ERJ Brainteaser - November

Question 3: United nations

Find the common link between:

Democratic Republic of the Congo

Dominican Republic

Equatorial Guinea

Mozambique

United States of America

Answer: Many thanks for all the innovative and imaginative replies to this week’s teaser. We usually publish all possibly correct answers, but as the list would be so lengthy this week, better just to spotlight the one reader who recognised the official solution. Extra well done, so, to Kamila Staszewska, R&D / quality lead, Abcon Industrial Products Ltd, Cootehill, Co. Cavan, Ireland, who saw that “these are all countries whose name contain all five vowels (a e i o u) at least once.”

New teaser on Monday.

An equilateral triangle and regular hexagon have perimeters of the same length. What is the ratio of the area of the triangle to the area of the hexagon?

Answer: In top shape (see Solutions) this week were: Bharat B Sharma, technical director, Rajsha Chemicals Pvt. Ltd, (TWC Group), Vadodara (Guj), India; Hans-Bernd Luechtefeld, consultant, Germany; Eur Ing John Bowen BSc [Hons], CEng, MIMMM, consultant, Bromsgrove, UK; Kamila Staszewska, R&D / quality lead, Abcon Industrial Products Ltd, Cootehill, Co. Cavan, Ireland; Sudi Sudarshan, principal consultant, Global Mobility Strategies, USA; Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands. Well done to all and everyone else who had a go.

PLEASE NOTE: Due to the upcoming Future Tire conference in Prague, the next teaser will be issued on Monday, 24 Nov. 

SOLUTIONS

Bharat B Sharma

Area ratio is 2:3  (Triangle : Hexagon)

Given that Perimeter of equilateral triangle = Perimeter of regular hexagon

Which is therefore , 3a=6b hence a=2b. (a is side of the triangle and b is side of regular hexagon. ..(i)

Areal of equilateral triangle = (√3/4) ×side(a)2  = (√3/4)×(2b)2 = (√3/4)×4b2 = √3×b2

Area of a regular hexagon = (3×√3/2)×side2 = 3×√3/2×b2 = 3/2 (area of triangle) =  3/2 area of triangle

Hence the ratio of areas of Tirangle: Hexagon is  2:3  (or 2/3 the area of Hexagon).

Ratio = 2 : 3 answer.

Hans-Bernd Luechtefeld

Let side length of the equilateral triangle be a, and side length of the regular hexagon be s. Equal perimeters:

3a = 6s ⇒ a = 2s.

Area of an equilateral triangle with side a:

A_triangle = (√3/4) a^2 = (√3/4) (2s)^2 = (√3/4) · 4s^2 = √3 s^2.

Area of a regular hexagon with side s:

A_hexagon = (3√3/2) s^2 (standard formula: six equilateral triangles of side s).

Ratio (triangle : hexagon):

A_triangle / A_hexagon = (√3 s^2) / ( (3√3/2) s^2 ) = 1 / (3/2) = 2/3.

Therefore, the ratio of areas is 2:3 (triangle : hexagon)

John Bowen

The area of an equilateral triangle is given by the formula {[Sq Root 3]/4} x L where L = length of a side

Our triangle of side length 1 will have an area of {[Sq Root 3]/4} x 1

In our case let the triangle side length = 1, so our regular hexagon of same perimeter length will be composed of 6 equilateral triangles each of side length 0.5L so the TOTAL area of the hexagon is 6 x {[Sq Root 3]/4} x 0.25 = {[Sq Root 3]/4} x 1.5

So the ratio of the triangle's area to the hexagon's area is 1:1.5

Kamila Staszewska

Triangle perimeter = Hexagon perimeter = P

Triangle side = P/3

Hexagon side = P/6

Triangle area =1/4*(√3)*(P/3)^2=1/36*√3*P^2

Hexagon area =3/2*√3*(P/6)^2=3/72*√3*P^2

Ratio: Triangle area/Hexagon area=( 1/36*√3*P^2) / (3/72*√3*P^2) = 1/36 * 72/3 = 2/3.

Sudi Sudarshan

Let the side of the hexagon be 2x. So perimeter of the hexagon is 6*2x = 12x

The equilateral triangle with the same perimeter has sides of 12x/3 = 4x

Area of equilateral triangle with sides 4x = 1/2* (4x)*(sqrt(12)x) = 4*sqrt(3)*x*x

Area of hexagon with side 2x = 6*area of 6 equilateral triangles of side 2x = 6*(1/2)*2x*sqrt(3)*x = 6*sqrt(3)*x*x

Ratio of area of triangle to area of hexagon = (4*sqrt(3)*x*x) / (6*sqrt(3)*x*x) = 2/3 = 0.667

Andrew Knox

Let the triangle have sides of length X, then the sides of the hexagon must be each of length X/2.

