ERJ Brainteaser: May

Which country completes the following series?

US + Japan = 82; Germany + Spain = 83; ???+ Italy = 84.

Answer: With a little bit of lateral thinking, a select number of readers worked out the answer as Denmark, which has an international telephone dialling code of 45. Well done, in order of correct reply, to: Flemming Vorbeck, purchasing manager commercial, NDI Group A/S, Brørup, Denmark; John Bowen, rubber industry consultant, Bromsgrove, Worcs, UK; David Mann, key account manager, SPC Rubber Compounding, UK; Michele Girardi, Scame Mastaf Spa, Suisio, Italy; Stephan Paischer, head of product management special products, Semperit AG Holding, Vienna, Austria; Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; France Veillette: chef environnement, Usine de Joliette, Bridgestone Canada Inc., Canada; Marion Jourdain, industrial VBelts PDG development engineer, Gates S.A.S., Nevers, France; and everyone else who had a go. 

New teaser on Tuesday.

At a rubber parts manufacturing plant, Simon has an important job of loading consignments of raw materials into the warehouse, which he always completes in three hours exactly. But today Pete has come along to give him a hand. How long will it take both to complete the job is Pete works half as fast as Simon?

Answer: Taking it a bit easy this week, with the answer working out quite straightforwardly as 2 hours. Well done to: Michael Easton, director, Globus Group, Trafford Park, Manchester, UK; Michele Girardi, Scame Mastaf Spa, Suisio, Italy; Stephan Paischer, head of product management special products, Semperit AG Holding, Vienna, Austria; Flemming Vorbeck, purchasing manager commercial, NDI Group A/S, Brørup, Denmark; Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; France Veillette: chef environnement, Usine de Joliette, Bridgestone Canada Inc., Canada; and everyone else who replied.

New, probably trickier, teaser on Monday.

Can you work out the ratio of the area of the equilateral triangle to the area of the regular hexagon shown, assuming both have perimeters of the same length?

Answer: To get to this week’s answer, 3:2 (or 1.5), readers came up with an impressive range of solutions (see below). Very well done, in order of reply, to: Stephan Paischer, head of product management special products, Semperit AG Holding, Vienna, Austria; John Bowen, rubber industry consultant, Bromsgrove, Worcs, UK; Flemming Vorbeck, purchasing manager commercial, NDI Group A/S, Brørup, Denmark; Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; Michele Girardi, Scame Mastaf Spa, Suisio, Italy; France Veillette, chef environnement, Usine de Joliette, Bridgestone Canada Inc., Canada; Michael Easton, director, Globus Group, Trafford Park, Manchester, UK; and everyone else who had a go. 

Solutions

Stephan Paischer
The answer is: area B of hexagon = 1.5 x area A of triangle
Triangle A with area A and side a
Hexagon B with area B and side b
Perimeter triangle: 3a -> equals to 
Perimeter hexagon: 6 b
So: 3 a= 6 b
a = 2b
Area triangle:
A = (a^2)/4 x sqrt(3)
Area hexagon:
B = 6 x (b^2)/4 x sqrt(3)
The hexagon is made up of 6 equal-sided triangle with side length b
Remodel area triangle by inserting a = 2b    to
A = 4 x (b^2) /4 x sqrt(3)
(b^2) = a / sqrt(3)
Insert into hexagon formula:
B = 3/2 x A / sqrt(3) x sqrt(3)
B = 3/2 x A
B = 1.5 x A

John Bowen
Area of an equilateral triangle is given by  [Sq Rt 3* [D sqd]/4  where D = length of side.
So for our Triangle, D = L/3 where L is perimeter
So the area of our triangle is [Sq Rt 3] *[L/3] Sqd /4 = [Sq Rt 3] * L Sqd/36.
For our hexagon, this is made up of 6 equilateral triangles each of side length L/6
So the total area is 6 *  [ Sq Rt 3 * {L/6} Sqd}]/4 which simplifies to 1.5 [Sq Rt 3] *L Sqd/36
So the area of the hexagon is 1.5 times the area of the triangle with the same perimeter.

Flemming Vorbeck
To work in actual numbers, let’s start by setting the perimeter to a given length, let’s say 30. Unit not important to solve the task.
Each side (s) in the two shapes will then be:
Triangle:  30/3 = 10;  Hexagon: 30/6 = 5
The area of an equatorial triangle is calculated as follows:
A = ?3/4 s^2
This gives our triangle with a perimeter of 30 the following area:
?3 = 1.732051
1,732051/4 = 0.433013
s^2 = 10^2 = 100
Triangle area: 0.433013 x 100 = 43.30127
The area of a hexagon is calculated as follows:
A = (3?3  s^2)/2
This gives our hexagon also with a perimeter of 30 the following area:
?3 = 1.732051
s^2 = 5^2 = 25
Hexagon area: 3 x 1.732051 x 25 divided by 2  = 129.9038106/2 = 64.95191
The ratio between the area of the triangle and the hexagon would then be 1 to (64.95191)/(43.30127) = 1.5

Andrew Knox
Answer: Ratio of Area Hexagon to Area Triangle is 6 : 4 (or 1.5).
Assuming both are regular polygons, then if the sides of hexagon are all of length b, then the sides of the triangle (if the perimeters are of equal length) will each be of length 2b.
The triangle can be divided into 4 four equal equilateral triangles each with sides of length b, by drawing lines from the centre of each side to the centre of the adjacent sides. 
The hexagon can be divided into 6 equilateral triangles by slicing it up like a cake from each corner to the centre. Each of these 6 equilateral triangles will also have sides of length b.
So the ratio of the area of the hexagon to the area of the triangle is simply 6:4, or 1.5.
I must admit that by resorting to published formula's (and without going back to first principles of sine/cosine, internal angles etc., which one ought to do?), I got the answer wrong, so this time I have gone for the stylish answer?.

