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**ganesh****Moderator**- Registered: 2005-06-28
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Problem #6

A salesman's commission is 5% on all sales up to $10,000 and 4% on all sales exceeding this. He remits $31,100 to his parent company after deducting his commission. His sales was worth $_________.

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**mathsyperson****Moderator**- Registered: 2005-06-22
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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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S = Sales

C = Commission

R = Remittance to Company after sales

For S <= $10,000, C = 0.05S

For S > $10,000, C = 500 + 0.04(S-10,000)

R = S - C

Now, we know that R = 31,100

So ... sorry, gotta go!

"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
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31,100 = S - [500+0.04(s-10,000)]

31,100 = S -[500 + 0.04s - 400]

31,100 = S - 100 - 0.04s

31,200 = 0.96 S

S = 31,200 / 0.96

S = 32,500

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**ganesh****Moderator**- Registered: 2005-06-28
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#7

The ratio of annual incomes of two persons is 9:7 and the ratio of their annual expenditure is 4:3. Each of them saves $2000 yearly. Find their annual incomes.

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**mathsyperson****Moderator**- Registered: 2005-06-22
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Wow, four simultaneous equations...

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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BY THE WAY, since my little {hide}{/hide} tag is getting used a bit, it will soon become too late to rename it to {popup} or {boing} or whatever.

So, is "hide" the best name, then (even though it is just hidden under a little button)

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

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**im really bored****Member**- Registered: 2005-05-12
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For MathIsFuns lake problem I got 6,931,471.8055994530941723212145818 liters of water that needs to be added.

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**Zach****Member**- Registered: 2005-03-23
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There's a Hide tag?!

Boy let me tell you what:

I bet you didn't know it, but I'm a fiddle player too.

And if you'd care to take a dare, I'll make a bet with you.

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**mathsyperson****Moderator**- Registered: 2005-06-22
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im really bored wrote:

For MathIsFuns lake problem I got 6,931,471.8055994530941723212145818 liters of water that needs to be added.

Was that with calculus or a more accurate version of my method?

Why did the vector cross the road?

It wanted to be normal.

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**im really bored****Member**- Registered: 2005-05-12
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Nope no calculas or your method, I just got a tip from a friend that its related to the natural log of 2, which is 0.69314718055994530941723212145818.

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**ganesh****Moderator**- Registered: 2005-06-28
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mathsyperson wrote:

Wow, four simultaneous equations...

There are only two:-

9x - 4y = 2,000

7x - 3y = 2,000

Solving for 'x', the annual income of the two can be calculated. They would be 9x and 7x.

*Last edited by ganesh (2005-07-20 19:27:58)*

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**ganesh****Moderator**- Registered: 2005-06-28
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Which is greater?

Cos (Sin A) or Sin (Cos A)?

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**mathsyperson****Moderator**- Registered: 2005-06-22
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Degrees, radians or grads?

I can tell just by thinking about it the answer for degrees and grads, but if you're working with radians, there might be a few exceptions, though I doubt it.

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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Oooh ... (reaches for Excel) ...

cos(sin(a)) fluctuates between 0.734664609.. and 1

sin(cos(a)) fluctuates between 0.841470985.. and -0.841470985..

They cross over first at about 28.9° (0.5031796.. radians), then back again at 331.1° (5.779671333.. radians)

Now, are 0.734664609.., 0.841470985.. or 0.5031796.. magical numbers?

"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
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Plotting cos(sin a) against sin(cos a) in Excel gives a smiley mouth. The important thing, though, is to see whether one is always bigger than the other, and this graph cannot do that.

I plotted radian value against cos(sin a)-sin(cos a) and I got a beautiful swirly pattern. This pattern always stayed above 0, so cos(sin a) is always bigger than sin(cos a).

Why did the vector cross the road?

It wanted to be normal.

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**ganesh****Moderator**- Registered: 2005-06-28
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Cos(Sin A) is always greater than Sin (Cos A),

as rightly pointed out by Mathsy!

We shall test this for three values in the first quadrant, in degrees.

First, 0.01°.

Cos (Sin A) > Sin(CosA).

Next, for 45°

Again, Cos (Sin A) > Sin(CosA).

Finally, for 89.9°

Yet again, we get the same result.

I shall try to search for a proof, or prove it myself!

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**MathsIsFun****Administrator**- Registered: 2005-01-21
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Ooops, my bad in some calculations.

Yes, I see same result now.

I will make up for my mistake by posting this nice graph that mathsy talked about:

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**ganesh****Moderator**- Registered: 2005-06-28
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Thank you, Admin, for the neat graph.

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**ganesh****Moderator**- Registered: 2005-06-28
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I just remembered a question asked to me when I participated in a quiz when I was 14 :

A three digit number is written on a window glass,

the difference between the number and as it can be seen from the other side is 693,

what's the number?

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**mathsyperson****Moderator**- Registered: 2005-06-22
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Well, it has to be made up of 0's 1's and 8's (and maybe 2's and 5's that swap when reversed, depending on the writing style)

I've got

, but there might be others.Why did the vector cross the road?

It wanted to be normal.

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**ganesh****Moderator**- Registered: 2005-06-28
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Two candles A and B, of equal height but different circumferences, burn for 4 hours and 3 hours respectively. If the two are lit at the same time, after what time would one candle be half the height of the other?

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**mathsyperson****Moderator**- Registered: 2005-06-22
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This is the closest answer I could find. I get about 3 minutes more than what the picture says.

Why did the vector cross the road?

It wanted to be normal.

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**ganesh****Moderator**- Registered: 2005-06-28
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As always, you are corect, Mathsy!

I shall show how I did it.

Assume both candles are 'h' inches in height.

The height of the first candle reduces by h/4 every hour and that of the second reduces by h/3.

Lets assume in time 't' hours, the first is twice the height of the second.

Therefore,

h - t(h/4) = 2 [h - t(h/3)]

h - ht/4 = 2h - 2ht/3

h = 2ht/3 - ht/4

h = 5ht/12

Cancelling h on both sides,

1 = 5t/12

or t = 12/5

The unit we had taken was hours,

therefore, in 12/5 hours, that is 2 2/5 hours, one will be double the height of the other.

Simplified, it is 2 hours and 24 minutes.

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**ganesh****Moderator**- Registered: 2005-06-28
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Problem #n

The distance between two cities is 840 kilometres.

If I start driving from one city to the other at a speed

5 kilometres/hour more than my normal speed,

I save 3 hours. What is my normal speed?

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