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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 17,263

In this topic, I shall give a problem and give the (proper mathematical) solution after a week.:D

The first one:-

The population of a town increases by 5% annually. If its population in 1995 was 138915, what was it in 1992?

Character is who you are when no one is looking.

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**tt****Member**- Registered: 2005-07-03
- Posts: 32

In 1992, the population was 120,000.

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 17,263

Outstanding! You are really supersmart!

Try this one....But don't post your reply immediately.

Let others too try.

(2) A mixture of 40 liters of milk and water contains 10% water. How much water must be added to make water 20% in the new mixture?

Character is who you are when no one is looking.

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**mathsyperson****Moderator**- Registered: 2005-06-22
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Work out how much milk there is, then make an equation.

To take a leaf out of MathsIsFun's book...

This much!

Why did the vector cross the road?

It wanted to be normal.

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**MathsIsFun****Administrator**- Registered: 2005-01-21
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LOL ... I think about 3 seconds of that would be enough!

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 17,263

That is close to a 1,000 gallons!

Character is who you are when no one is looking.

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**im really bored****Member**- Registered: 2005-05-12
- Posts: 76

I think you would have to add 5 liters of water to make it 20%

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**mathsyperson****Moderator**- Registered: 2005-06-22
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That's what I got too.

Why did the vector cross the road?

It wanted to be normal.

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 17,263

Thats right!

This is how it is solved.

40 litres of milk and water contain 10% water.

Therefore, the milk content is 36 litres and water is 4 litres.

Lets assume x litres of water are added to make water 20%.

The ratio of water to the total becomes

4+x/40+x

this is equal to 20/100 or 1/5.

4+x/40+x = 1/5,

5(4+x) = 40+x

20+5x = 40+x

4x=20

x=5 litres

Character is who you are when no one is looking.

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**MathsIsFun****Administrator**- Registered: 2005-01-21
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Oh, silly me.

I would have added another 4 litres to go from 10% to 20%, and then realised there were now 44 litres, so I would have added another litre for luck.

Hey, 4 litres plus one for luck = 5 litres!

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman

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**MathsIsFun****Administrator**- Registered: 2005-01-21
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I seem to recall that there was a puzzle where there was a lake that contained too much salt.

The government decided to pump fresh water in until the salt content was low enough, but the lake was already full, so when you add water, the existing water spills out.

If the lake contained 10 Million liters, how much water would you need to pump in to lower the salt content from 2% to 1%

This, I think, may need some advanced maths.

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman

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**mathsyperson****Moderator**- Registered: 2005-06-22
- Posts: 4,900

mathsyperson wrote:

This much!

I think that still applies. It's an interesting puzzle though, I might just have to do that one.

Why did the vector cross the road?

It wanted to be normal.

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 17,263

I shall try to solve that when I start the next week,

here's my problem for the day

Using the numbers 1, 3, 4, and 6, together with the operations +, -, ×, and ÷, and unlimited use of brackets, make the number 24. Each number must be used precisely once. Each operation may be used zero or more times. Decimal points are not allowed, nor is implicit use of base 10 by concatenating digits, as in 64 - 31.

Character is who you are when no one is looking.

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**mathsyperson****Moderator**- Registered: 2005-06-22
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I've seen it before, and it is HORRIBLE.

If you're really stuck, try looking here for inspiration.

Why did the vector cross the road?

It wanted to be normal.

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 17,263

I saw this problem and I was impressed. I didn't know someone had posted a similiar problem earlier. Okay, I shall give the solution and post a new problem instead.

The solution :- 6 /(1-3/4)

The new problem #3

Imagine you are writing numbers from 1 to 1000 on a piece of paper. How many times would you write the number'9'?

Character is who you are when no one is looking.

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**MathsIsFun****Administrator**- Registered: 2005-01-21
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Hmmm ... you write 9 once every 10 numbers ...

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**mathsyperson****Moderator**- Registered: 2005-06-22
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My guess

Don't click that unless you're ganesh, you want to check your own answer or you want it spoiled. You have been warned!

With MathsIsFun's problem, I tried using Excel to work out how much salt is lost whenever you add a certain amount of water, and repeating that stage until you get to less than 1% salt. The value for the amount needed could be improved by adding less and less each time, but it's still only an approximation. I'm sure there's some complex way to work out the exact amount, but I don't know it.

I got down to adding 200 litres of water each time (any less than that and Excel gets really slow) and found that the amount needed (hopefully) lies between 6,931,400 and 6,931,600 litres. I can't improve on that.

Why did the vector cross the road?

It wanted to be normal.

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 17,263

You are right, Mathsy!

#4 (Without paper, pencil; Time limit : 1 minute)

The sum of three numbers is 174. The ratio of the second number to the third is 9:16 and the ratio of the first number to the third is 1:4. The second number is _______.

Character is who you are when no one is looking.

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**mathsyperson****Moderator**- Registered: 2005-06-22
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Hmm. It's not too hard to solve, but I could never do it with those limitations, so I fail.

There must be a shortcut somewhere that I've missed...

Why did the vector cross the road?

It wanted to be normal.

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**Roraborealis****Member**- Registered: 2005-03-17
- Posts: 1,594

Dammit! I had the answer to the nines one, but then I forgot it again!!!!

School is practice for the future. Practice makes perfect. But - nobody's perfect, so why practice?

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**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,560

The Lake Problem.

Yes, mathsy, that is a valid way to solve it, it comes under the broad heading "Numerical Methods". Excel is great at that.

Now, with very coarse steps (say 1 million liters) what did you come up with? And then as they got smaller and smaller how did they change? And if you could make the steps really tiny where would the answer end up?

When x becomes *very small* we call it "dx" and you end up doing Calculus!

I posed this problem from memory, and believe that using Calculus I can come up with an exact answer. When I get a few minutes I will have a try and post the answer here (unless Ganesh...?)

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 17,263

I um unable to reply. I shall wait for your post and comment on that later, MathsIsFun.

ganesh wrote:

#4 (Without paper, pencil; Time limit : 1 minute)

The sum of three numbers is 174. The ratio of the second number to the third is 9:16 and the ratio of the first number to the third is 1:4. The second number is _______.

Since both the ratios have 4 or a multiple of 4 for C,

the ratio A:B:C = 4:9:16.

The sum of these is 29, and it can be noticed 174 is 29 x 6.

Therefore, the second number is 9 x 6, that is 54.

Character is who you are when no one is looking.

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**ganesh****Moderator**- Registered: 2005-06-28
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A cistern can be filled by two pipes A and B separately in 45 minutes and 36 minutes respectively. A tap C at the bottom can empty the cistern in 30 minutes. If the tap C is opened 7 minutes after the two pipes A and B are opened, in how many minutes will the cistern be full?

Character is who you are when no one is looking.

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**mathsyperson****Moderator**- Registered: 2005-06-22
- Posts: 4,900

Why did the vector cross the road?

It wanted to be normal.

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 17,263

You got it √

Character is who you are when no one is looking.

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