Michele Girardi 
The ratio of the area of the triangle to the one of the hexagon is 2/3 .
This can be seen graphically by :
- subdiving the triangle in 4 units by uniting the middle points of the sides
- subdividing the hexagon in 6 units by uniting the vertexes
- veryfing that the subtriangles have the same area , so the ratio is 4/6 = 2/3
Alternatively :
- the area of a polygon is given by perimeter x  apothem / 2
- since the triangle and the hexagon have the same perimeter, the ratio in given by the ratio of the apothems
- setting the  side of triangle  = 1, it's ap.Tria. = 1/2*1*tan 30° = 1/2*sq.root(3) /3 ap. Hex = 1/4 *1*tan 60°= 1/4*sq.root(3) so area ratio = ap.Tria/ ap.Hex = 2/3.

France Veillette
The ratio of the area of the triangle to the area of the hexagon is 2/3.
Given that the perimeter is the same, one side of the triangle is double one segment of the hexagon.  Using the formula for the area of a triangle being side2*?3/4 and the area of the hexagon being 3*?3*segment2/2; the ratio triangle/hexagon is 2/3.
 

New teaser on Monday.

On a randomly chosen day, the probability that Pete travels to work by car, by bicycle or on foot is 1/2 , 1/6 and 1/3 respectively.

The probability of him, being late when using these methods of travel is 1/5, 2/5 and 1/10 respectively.

Find the probability that on a randomly chosen day: a) Pete travels by foot and is late; b) Pete is not late.

Answer: Firstly, apologies for the name mix-up on Monday: the question was challenging enough and hopefully this did not confuse readers. The answers were 1/30 and 4/5, as expertly worked out (partly or fully) by: John Bowen, rubber industry consultant, Bromsgrove, Worcs, UK; Flemming Vorbeck, purchasing manager commercial, NDI Group A/S, Brørup, Denmark;  Michele Girardi, Scame Mastaf Spa, Suisio, Italy; Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands;  Stephan Paischer, head of product management special products, Semperit AG Holding, Vienna, Austria; Stefano Carletti, R&D Indorama Mobility Group, Office address: Glanzstoff Sicrem Spa, Pizzighettone (CR), Italy. Well done to all, and everyone else who had a go.

Solutions

Flemming Vorbeck
The probability of Pete travelling by foot 1/3 = 33.33% of the days. The probability of Pete being late when travelling by foot is 1/10 = 10%.
So, the probability of Pete travelling by foot AND being late is 10% of 33.33% = 3.33%
To find out, when Pete is NOT late, we must do a bit of backwards calculation.
Travel by car: 1/2 = 50%. Late by car: 1/5 = 20%. Not late by car: 50% * (100%-20%) = 40%
Travel by bicycle: 1/6 = 16.67%. Late by bicycle: 2/5 = 40%. Not late by bicycle: 16.67% * (100%-40%) = 10%
Travel by foot: 1/3 = 33.33%. Late by foot: 1/10 = 10%. Not late by foot: 33.33% * (100%-10%) = 30%
So, the probability of Pete not being late regardless of method of travel is: 40%+10%+30% = 80%.

Michele Girardi
The probability of Pete being on time in the 3 cases is 4/5, 3/5, 9/10, respectively. So, the answers are:
a) The probability of traveling by foot and being late is   1/3*1/10 = 1/30= 3.3%
b) the probability of being on time is   1/2*4/5+1/6*3/5+1/3*9/10 = 0, 8 = 80%

Stephan Paischer 
A: 1/3x1/10 = 3,3%
B: 1/2x4/5 + 1/6x3/5 + 1/3x9/10
I use the mathematical operator “x” (multiplication) to express the stochastic “and”: how big is the probability to come by foot (1/3) AND be late (1/10).
The same for case B, just that I use probabilities for being on time: (1-1/5 = 4/5 …. As 1/5 is the probability to be late by car, the probability to be on time is 4/5).

Andrew Knox
Answer a): The chance that Peter has gone by foot and is late is 1/3 x 1/10 = 1/30.
Answer to part b): 
If the question were b) Pete is not late, then the answer is 4/5
The chance that he went by car and was not late is 1/2 x 4/5 = 12/30
The chance that he went by bicycle and was not late is 1/6 x 3/5 = 3/30
The chance that he went on foot and was not late is 1/3 x 9/10 = 9/30
So the chance he was not late is the sum of these i.e. 24/30 or 4/5.
(Just by way of a check, the chance that he would be late is:
By car: 1/2 x 1/5 = 3/30
By bicycle 1/6 x 2/5 = 2/30
On foot 1/3 x 1/10 = 1/30
The sum of these is 6/30 or 1/5. 
1/5 + 4/5 = 1, which is correct, i.e. it is a certainty (probability of 1) that he will be either late or not late).
 

New teaser on Monday