Parallelogram 2 Year 10 19 Sep 2019Musical Mathematics

This is a preview of Parallel. You have to login or create an account, to be able to answer questions and submit answers.

Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

  1. a portmanteaux word combining parallel and telegram. A message sent each week by the Parallel Project to bright young mathematicians.

Welcome to the second of our Parallelograms designed for Year 10 students, a collection of mathematical challenges designed to stretch your brain and make your neurons more squiggly.

These challenges are a random walk through the mysteries of mathematics, most of which will be nothing to do with what you are doing at the moment in your classroom. Be prepared to encounter all sorts of weird ideas, including a few questions that appear to have nothing to do with mathematics at all.

  • Tackle each Parallelogram in one go. Don’t get distracted.
  • When you finish, remember to hit the SUBMIT button.
  • Finish by Sunday night if your whole class is doing parallelograms.

IMPORTANT – it does not really matter what score you get, because the main thing is that you think hard about the problems... and then examine the solution sheet to learn from your mistakes.

1. Musical mathematics

It is extraordinary that so much of the universe can be explained using mathematical equations. Indeed, it is often said that mathematics is the language of the universe. It is certainly the language of science.

In this clip, from a documentary in the American Nova series, the jazz musician Esperanza Spalding explains how maths is also at the heart of music. Pay attention to the way that numbers relate to musical intervals (an octave, a fifth and a fourth).

2 marks

1.1. What ratio in lengths describe a fourth?

  • 4:4
  • 4:3
  • 4:2
  • 4:1
  • 1:4
2 marks

1.2. Pythagoras is said to have noticed the relationship between mathematics and music. According to Pythagoras, the universe is based on numbers, but only whole numbers and their ratios. So, when one of his followers discovered a different type of number, Pythagoras executed him. What sort of number did the follower discover? You might need to do some googling to get the answer.

Hint: Pythagoras acted irrationally when he executed the follower.

  • Negative numbers
  • Zero
  • 2
  • Fractions
  • Quaternions

The square roof of 2 is an irrational number, which means it can never be written as a fraction or ratio (ir-ratio-nal). Pythagoras felt that such numbers were evil or should not exist, which is why he is said to have executed the young mathematician who discovered that 2 is irrational.

2. Do Unto Caesar

4 marks

2.1 At Caesar's Palace Casino in Vegas, there were three poker players, Alice, Bernie and Craig. At the beginning of the night the amount of money each had was split in the ratio 7 : 6 : 5, with Alice starting with the most and Craig the least. At the end of the night, Alice was still ahead and Craig still last, but the money was divided according to the ratio 6 : 5 : 4.

This meant that one of the players was up by $12.

What were the combined assets of the players at the beginning of the evening, in dollars? (No need to put the currency mark in your answer, just put your answer in numbers!)

Correct Solution: 1,080

Show Hint (–1 mark)
–1 mark

No money entered or left the game. The ratios of how the money was divided changed, but the total amount of money was unchanged.

Show Hint (–1 mark)
–1 mark

Turn each set of ratios into fractions.

Show Hint (–1 mark)
–1 mark

Bernie still had 1/3 of the money, while Craig lost money, so it must have been Alice who had won the $12.

This problem is from the terrific NRICH website, a treasure trove of mathematical ideas and questions. Go and explore.

7:6:5=718:618:518
6:5:4=615:515:415

The lowest common multiple of 15 and 18 is 90, so

3590:3090:2590
3690:3090:2490

We can see that Craig lost 190 of the money and Alice gained it. We are told that the sum of money gained is $12. Therefore 1/90 of the money = $12.

Therefore the total amount of money is 90 × $12 = $1,080.

And don’t forget to explore NRICH soon.

3. Intermediate Maths Challenge Problem (UKMT)

Every Parallelogram will contain two or more UKMT Intermediate Maths Challenge problems. This one is fairly simple, but others will be much trickier... and for all of them, remember, NO CALCULATORS!

3 marks

Which of the following fractions is closest to 1?

  • 78
  • 87
  • 910
  • 1011
  • 1110

We tackle this question by calculating the differences between the given options and 1.

We have 178=18,
871=17,
1910=110,
11011=111, and
11101=110.

The smallest of these differences is 111.

Therefore, of the given options, 1011 is closest to 1.

4. Intermediate Maths Challenge Problem (UKMT)

4 marks

The world’s fastest tortoise is acknowledged to be a leopard tortoise from County Durham called Bertie. In July 2014, Bertie sprinted along a 5.5m long track in an astonishing 19.6 seconds.

What was Bertie’s approximate average speed in km per hour? Remember, NO CALCULATOR!

  • 0.1
  • 0.5
  • 1
  • 5
  • 10

It took Bertie 19.6 seconds to travel 5.5m. So Bertie went 5.519.6 m in 1 second. There are 3600 seconds in one hour. Therefore Bertie travelled

5.519.6×3600 m in one hour.

There are 1000m in 1 km. Therefore Bertie’s average speed in km per hour was

5.519.6×36001000=5.519.6×185.

This is approximately equal to

518×185=1.

Therefore Bertie’s approximate average speed was 1 km per hour.

I hope you enjoyed the second Parallelogram of the year. There will be more next week, and the week after, and the week after that. So check your email or return to the website on Thursday at 3pm.

In the meantime, you can find out your score, the answers and go through the answer sheet as soon as you hit the SUBMIT button below.

When you see your % score, this will also be your reward score. As you collect more and more points, you will collect more and more badges. Find out more by visiting the Rewards Page after you hit the SUBMIT button.

It is really important that you go through the solution sheet. Seriously important. What you got right is much less important than what you got wrong, because where you went wrong provides you with an opportunity to learn something new.

Cheerio, Simon.