Brain Teaser No 1

40 kilos.

-Dan

Quadrilateral observation
It's easy to see that connecting the mid-points of each side of the quadrilateral gives you a parallelogram. Each side is parallel to one of the lines connecting opposite corners of the quadrilateral. However, showing that its area is 1/2 the original quadrilateral needs to be proved.

cheers

Dan

However, showing that its area is 1/2 the original quadrilateral needs to be proved.

This IS proved in my method viz,

By showing that the LARGE parallelogram is TWICE the area of the quad and that the area of the SMALLER parallelogram is ONE QUARTER the area of the large paralellogram.

BTW it also PROVES that joining the mid-points of a quadriateral forms a parallelogram, which otherwise isn't too obvious........eg just having two pairs of parallel sides isn't enough, they also have to join at or inside the quadrilateral. (In fact they do join at the mid-points of the quadrilateral.)

I will post some pictures tomorrow......

Cheers

Don

BTW Dan, 40 kilo is correct, so, without sounding patronising, WELL DONE.

Cheers

Don

Don,

Sorry, didn't mean to suggest your proof wont work - I'm curious how you did it. I'm sure a picture is worth a thousand words. I look forward to seeing how you relate the sizes of the //gram to each other and the large //gram to the quad.

While it isn't obvious that the quad with vertices at the midpoints of the original quad is indeed a //gram, I can prove it this way (hopefully more clearly than before).

1. Given quad with verticies ABCD. Draw a line between the mid points of two adjacent sides (ABC) of the original quad. The triangle created is similar to one of the triangles created by joining opposite corners (AC). Hence the line between the mid points is // to AC.

2. Repeat for sides opposite to vertex B. Since both lines are // to AC, they are parallel to each other.

3. Repeat for pairs of sides either side of line BD, and you have another set of // lines, intersecting the mid-points.

4. The resulting quad must then be a //gram.

As you can see, this proof doesnt require construction of an outside //gram, but got me no closer to showing that the contained //gram is half the area of the original quad!

cheers,

Dan

I'm sure a picture is worth a thousand words

re the Quadrilateral of real estate.

So, we'll try the picture below, (Stallion would have posted a thousand words !)

Worth noting the following points

ABCD is an iregular quadrilateral

The lines EF etc pass through the corners of the quad and are parallel to the respective diagonals

The quad diagonals intersect at O which forms the common point of 4 parallelograms which in turn form the large parallelogram EFGH.

Each of the 4 parallelograms comprises two congruent triangles eg ABO and ABF. Hence EFGH has twice the area of ABCD.

The next picture will show the construction of the (required) smaller parallelogram, but the whole proof should be getting pretty obvious by now.........

Cheers

Don

Another thousand words continued..............

Draw lines parallel to db to intersect ao and ac at their mid-points
Draw lines parallel to ac to intersect do and bo at their mid-points.

Similar triangles allows us to see that these new lines intersect at the mid-points of ab; bc; cd and da and form a new parallelogram WXYZ with dimensions one half of efgh and hence an area of one quarter efgh.

QED

Now I'm still wondering WHY a real estate agent would ever what to be able to do this......

Cheers

Don

Another trivial one, so it only counts if you do it in your head !

A cup of olive oil is added to a three cup mixture composed of two fifths salad dressing (vinegar !) and three fifths olive oil. What percentage of the four cup mixture is olive oil ?

Remember, your answer will only be accepted if you did this one in your head

Cheers

Don

25 + 0.6*75

Paul

Paul,

Mental arithmetic isn't your strong point then.......

You are, of course, absolutely right, but I had expected the answer to be presented in its most simple form.

Quite a few (I have no doubts at all) would still need pencil and paper to work out your sum...

Cheers

Don

When you seek "simplest form," I am always intrigued

Lynn,

I think we can safely say that 70% is the correct answer in its simplest form

And we all know that Paul R was teasing.........well, they are brain teasers


Anyway, glad to see you had a good time with the old wedding bells and that you now know how to divide the display area in your shop in half......

