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Tuesday, 13 October 2015
Kids and Wigs: Hint
Once more, just express it all by means of a mathematical system. A bit of Algebra and done!
Back to the problem
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Kids and Wigs: Solution
The problem is easily solved with a system formed by three equations with three variables.
We will call the number of wigs X.
Y will be the number of her kids.
Z will be the exact number of her grandchildren.
She said that the sum of these three numbers was 19. Thus:
X + Y + Z = 19 (A)
We know that the number X, the number of wigs, was equal to twice as much as the number of grandchildren less 1.
Thus:
Thus:
X = 2(Z-1) (B)
The number of grandchildren is twice as much as the number of kids:
Z = 2Y (C)
Solving this system, formed by A, B and C, we find out that
X = 10 Y = 3 Z = 6
Back to the problem
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Kids and Wigs
You have just entered a train. You hear a girl wearing a black costume asking a lady wearing a blue dress about her wigs.
- But, in short, how many wigs do you have?
The lady, in a joking tone, shaking her earrings (which matched the dress), answered:
- The number of my wigs, my dear, is nothing out of the ordinary. It is very reasonable. It is equal to twice as much as the number of my grandchildren less one.
- Less one? How? – inquired the girl who was showing a lack of understanding in terms of calculations.
- Less one, for sure – confirmed the lady. From the total number of my grandchildren, take out one and multiply the result by two. You will then get exactly the number of my wigs. Do you understand?
- And, lady, have you got many grandchildren? – asked the indiscreet girl in black.
- Well – replied the lady, very sweetly. My kids are already all married (and well married!) and each one of them has already given me two grandchildren. I noticed, the other day, by chance, that if I add the number of my wigs to the number of my kids and the number of my grandchildren, I would get 19. Do you understand?
We ask:
Can you, my friend, having read this curious and indiscreet conversation, calculate the number of wigs of the lovely lady in the blue dress?
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Source of the Problem
Solution: Anatole, Carla, and Border
Take X to be the price of one barrel. For the 59 that crossed the border, Anatole paid 5 barrels plus 40 dollars. Thus, he paid 5X + 40.
This was the tax due for the 59 barrels; to find out the value of the tax paid for one barrel we need to divide it by 59. So the tax paid by Anatole per barrel was
(5X+40)/59
For the 18 barrels that she crossed the border with, Carla paid 2 barrels and received 40 dollars in return. Thus, Carla paid 2X-40 for the 18 barrels.
This way, the tax paid by Carla per barrel was
(2X-40)/18
Since both fractions represent the value paid for one barrel, they must be equal. Thus, we have:
(5X+40)/59 = (2X-40)/18
∴ 90X + 720 = 118X – 2,360 =>28X = 3,080 => X = 110
Thus, 110 dollars is the price of one barrel.
The total amount paid in tax for the 77 barrels was 7 barrels. Thus, the total amount of tax charged was 770 dollars. We know that 77 barrels crossed the border. Thus, each barrel had
770 / 77
of tax on them, that is, each barrel was taxed 10 dollars.
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Hint: Anatole, Carla, and the Border
Try to express all by means of a mathematical system.
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Anatole, Carla, and the Border
Two wine merchants, Anatole and Carla, arrived at the border carrying barrels. Anatole carried 64 barrels and Carla carried 20 barrels. The 84 barrels were of equal weight.
Since they did not have enough money to pay the taxes for the crossing, they asked Marsha, the customs officer, whether they could use some of the barrels to pay the taxes they owed (each barrel had a value assigned by law).
Marsha was an understanding woman, so she agreed.
Because she was also very honest, example of employee, very righteous, she wanted to make sure that the correct tax was paid, so that the transaction would be perfectly legal.
After making some calculations (taking into account the price of each barrel and the value of the taxes), Marsha said:
- Anatole will pay 5 barrels plus 40 dollars. He will cross the border with 59 barrels. Carla will pay 2 barrels and will receive 40 dollars. She will cross the border with 18 barrels.
And so they did, both merchants agreeing with the calculations.
Anatole paid the 5 barrels plus 40 dollars and Carla paid 2 barrels and received 40 dollars.
They crossed the border with 59+18 barrels, that is, 77 barrels, and the taxes paid (for the 77 barrels) was 7 barrels because the Customs did not receive any part in money. The 40 dollars paid by Anatole were given to Carla.
We ask:
What was the price of each barrel? What was the tax paid over each barrel?
Wise Geometrician: Solution
The male donkey was carrying 7 loads, and the female donkey was groaning under the weight of 5 loads.
See:
Take Y to be the male donkey’s loads, X to be the female donkey’s loads.
We know that:
Y+1 = 2(X-1) and Y-1 = X+1
This implies that (adding them up, after multiplying the second by –1)
2 = X-3 => X = 5=>Y = 7
Back to the problem
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Wise Geometrician: Hint
Express the problem by means of a mathematical system.
Solution
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Wise Geometrician
You are now a wise geometrician and, as such, you might be able to solve the problem that the donkey poses to you.
A male and a female donkey, loaded with bags of wheat, were heading towards the market. The female donkey was groaning because of the heavy load. Each of the bags weighted the same.
- What are you complaining about? – said the male – If you gave me one of your bags, I would have twice as many as you have; if I gave you one of mine, our load would be the same.
- Things being so, tell me, wise geometrician, how many bags of wheat do each of us carry? – said the female.
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Solution: Curved Figures
Some people might think that the second figure has the longest upper border, because looks disguise. But they are exactly of the same size! They were built using a copy and paste procedure.
Hint: Curved Figures
Perhaps use your finger to accompany the borders of the figures. You may wish to copy and paste and then put one over the other.
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How Curved Can a Curved Figure Be?
Monday, 12 October 2015
Source of the Problem
The problems we present on this blog came from a Brazilian Mathematician called Malba Tahan. Malba Tahan is actually a pseudonym. According to the source we here mention, his original name was Julio Cesar de Mello e Souza.
These problems were presented at the Science Fair that happened in Melbourne in 2001. They were part of a piece of software built by Marcia R. Pinheiro. This piece of software was displayed at a kiosk where everyone representing VUT was supposed to be.
Coordinating the kiosk was Prof. Neil Diamond, who worked in the same department as Marcia R. Pinheiro back then.
The problems are based on the book As Maravilhas da Matemática (The Wonders of Mathematics) by Malba Tahan (6th edition, Bloch, 1987, Rio de Janeiro, Brazil).
Marcia R. Pinheiro has translated and adapted them.
These problems were presented at the Science Fair that happened in Melbourne in 2001. They were part of a piece of software built by Marcia R. Pinheiro. This piece of software was displayed at a kiosk where everyone representing VUT was supposed to be.
Coordinating the kiosk was Prof. Neil Diamond, who worked in the same department as Marcia R. Pinheiro back then.
The problems are based on the book As Maravilhas da Matemática (The Wonders of Mathematics) by Malba Tahan (6th edition, Bloch, 1987, Rio de Janeiro, Brazil).
Marcia R. Pinheiro has translated and adapted them.
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Solution: Queen's Soldiers
You distribute them equally over the vertices and sides of a star-shaped pentagon!
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Hint: Queen's Soldiers
Think of a geometric shape that would allow you to do that. It might have to do with triangles or something that uses triangles.
Solution
Back to the problem
Back to the problem
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Queen's Soldiers
Tell me, wise individual,
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Source of the Problem
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