now rewrite the opposite percentages required as fractions: 30, 40, 60 and 70 per cent are 3/10, 2/5, 3/5 and seven/10.
Let’s say that after profitable 1/2, Arun wins one of many subsequent three. That get’s him to 2/5, when he may have gained 40 per cent. If he wins one of many subsequent 5, that get’s him to three/10, which is 30 per cent. If he wins 9 of the subsequent ten, that get’s him to 12/20, which is 60 per cent. If he wins one other 9 of the subsequent ten, he finishes with 21/30, which is 70 per cent.
We’re carried out! G = 30.
I hope you loved right now’s puzzle. I’ll be again in two weeks.
Because of the UK Arithmetic Belief for right now’s puzzle. The belief organises the BMO, in addition to many different college maths competitions. For extra data, please go to ukmt.org.uk.
I set a puzzle right here each two weeks on a Monday. I’m all the time on the look-out for nice puzzles. If you want to counsel one, e mail me.
I’m the writer of a number of books of puzzles, most lately the Language Lover’s Puzzle E-book. I additionally give college talks about maths and puzzles (on-line and in particular person). In case your college is please get in contact.