Cheers

Don

One of those co-incidences occurred today. Nothing spectacular you understand, just a minor co-incidence. I put my hand in my pocket to pay for lunch and noticed that I had three times as many pennies, plus 1p, as I had whole pounds. My colleague on the other hand, noticed that he had as many whole pounds as I had pennies, and as many pennies as I had whole pounds. He also had three times as much as me, plus 1p.

How much did we each have just before lunch?

Cheers

Don

Don,
You had £12.37. The other person had £37.12. I hope the other person paid!
Steve D

Anybody care to fill in the algebra (allgiberish?) between my teaser and Steved's answer?

Cheers

Don

This one appeared recently in one of those free newspapers (the Metro) that flood the UK. I hope I am not breaking any copyright rules....

I could, of course, change the names to avoid copyright prosecution, but then if I chose arbitrary names such as "Mr Pig", "Stallion", "Belzebub" or "Mr Parry" the risk of prosecution would become a certainty.....

Mr Meeke never tells lies. Mr Brashe always lies. One day, one of them pointed to the other and said to me "He has just said that he is called Mr Brashe".

Which one of them was speaking?

Cheers

Don

Liar, Liar, (Pants on Fire)!

It must have been Mr Brashe speaking. He could have been lying on two points.

1) The lie is that Mr Meake said anything at all.
2) Mr Meake did say something, but the lie was what Mr Meake actually said.

Steve D

Don,

The algebra

You had y pennies and x pounds

Therefore y = 3x + 1

Also, as you colleague had 3 times as much as you, plus 1p and his number of pennies was equal to your number of pounds and vice-versa

3*(100x+y)=100y+x

or

299x-97y+1=0

substiute in first equation and get

299x-291x-97+1=0

therefore x=12 and y= 37

Mr Brashe was telling you fibs.

Matthew

This one is little more than a simple high school geometry question (worth about 5 marks), but has, what I think, a neat answer........

In the diagram below, each side of the triangle is 3 units long.

What is the area of the circle ?

Cheers

Don

Steved and Matthew,

You are both right that it was Mr Brashe

My logic is based on the following :-

Neither man would say "My name is Mr Brashe"

Cheers

Don

3 * pie

Matthew

Triangle in a circle,

Sorry, forgot to mention it was in the "mental arithmetic" section.....

Matthew, your answer is only acceptable if you did it in your head.....please confirm....


If not, i'm afraid you are going to have to show all your working out.........

Cheers

Don

Don,

I definitely didn't use an electronic calculator, I may have resorted to a piece of paper and pencil though!

Lynn,

The equation that I get goes like this...

Area of circle = 9 * pie / (4 * (sin(180/n))^2)

Where n is the number of sides of the inscribed figure.

Matthew

A warm up question (or is a wind up, as Del Boy would say)

Del Boy moves into Hifi....

Now, for the benefit of our American cousins, French, Arab and African friends,.....(ok, lets cut the bullsh*t)...Del Boy, alias Derek Trotter, of Trotters Independent Traders, and his brother Rodney, both of 'Only Fools and Horses' fame, have moved into the hifi delivery market. Their delivery vehicle of choice is, of course, a bright yellow Reliant Robin three wheeled van.

In their first year of business they cover ten thousand miles. They have two spare tyres and changed them around so that each tyre covered the same distance.

How many miles did each tyre travel?

Cheers

Don

I may be a "dozy old twonk" or a "disptick", but ......

I think the answer is each tyre "worked" for 6000 miles, but all tyres travelled 10000 miles (ie 6000 working, 4000 being carried).

Steve D

the answer is each tyre "worked" for 6000 miles,

ergo: i go w/ steveD

Good answers. Appologies about the confusion Steved between travel distance and worked distance.

Now a number of people that I've put this question to answered, rather hastilty, 2,000 miles. Hands up anybody who was going to say 2,000 miles at some stage.....

Cheers

Don