Constructing a line of length d perpendicular to the middle of one side of the triangle to the opposite point, then length d is X/2 .tan 60, and the area of the whole triangle is (X/2).d or (Xsquared/4).Tan60 or (1.732051/2).X = 0.86603 X

Similarly by dividing the hexagon into 6 equilateral triagles and constructing a line c perpendicular to the centre of one side of the hexagon to the centre, length c = (X/4).Tan60 = 0.433015 X, and the area of one of the 6 equilateral triangles so created is (X/4).0.433015 X.

Substituting X = 2 is the easiest way (on a mobile phone at least) to demonstrate that the ratio of the two areas is 1.5 in favour of the hexagon.

I came across the description of the puzzle again this morning and see that (the way the question is phrased) the answer should in fact be 2/3rds....

…  In fact there is a much simpler approach to this problem.
Divide the equilateral triangle into the four equal-sized equilateral triangles that will fit into it, and similarly divide the hexagon into the 6 equilateral triangles that will fit into it. As all the small triangles are the same size, the ratio of the areas of the triangle to the hexagon is simply 4:6 or 2/3.

Next teaser on Monday 24 Nov.

35.6, 37.4, 41.0, 44.6,  ?  ,

Clues: Getting warmer?, prime time... 

Answer: Well done, in order of reply, to this week’s prime movers: Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; John Coleman, membership manager, Circol ELT, Dublin, Ireland; Sudi Sudarshan, principal consultant, Global Mobility Strategies, USA; Kamila Staszewska, R&D / quality lead, Abcon Industrial Products Ltd, Cootehill, Co. Cavan, Ireland; Amparo Botella, responsable de Compras y Calidad, Ismael Quesada SA, Elche, Alicante, Spain; Eur Ing John Bowen BSc [Hons], CEng, MIMMM, consultant, Bromsgrove, UK; Hans-Bernd Luechtefeld, consultant, Germany; Peter D. Talbot, research scientist, Chem-Trend LP, Howell, MI, USA; and everyone else who had a go. 

SOLUTIONS

Andrew Knox

Answer: 51.8

Sequence is ascending prime numbers (2, 3, 5, 7, 11) as values in degC, converted into degrees Fahrenheit. So, 11 x 9/5 = 51.8.

John Coleman

Question 1: Sequence-wise

35.6, 37.4, 41.0, 44.6,  ?  ,   

These are the Fahrenheit equivalent temperatures of the first four prime numbers in centigrade, i.e.

Therefore, the next number in the sequence is 51.8 which is the Fahrenheit equivalent of 11°C.

Sudi Sudarshan

The numbers, if regarded as temperature readings in Fahrenheit,  when translated to Celsius are:

(35.6 - 32)*5/9 = 2

(37.4 - 43)*5/9 = 3...

We get the sequence

2, 3, 5, 7, ...

This is the sequence of prime numbers. The next prime number is 11.

11°C  translates to 11*9/5 + 32 = 51.8°F 

Kamila Staszewska

These are temperatures in Fahrenheit that corresponds with "prime" degrees in Celsius:

35.6(°F) = 2(°C)

37.4(°F) = 3(°C)

41.0(°F) = 5(°C)

44.6(°F) = 7(°C)

Next in the sequence is 11(°C) which is 51.8 (°F).

Amparo Botella

Very complicated this time, but the clue, seems to have clarify something.

I believe the numbers given are Fahrenheit grades that correspond to prime-number Celsius grades:

Following the conversion formula: ºC=(ºF-32)/1.8

35.6°F → (35.6−32) ÷ 1.8 = 3.6 ÷ 1.8 = 2°C

37.4°F → (37.4−32) ÷ 1.8 = 5.4 ÷ 1.8 = 3°C

41.0°F → (41.0−32) ÷ 1.8 = 9.0 ÷ 1.8 = 5°C

44.6°F → (44.6−32) ÷ 1.8 = 12.6 ÷ 1.8 = 7°C

Next prime after 7 is 11°C. Convert 11°C to °F:

ºF=(ºCx1.8)+32

ºF=(11x1.8)+32=19.8+32=51.8

So the missing term is 51.8.

John Bowen

The numbers given are the Farenheit values of  2,3,5,7 Celsius

Next would be 9C or 48.2 in Farenheit

Hans-Bernd Luechtefeld

We are looking for the next prime number in Fahrenheit: 51.8

Sequence:

35.6 F = 2°C, 37.4 F = 3°C, 41 F = 5°C, 44.6 F = 7°C => 51.8 F = 11°C

Peter Talbot

This series comprises temperatures expressed in °F derived from a series of values originated in °C.

On examination, 35.6°F = 2°C, 37.4°F = 3°C, 41.0°F = 5°C , 44.6°F = 7°C.

2,3,5 & 7 are the first 4 prime numbers. The next and 5th member of this series is 11.

Therefore, the value we are looking for in this brainteaser is 11°C expressed in °F = 51.8.

News teaser on Monday.